LEVEL STRUCTURES ON FINITE GROUP SCHEMES AND APPLICATIONS
                                   By
                          Chuangtian Guan
                         A DISSERTATION
                             Submitted to
                     Michigan State University
              in partial fulfillment of the requirements
                           for the degree of
                Mathematics – Doctor of Philosophy
                                  2022


                                         ABSTRACT
     LEVEL STRUCTURES ON FINITE GROUP SCHEMES AND APPLICATIONS
                                               By
                                       Chuangtian Guan
The notion of level structures originates from the study of the moduli of elliptic curves. In this
thesis, we consider generalizing the notion of level structures and make explicit calculations
on different moduli spaces.
    The first moduli space we consider is the moduli of finite flat (commutative) group
schemes. We give a definition of Γ(𝑝)-level structure (also called the “full level structure”)
over group schemes of the form 𝐺 × 𝐺, where 𝐺 is a group scheme or rank 𝑝 over a ℤ𝑝 -
scheme. The full level structure over 𝐺 × 𝐺 is flat over the base of rank | GL2 (𝔽𝑝 )|. We also
observe that there is no natural notion of full level structures over the stack of all finite flat
commutative group schemes.
    The second moduli space we consider is the moduli of principally polarized abelian sur-
faces in characteristic 𝑝 > 0 with symplectic level-𝑛 structure (𝑛 ≥ 3), which is known as the
Siegel threefold. By decomposing the Siegel threefold using the Ekedahl–Oort stratification,
we analyze the 𝑝-torsion group scheme of the universal abelian surface over each stratum.
To do this, we establish a machinery to produce group schemes from their Dieudonné mod-
ules using a version of Dieudonné theory due to de Jong. By using this machinery, we give
explicit local equations of the Hopf algebras over the superspecial locus, the supersingular
locus and ordinary locus. Using these local equations, we calculate explicit equations of the
Γ1 (𝑝)-covers over these strata using Kottwitz–Wake primitive elements. These equations
can be used to prove geometric and arithmetic properties of the Γ1 (𝑝)-cover over the Siegel
threefold. In particular, we prove that the Γ1 (𝑝)-cover over the Siegel threefold is not normal.


      Copyright by
CHUANGTIAN GUAN
              2022


                               ACKNOWLEDGEMENTS
It has been a long journey through my six-year study at Michigan State University. I would
not have been able to finish it without the tremendous help I received. I want to thank
everyone who supported me along this trip.
    First and foremost, I want to thank my advisor George Pappas. Thank you for teaching
me hour after hour, supporting me when I struggle, and encouraging me when I want to give
up. Thank you for all the helpful and productive meetings we have together. It is always
relaxing and enjoyable to meet with you. Your wisdom and work ethic continue to inspire
me in my life.
    I am very grateful to Preston Wake for extremely helpful conversations on my projects,
papers and statements. Without your help, my paper and thesis will not exist in current
form. I want to thank Rajesh Kulkarni for all the courses I took with you and your helpful
comments on my statements. I also want to thank Aaron Levin and Michael Shapiro for
being in my committee and the guidance through different stages of my Ph.D. career.
    Teaching is an important part of my experience at Michigan State University and I want
to thank all the faculty, staff and students that I meet during my teaching. I want to
especially thank Tsveta Sendova for consistent help on my teaching and life in all aspects
these years, and Shiv Karunakaran for bringing me to the wonderful teaching experience in
MTH 299.
    I also want to thank all my friends accompanying me in this journey. I especially want to
thank Nick Rekuski, Joshua Ruiter, Zheng Xiao, Ioannis Zachos, Yizhen Zhao and Zhihao
Zhao for countless interesting conversations on math and life. I want to thank all fellow
coorganizers and participants of Student Algebra Seminar and Student Arithmetic Geometry
Seminar, from whom I learn so much. I want to thank my friends Keping Huang, Mihalis
Paparizos, Jian Song, Zhixin Wang, Kang Yu, Chen Zhang, Rui Zhang, Zhe Zhang, Yunlu
Zhang for the great time we have together. I want to thank Jeanne Wald for your amazing
                                             iv


new year parties and your heartwarming concerns that make me feel home.
   Last, I thank a lot to my parents for all the encourage and support behind. My special
thanks goes to my girlfriend Junnuo Yu, who is always next to me.
                                              v


                              TABLE OF CONTENTS
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    vii
CHAPTER 1      INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . .          1
CHAPTER 2 PRELIMINARIES . . . . . . . . . . . . . . . . .             . . . . . . . . . . . 10
   2.1 Group Schemes and Classification Theorems . . . . . . .        . . . . . . . . . . . 10
   2.2 Siegel Modular Varieties and Ekedahl–Oort Stratification       . . . . . . . . . . . 14
   2.3 Kottwitz-Wake Primitive Elements . . . . . . . . . . . .       . . . . . . . . . . . 17
CHAPTER 3 FULL LEVEL STRUCTURE ON FINITE GROUP                          SCHEMES       . . . 23
   3.1 Full level structure on 𝐺 × 𝐺 . . . . . . . . . . . . . . . .    . . . . . . . . . . 26
   3.2 Proof of Theorem 3.1.5 . . . . . . . . . . . . . . . . . . . .   . . . . . . . . . . 29
   3.3 Nonexistence of full level structure over the stack . . . . .    . . . . . . . . . . 33
   3.4 Full level structure on 𝐺 × 𝐺 × 𝐺 . . . . . . . . . . . . . .    . . . . . . . . . . 35
CHAPTER 4      CONSTRUCTIONS OF GROUP SCHEMES USING DIEUDONNÉ
               MODULES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      . 37
   4.1 A Version of Dieudonné Theory by De Jong . . . . . . . . . . . . . . . . .         . 38
   4.2 Group Schemes Annihilated by 𝑉 . . . . . . . . . . . . . . . . . . . . . . .       . 43
   4.3 Group Schemes Annihilated by 𝑉 2 . . . . . . . . . . . . . . . . . . . . . .       . 44
   4.4 Group Schemes Annihilated by 𝑉 3 . . . . . . . . . . . . . . . . . . . . . .       . 45
CHAPTER 5 Γ1 (𝑝) -COVER OVER THE SIEGEL                 THREEFOLD         . . . . . . . . . 55
   5.1 Γ1 (𝑝) -cover over the Superspecial Locus . .    . . . . . . . . . . . . . . . . . . 55
   5.2 Γ1 (𝑝) -cover over the Supersingular Locus .     . . . . . . . . . . . . . . . . . . 57
   5.3 Γ1 (𝑝) -cover over 𝑝-rank-1 Locus . . . . . .    . . . . . . . . . . . . . . . . . . 61
       5.3.1 Mixed extensions . . . . . . . . . . .     . . . . . . . . . . . . . . . . . . 62
       5.3.2 Calculations on extensions . . . . . .     . . . . . . . . . . . . . . . . . . 63
   5.4 Γ1 (𝑝) -cover over the Ordinary Locus . . . .    . . . . . . . . . . . . . . . . . . 70
   5.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .      77
                                             vi


                                 LIST OF TABLES
Table 1.1 Regularity Properties of the Γ1 (𝑝)-cover over the Siegel Threefold . . . . . 9
                                           vii


                                         CHAPTER 1
                                     INTRODUCTION
Let ℋ be the complex upper half plane. The special linear group SL2 (ℤ) and its subgroups
act on ℋ by Möbius transformations:
                                       𝑎 𝑏          𝑎𝑧 + 𝑏
                                     (      )⋅𝑧 ≔          .                              (1.1)
                                       𝑐 𝑑          𝑐𝑧 + 𝑑
Let 𝑁 ≥ 1 be an integer. Let Γ0 (𝑁 ), Γ1 (𝑁 ) and Γ(𝑁 ) be subgroups of SL2 (ℤ) defined as
follows:
                                         𝑎 𝑏       ∗ 0
                          Γ0 (𝑁 ) ≔ {(       )≡(        )    mod 𝑁 } ,                    (1.2)
                                         𝑐 𝑑       ∗ ∗
                                         𝑎 𝑏       1 0
                          Γ1 (𝑁 ) ≔ {(       )≡(        )    mod 𝑁 } ,                    (1.3)
                                         𝑐 𝑑       ∗ 1
                                        𝑎 𝑏        1 0
                          Γ(𝑁 ) ≔ {(        )≡(        )     mod 𝑁 } .                    (1.4)
                                        𝑐 𝑑        0 1
    In general, we call a subgroup Γ a congruence subgroup (of SL2 (ℤ)) if Γ contains Γ(𝑁 )
for some integer 𝑁 ≥ 1. The subgroups Γ0 (𝑁 ), Γ1 (𝑁 ) and Γ(𝑁 ) are the most interest-
ing congruence subgroups. In particular, the quotients of the upper half plane by these
congruence subgroups have the following moduli interpretations:
                 ⎧ isomorphism class of pairs (𝐸, 𝐺), where 𝐸 is an elliptic curve      ⎫
                 {                                                                      }
     Γ0 (𝑁 )\ℋ = ⎨                                                                      ⎬,
                 {               over ℂ and 𝐺 ⊂ 𝐸 a subgroup of order 𝑁                 }
                 ⎩                                                                      ⎭
                 ⎧ isomorphism class of pairs (𝐸, 𝑃 ), where 𝐸 is an elliptic curve     ⎫
                 {                                                                      }
     Γ1 (𝑁 )\ℋ =                                                                          ,
                 ⎨                                                                      ⎬
                 {          over ℂ and 𝑃 ∈ 𝐸 generates a subgroup of order 𝑁            }
                 ⎩                                                                      ⎭
                 ⎧ isomorphism class of triples (𝐸, 𝑃 , 𝑄), where 𝐸 is an elliptic      ⎫
                 {                                                                      }
                 {                                                                      }
      Γ(𝑁 )\ℋ = ⎨ curve over ℂ and 𝑃 , 𝑄 ∈ 𝐸 generate the 𝑁 -torsion points 𝐸[𝑁 ]       ⎬.
                 {                                                                      }
                 {             with ⟨𝑃 , 𝑄⟩ = 𝑒2𝜋𝑖/𝑁 for the Weil pairing ⟨⋅, ⋅⟩        }
                 ⎩                                                                      ⎭
    These quotients are denoted by 𝑌0 (𝑁 ) (resp. 𝑌1 (𝑁 ), 𝑌 (𝑁 )). The modular curves 𝑋0 (𝑁 )
(resp. 𝑋1 (𝑁 ), 𝑋(𝑁 )) are constructed by compactifying 𝑌0 (𝑁 ) (resp. 𝑌1 (𝑁 ), 𝑌 (𝑁 )). These
modular curves are known to admit smooth models over ℤ[1/𝑁 ].
                                                1


    As we can see above, modular curves arise as the moduli spaces of elliptic curves with
some extra structures. These extra structures are called level structures. In particular, the
extra structures that appear in the moduli description of 𝑌0 (𝑁 ) (resp. 𝑌1 (𝑁 ), 𝑌 (𝑁 )) are
called Γ0 (𝑁 ) (resp. Γ1 (𝑁 ), Γ(𝑁 ))-level structures.
    The first systematic study of integral models of modular curves over ℤ was done by
Deligne–Rapoport [8], who construct models of 𝑋0 (𝑝) and 𝑋1 (𝑝) over the 𝑝-adic integers
ℤ𝑝 . In [19], Katz and Mazur construct integral models of 𝑋0 (𝑁 ), 𝑋1 (𝑁 ) and 𝑋(𝑁 ), by
carefully defining the moduli problems of elliptic curves with level structures. For example,
following an idea of Drinfeld in [10], Katz and Mazur define a set of sections {𝑃1 , … , 𝑃𝑁 2 }
of 𝐸[𝑁 ] to be a “full set of sections”, if the points generate the group scheme 𝐸[𝑁 ] as
Cartier divisors. Using this notion, the Γ(𝑁 )-level structure on 𝐸[𝑁 ], also called “full level
structure”, is defined to be the maps in Hom((ℤ/𝑁 ℤ)2 , 𝐸[𝑁 ]) whose images form a full set of
sections. Katz and Mazur use this notion of full level structure to construct integral models
of 𝑋(𝑁 ).
    In Chapter 2, we review some preliminaries such as the Oort–Tate and Raynaud theory
of group schemes of order 𝑝, the Ekedahl–Oort stratification and the definition of primitive
elements of group schemes due to Kottwitz–Wake. Throughout the rest of this thesis, the
central questions we consider are to generalize these notions of level structures, and make ex-
plicit calculations on different moduli spaces. Particularly, we consider Γ(𝑝)-level structures
on moduli of finite flat group schemes and Γ1 (𝑝)-level structures on the moduli of principally
polarized abelian surfaces with symplectic level structure. For each of the case, we consider
the following questions:
(A) How to define a good notion of level structures over the moduli space?
(B) Given a good notion of level structure over the moduli, find equations that describe the
      universal covers, at first locally.
(C) What arithmetic/geometric properties can we obtain from the above descriptions?
                                                 2


    First we consider the moduli of finite flat group schemes (over ℤ𝑝 ) and consider Question
(A) for Γ(𝑝)-level structures (also called “full level structure”) on it. Note that although
there is no “moduli space” of finite flat group schemes, we can still consider the “moduli
stack” instead. Here by a stack, we simply mean a category fibered in groupoids over
Schℤ𝑝 as in [6]. More precisely, let C be a stack of group schemes 𝐺/𝑆 of certain type
(for example, finite flat commutative so that 𝐺[1/𝑝] is étale locally isomorphic to (ℤ/𝑝𝑛 ℤ)𝑔 )
over Schℤ𝑝 . The objects in C are group schemes 𝐺/𝑆 of the fixed type and the morphisms
are Cartesian squares. By a “good” Γ(𝑝)-level structure over C, we mean a fibered functor
ℱ ∶ C → Sch, such that ℱ(𝐺/𝑆) is a closed subscheme of Hom𝑆 ((ℤ/𝑝𝑛 ℤ)𝑔 , 𝐺) and such
that for 𝑓 ∶ 𝐺/𝑆 → 𝐺′ /𝑆 ′ , the morphism ℱ(𝑓) ∶ ℱ(𝐺/𝑆) → ℱ(𝐺′ /𝑆 ′ ) is the restriction of
the morphism Hom𝑆 ((ℤ/𝑝𝑛 ℤ)𝑔 , 𝐺) → Hom𝑆′ ((ℤ/𝑝𝑛 ℤ)𝑔 , 𝐺′ ) induced by 𝑓, and satisfies the
following conditions:
(1) ℱ(𝐺/𝑆) is flat over 𝑆 and of rank |GL𝑔 (ℤ/𝑝𝑛 ℤ)|.
(2) ℱ(𝐺/𝑆) is invariant under the right GL𝑔 (ℤ/𝑝𝑛 ℤ)-action on Hom𝑆 ((ℤ/𝑝𝑛 ℤ)𝑔 , 𝐺). When
     inverting 𝑝, we have an identification
                            ℱ(𝐺[ 𝑝1 ]/𝑆[ 𝑝1 ]) = Isom𝑆[ 𝑝1 ] ((ℤ/𝑝𝑛 ℤ)𝑔 , 𝐺[ 𝑝1 ])
     as closed subschemes of Hom𝑆[ 1 ] ((ℤ/𝑝𝑛 ℤ)𝑔 , 𝐺[ 𝑝1 ]).
                                      𝑝
(3) When identifying
                        Hom𝑆 ((ℤ/𝑝𝑛 ℤ)𝑔 , 𝐺) ×𝑆 𝑇 = Hom𝑇 ((ℤ/𝑝𝑛 ℤ)𝑔 , 𝐺𝑇 )
     in the natural way, we have
                                     ℱ(𝐺/𝑆) ×𝑆 𝑇 = ℱ(𝐺𝑇 /𝑇 )
     as closed subschemes, for any 𝑆-scheme 𝑇 .
                                                ∼
(4) For any group scheme isomorphism 𝐺 −        → 𝐺′ , the induced isomorphism
                                                     ∼
                           Hom𝑆 ((ℤ/𝑝𝑛 ℤ)𝑔 , 𝐺) −    → Hom𝑆 ((ℤ/𝑝𝑛 ℤ)𝑔 , 𝐺′ )
                                                   3


     restricts to an isomorphism
                                                 ∼
                                      ℱ(𝐺/𝑆) −  → ℱ(𝐺′ /𝑆).
The condition (4) is automatic from being a functor.
    There have been many attempts by other mathematicians to give such a construction.
In [19], Katz and Mazur suggest a construction of full level structures for general group
schemes. Unfortunately, this is shown to be badly behaved (for example, not flat) by Chai
and Norman in [3]. In [42], Wake gives a good notion of full level structure for 𝜇𝑝 × 𝜇𝑝 . In
his paper, Wake also gives an alternative description of his full level structure on 𝜇𝑝 × 𝜇𝑝
that can be defined for general group schemes. However, this alternative description still
fails to behave well for general group schemes.
    In [21], Kottwitz and Wake construct a general notion of Γ1 (𝑝)-level structure on finite
flat group schemes, given by so-called primitive elements (see Section 2.3). In [13], we extend
the result of Wake and give a definition of full level structure on all group schemes of the
form 𝐺 × 𝐺 using the notion of Kottwitz-Wake primitive elements. Here is the main theorem
in Chapter 3:
Theorem 1.0.1. Let 𝑆 be a ℤ𝑝 -scheme and let 𝐺 be a finite flat commutative group scheme
of rank 𝑝 over S. The full level structure on 𝐺 × 𝐺 defined in Definition 3.1.2 satisfies
condition (1)-(3).
    Back to the language of stacks, Theorem 1.0.1 implies that we defined a well-behaved
notion of full level structure over the stack 𝑂𝑇 , whose objects are group schemes of the
form 𝐺 × 𝐺 where 𝐺 is an Oort–Tate scheme (see Section 2.1) over a ℤ𝑝 -scheme 𝑆, and
morphisms are diagonal group scheme isomorphisms 𝐺 × 𝐺 → 𝐺′ × 𝐺′ induced by two
identical isomorphisms of Oort–Tate group schemes 𝐺 → 𝐺′ .
    One might hope to extend this result over the stack of finite flat commutative group
schemes. Unfortunately, we record the following negative result in Chapter 4:
                                               4


Theorem 1.0.2. There is no “natural” notion of full level structure over group schemes
which is flat of the expected rank over the base.
    Here, by “natural”, we mean one is actually defined over the stack, i.e. satisfying con-
dition (4) as above. In fact, we observe that there is no good notion of full level structure
even over the substack 𝑂𝑇 × 𝑂𝑇 , whose objects are 𝐺 × 𝐺′ where 𝐺, 𝐺′ are Oort–Tate
group schemes. We show that the full level structure defined in Theorem 1.0.1 is the only
possible definition that gives a flat model. The full level structure we define is preserved
by the diagonal group scheme isomorphisms 𝐺 × 𝐺 → 𝐺′ × 𝐺′ induced by two identical
isomorphisms of Oort–Tate group schemes 𝐺 → 𝐺′ . However, there exists group schemes
for which the full level structure scheme is not preserved by all automorphisms. It is notable
that the counterexamples come from non-𝑝-divisible groups. So there is still a possibility of
a positive result for truncated 𝑝-divisible groups.
    Next, we consider Γ1 (𝑝)-level structures over the moduli of principally polarized abelian
surface with symplectic level-𝑁 structure. For 𝑁 ≥ 3, this is a fine moduli space by [28].
This moduli space is called the Siegel threefold and is denoted by 𝒜 ≔ 𝒜2,1,𝑁 . We are
particularly interested in the level structures over the special fiber 𝒜 ̄ ≔ 𝒜 ⊗ 𝔽𝑝 . There
is already a well-behaved notion of Γ1 (𝑝)-level structure defined by taking Kottwitz–Wake
primitive elements on the 𝑝-torsion of the universal abelian surface 𝒳. The Γ1 (𝑝)-cover is
then given by 𝒳× [𝑝] ≔ (𝒳[𝑝])× over 𝒜. (Some mathematicians use the name “Γ1 (𝑝)-cover”
differently. For example, Haines and Rapoport use “Γ1 (𝑝)-cover” for the pro-𝑝 Iwahori
structure in [16].) So in this case, Question (A) has already been resolved.
    Consider Question (B). In [30], Oort defines a stratification of 𝒜 (for general dimension 𝑔),
now called the Ekedahl–Oort stratification. The Ekedahl–Oort stratification is parametrized
by “elementary sequences”. For each elementary sequence 𝜑, we denote the associated stra-
tum by 𝑆𝜑 . We want to give the local equations of the Γ1 (𝑝)-cover 𝒳× [𝑝] over each stratum
𝑆𝜑 . To do this, we need some machinery to systematically produce group schemes.
    This machinery is built in Chapter 4. The tool we use is a version of Dieudonné theory
                                               5


due to de Jong [4]. We review this version of Dieudonné theory and compare it with the
crystalline Dieudonné theory. Although this version of Dieudonné theory does not establish
an antiequivalence of categories, there are still bijections between morphisms and extensions.
Using these bijections, we construct group schemes killed by 𝑝 whose Dieudonné modules
have nilpotent Verschiebung. These group schemes are constructed by taking consecutive
extensions of group schemes with trivial Verschiebung, and their Hopf algebras are given
explicitly in terms of their Dieudonné modules. Here is a sample of such result:
Theorem 1.0.3. Let 𝑆 = Spec 𝑅, where 𝑅 is a local ring in characteristic 𝑝. Assume that
𝑆 and its Frobenius lift modulo 𝑝2 . (This is true, for example, when 𝑆 is smooth.) Let 𝐺/𝑆
be a (finite flat commutative) group scheme of rank 𝑝4 and killed by 𝑝. Suppose that the
Frobenius and Verschiebung on the Dieudonné module are given by matrices of the following
form:
                                        0 𝑎 1 𝑎2 𝑐              0 𝑏 1 𝑏2 𝑑
                                     ⎛
                                     ⎜0 0 0 𝑒1 ⎟        ⎞      ⎛
                                                               ⎜0 0 0 𝑓1 ⎞    ⎟
                              𝐹 =⎜   ⎜                  ⎟
                                                        ⎟ ,𝑉 = ⎜
                                                               ⎜              ⎟,                (1.5)
                                     ⎜0 0 0 𝑒2 ⎟               ⎜0 0 0 𝑓2 ⎟    ⎟
                                     ⎝0 0 0 0 ⎠                ⎝0 0 0 0 ⎠
where 𝑎1 , 𝑎2 , 𝑏1 , 𝑏2 , 𝑐, 𝑑, 𝑒1 , 𝑒2 , 𝑓1 , 𝑓2 ∈ 𝑅. Then
           𝐺 ≅ Spec 𝑅[𝑥, 𝑦1 , 𝑦2 , 𝑧]/ (𝑥𝑝 , 𝑦1𝑝 − 𝑎1 𝑥, 𝑦2𝑝 − 𝑎2 𝑥, 𝑧 𝑝 − 𝑐𝑥 − 𝑒1 𝑦1 − 𝑒2 𝑦2 ) (1.6)
and there are also explicit formulas for the coalgebra structure (see Equation (4.12)).
     In Chapter 6, we give some explicit calculations of the Γ1 (𝑝)-cover 𝒳× [𝑝] over each
stratum 𝑆𝜑 of the Siegel threefold 𝒜.̄ There are 4 Ekedahl–Oort strata. They are: the
superspecial locus, the supersingular (but not superspecial) locus, the 𝑝-rank-1 locus and
the ordinary locus. The loci have dimensions 0,1,2,3 respectively. Over each stratum, there
is a canonical group scheme filtration of the 𝑝-torsion of the universal abelian surface 𝒳.
     Let 𝒳𝜑 be the restriction of 𝒳 over 𝑆𝜑 . The Γ1 (𝑝)-cover 𝒳× [𝑝] → 𝒜 restricts to
𝒳×                    ×
   𝜑 [𝑝] ≔ (𝒳𝜑 [𝑝]) over 𝑆𝜑 . We want to calculate explicit descriptions of the Γ1 (𝑝)-cover
                                                          6


𝒳×𝜑 [𝑝]/𝑆𝜑 by finding (local) equations. To do this, we first want to get a (local) description
of the Hopf algebras of 𝒳𝜑 [𝑝]/𝑆𝜑 for each stratum.
    The superspecial locus is a union of discrete points corresponding to products of super-
singular elliptic curves. The Hopf algebra of 𝒳𝜑 [𝑝] over this locus can be easily calculated
using classical Dieudonné theory over perfect fields.
    Now consider the supersingular locus. Theorem 1.0.3 applies in this situation and it shows
that the group scheme 𝒳𝜑 [𝑝] over 𝑆𝜑 , where 𝑆𝜑 is the supersingular locus, is (Zariski-locally)
of the form (1.6).
    To make the Hopf algebra description of 𝒳𝜑 [𝑝]/𝑆𝜑 more precise, we will use some specific
constructions of the supersingular locus. Following an idea of Moret-Bailly [25] and Oort
[32], one can form families of supersingular abelian surfaces 𝒴 over ℙ1 . It is shown in [18]
that for any irreducible component 𝑊 of the supersingular locus, 𝒳𝑊 /𝑊 pulls back to 𝒴/ℙ1
via some surjective morphism ℙ1 → 𝑊 and the morphism is generically an immersion. In
[22], Kudla and Rapoport give nice descriptions of the Dieudonné modules of 𝒴/ℙ1 , whose 𝐹
and 𝑉 modulo 𝑝 are of the shape (1.5) in Theorem 1.0.3. Using the construction in Theorem
1.0.3, we can explicitly write down the Hopf algebra of 𝒴[𝑝].
    For the 𝑝-rank-1 locus, we are not able to obtain explicit descriptions of the Hopf algebras.
However, we give some partial results using the theory of mixed extensions in Theorem
5.3.7 due to Grothendieck. In particular, we explicitly calculate all extensions and ext
groups that are ingredients of the theory of mixed extensions. The only obstruction is to
construct an explicit mixed extension using the calculated data. Once we have one explicit
mixed extension, we can get all mixed extensions by applying all calculated ingredients to
Proposition 5.3.2.
    For the ordinary locus, Serre–Tate theory applies and we can get explicit expressions of
the Hopf algebras using Serre–Tate coordinates.
    All these group schemes that we construct over the three Ekedahl–Oort strata are written
explicitly as complete intersections. In this case, one can obtain explicit generators of the
                                                7


defining ideal of Kottwitz-Wake primitive elements as certain determinants. Putting all these
results together, we have the following result:
Theorem 1.0.4. Over each Ekedahl–Oort stratum 𝑆𝜑 , the Γ1 (𝑝)-cover 𝒳×                                   𝜑 [𝑝]/𝑆𝜑 has the
following description:
   1. Let 𝑆𝜑 be the superspecial locus. Over each point of 𝑆𝜑 , the Γ1 (𝑝)-cover 𝒳×                            𝜑 [𝑝]/𝑆𝜑 is
      given by
                                          Spec 𝔽̄𝑝 [𝑥, 𝑦]/(𝑥𝑝 , 𝑦𝑝 , 𝑥𝑝 −1 𝑦𝑝 −1 )
                                                                   2  2      2        2
      over Spec 𝔽̄𝑝 .
   2. Let 𝑆𝜑 be the supersingular stratum and let 𝑊 be an irreducible component of 𝑆𝜑 .
      The Γ1 (𝑝)-cover 𝒳̄ ×                                                ×          1
                                 𝑊 [𝑝]/𝑊 is the pullback of 𝒴 [𝑝]/ℙ𝔽̄ via some open immersion
                                                                                        𝑝
      𝑊 →       ℙ1𝔽̄ . Over each affine chart of the standard cover ℙ1𝔽̄ = 𝔸10 ∪ 𝔸1∞ , the restricted
                    𝑝                                                                     𝑝
                         ×
      Γ1 (𝑝)-cover 𝒴 [𝑝]|𝔸1̄       /𝔸1𝔽̄    is isomorphic to
                                𝔽𝑝       𝑝
                             Spec 𝔽̄𝑝 [𝜇, 𝑥, 𝑦]/(𝑥𝑝 , 𝑦𝑝 − (𝜇𝑝 − 𝜇)𝑥𝑝 , 𝑥𝑝 −1 𝑦𝑝 −1 )
                                                          2      2                        2      2
      over Spec (𝔽̄𝑝 [𝜇]).
   3. Let 𝑆𝜑 be the ordinary locus. Let 𝑆𝜑 be the ordinary locus and 𝑥 be a closed point of
      𝑆𝜑 . Let 𝒪𝑆̂ 𝜑 ,𝑥 be the completion of the local ring of 𝑆𝜑 at 𝑥. Then the base change of
      𝒳×                         ̂
         𝜑 [𝑝]/𝑆𝜑 to Spec 𝒪𝑆𝜑 ,𝑥 is isomorphic to
                                                                         𝑥𝑝                  𝑝
                                                                           1 −𝑃1 (𝑦1 ,𝑦2 ),𝑥2 −𝑃2 (𝑦1 ,𝑦2 ),
                       Spec 𝔽̄𝑝 J𝑡1 , 𝑡2 , 𝑡3 K[𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 ]/ (            𝑦1𝑝 −𝑦1 ,𝑦2𝑝 −𝑦2 ,         )
                                                                       (𝑦1𝑝−1 −1)(𝑦2𝑝−1 −1)Φ𝑝 (𝑥1 )Φ𝑝 (𝑥2 )
      over Spec 𝔽̄𝑝 J𝑡1 , 𝑡2 , 𝑡3 K. Here, Φ𝑝 denotes the cyclotomic polynomial, the polynomials
      𝑃1 , 𝑃2 ∈ 𝔽̄𝑝 J𝑡1 , 𝑡2 , 𝑡3 K[𝑦1 , 𝑦2 ] are certain interpolation polynomials and the variables
      𝑡1 , 𝑡2 , 𝑡3 are the Serre-Tate coordinates.
    Next, we consider Question (C), the geometric and arithmetic properties of the Γ1 (𝑝)-
cover over each stratum 𝑆𝜑 and over the whole moduli space 𝒜2,1,𝑁 in mixed characteristics.
                                                               8


    The first property we consider is the normality. Using the descriptions given in Theorem
(1.0.4), we prove that the Γ1 (𝑝)-cover 𝒳×    𝜑 [𝑝] is not normal in all three cases of Theorem 1.0.4
and the whole integral model 𝒳× [𝑝]/𝒜2,1,𝑁 is not normal as well.
    Next, we consider the regularity properties. The whole integral model 𝒳× [𝑝] is Cohen–
Macaulay since it is finite flat over a Cohen–Macaulay base. Over the superspecial and
ordinary locus, we prove that 𝒳×    𝜑 [𝑝] is Cohen–Macaulay, but not Gorenstein using the Hopf
algebra descriptions. Using computer programs like Macaulay2, we can check the same prop-
erties hold over the supersingular locus for fixed primes. In particular, over the supersingular
locus, the Γ1 (𝑝)-cover 𝒳× 𝜑 [𝑝] is also Cohen–Macaulay, but not Gorenstein. We summarize
these results in the following table:
         Table 1.1 Regularity Properties of the Γ1 (𝑝)-cover over the Siegel Threefold
                          superspecial           supersingular         ordinary         whole
                              locus                   locuss             locus     integral model
          Normal               No                       No                No             No
     Cohen–Macaulay            Yes          Yes (for fixed primes)        Yes            Yes
        Gorenstein             No           No (for fixed primes)         No
                                                     9


                                          CHAPTER 2
                                     PRELIMINARIES
In this chapter, we will review some important tools that will be used later.
2.1     Group Schemes and Classification Theorems
Let 𝑆 be a scheme. A group scheme 𝐺 over 𝑆 is a representable functor from the category
of 𝑆-schemes to the category of groups. Equivalently, a group scheme 𝐺 is an 𝑆-scheme
together with scheme morphisms 𝑚 ∶ 𝐺 ×𝑆 𝐺 → 𝐺, inv ∶ 𝐺 → 𝐺 and 𝜖 ∶ 𝑆 → 𝐺 so that the
following diagrams commute:
                                                    (𝑚,Id)
                                  𝐺×𝐺×𝐺                     𝐺×𝐺
                                      𝑆      𝑆                  𝑆
                                  (Id,𝑚)                          𝑚                (2.1)
                                                      𝑚
                                     𝐺×𝐺                        𝐺
                                          𝑆
                                                 (𝜖,Id)
                                     𝑆×𝐺                   𝐺×𝐺
                                         𝑆                   𝑆
                                      pr2
                                                                                   (2.2)
                                                               𝑚
                                         𝐺                  𝐺
                                                (inv,Id)
                                     𝐺×𝐺                   𝐺×𝐺
                                          𝑆                  𝑆
                                                                                   (2.3)
                                                               𝑚
                                                    𝜖
                                         𝑆                  𝐺
    A group scheme 𝐺/𝑆 is called commutative if the points 𝐺(𝑇 ) are commutative groups,
for all 𝑆-schemes 𝑇 . Equivalently, this means the diagram
                                               (pr2 ,pr1 )
                                𝐺×𝐺                             𝐺×𝐺
                                  𝑆                                𝑆
                                                                                   (2.4)
                                           𝑚                𝑚
                                                   𝐺
commutes.
                                                   10


    We say a group scheme 𝐺/𝑆 is locally free (resp. flat, finite) if 𝐺 → 𝑆 is a locally free
(resp. flat, finite) morphism. For a finite, locally free group scheme 𝐺/𝑆, we call the rank
of 𝒪𝐺 as a locally free 𝒪𝑆 -module “the rank of the group scheme 𝐺/𝑆” (suppose that 𝑆 is
connected).
    Throughout this paper, all base schemes are noetherian and all group schemes are as-
sumed to be commutative and flat over the base. Note that being flat and locally free are
the same when the morphism is locally of finite presentation. Therefore all group schemes
are locally free and commutative over the base.
    Let 𝑆 be a scheme over Spec ℤ𝑝 and let 𝐺 be a finite flat commutative group scheme over
𝑆 of rank 𝑝. In [33], Oort and Tate give a classification theorem for all group schemes 𝐺/𝑆
of this type. They define an anti-equivalence of categories as following:
  ⎧triples (ℒ, 𝑢, 𝑣), where ℒ is a line bundle over⎫
  {                                                     }      ⎧𝐺/𝑆, finite flat commutative⎫
  {                                                     }      {                            }
  ⎨  𝑆,  𝑢 ∈  Γ(𝑆,  ℒ ⊗(𝑝−1)
                             ), 𝑣 ∈ Γ(𝑆,  ℒ ⊗(1−𝑝)
                                                   ) so ⎬ ⟶    ⎨                            ⎬
  {                                                     }      {group schemes of rank 𝑝     }
  {that 𝑢 ⊗ 𝑣 = 𝑤                                       }      ⎩                            ⎭
  ⎩                   𝑝                                 ⎭
Here 𝑤𝑝 is a constant in 𝑝ℤ×     𝑝 ⊂ ℤ𝑝 . Specifically, when 𝑆 = Spec 𝑅 where 𝑅 is a local ring,
the line bundle ℒ is trivial. Therefore to give a rank 𝑝 group scheme over such 𝑆, it suffices
to give two elements 𝑢, 𝑣 ∈ 𝑅 satisfying 𝑢𝑣 = 𝑤𝑝 . For such a pair (𝑢, 𝑣), the corresponding
Hopf algebra is
                                        Spec 𝑅[𝑥]/(𝑥𝑝 − 𝑢𝑥),
where the coalgebra operations are given by
                                                              𝑝−1
                             ∗                            1       𝑣𝑥𝑖 ⊗ 𝑥𝑝−𝑖
                          𝑚 (𝑥) = 1 ⊗ 𝑥 + 𝑥 ⊗ 1 +             ∑              ,
                                                        1 − 𝑝 𝑖=1 𝑤𝑖 𝑤𝑝−𝑖
                         inv∗ (𝑥) = −𝑥,
                           𝜖∗ (𝑥) = 0.
Here 𝑤𝑖 ’s are also constants in ℤ𝑝 with 𝑤1 , … , 𝑤𝑝−1 ∈ ℤ×     𝑝 and 𝑤𝑝 = 𝑝𝑤𝑝−1 . The constants
𝑤1 , … , 𝑤𝑝−1 satisfy that 𝑤𝑖 ≡ 𝑖! mod 𝑝. For more details on the 𝑤𝑖 ’s, see [33, page 10].
                                                   11


    Haines and Rapoport express this result using stack language in [16, Theorem 3.3.1]. For
convenience, we give the result here:
Theorem 2.1.1 ([16]). The ℤ𝑝 -stack 𝑂𝑇 of finite flat commutative group schemes of rank
𝑝, satisfies the following properties:
  (i) 𝑂𝑇 is an Artin stack isomorphic to
                                      [(Spec ℤ𝑝 [𝑠, 𝑡]/(𝑠𝑡 − 𝑤𝑝 ))/𝔾𝑚 ].
       The action of 𝔾𝑚 is given by 𝜆 ⋅ (𝑠, 𝑡) = (𝜆𝑝−1 𝑠, 𝜆1−𝑝 𝑡) with 𝑤𝑝 ∈ 𝑝ℤ×     𝑝 as above.
 (ii) The universal group scheme 𝒢 over 𝑂𝑇 is
                                  𝒢 = [(Spec𝑂𝑇 𝒪[𝑥]/(𝑥𝑝 − 𝑡𝑥))/𝔾𝑚 ].
       The action of 𝔾𝑚 is given by 𝜆 ⋅ 𝑥 = 𝜆𝑥.
    In [38], Raynaud generalizes the notion of Oort–Tate group schemes to higher ranks. In
particular, let 𝔽𝑞 be a finite field, where 𝑞 = 𝑝𝑛 . Raynaud consider 𝔽𝑞 -vector space schemes
of rank 1, which are the same as group schemes of rank 𝑞 = 𝑝𝑛 together with a 𝔽𝑞 -action
on it. Let ℚ𝑞 ≔ ℚ𝑝 (𝜁𝑞−1 ) be the unique unramified extension of ℚ𝑝 of degree 𝑛, and let ℤ𝑞
be the ring of integers of ℚ𝑞 . The character group of 𝔽×        𝑞 is a cyclic group of order 𝑞 − 1.
                                                                                       𝑖−1
                                                                                      𝑝
Let 𝜒1 ∶ 𝔽× 𝑞 → ℤ𝑞 be the generator of the character group and we let 𝜒𝑖 ≔ 𝜒1              . Therefore
                                                            𝑛     𝑒
𝜒𝑛+1 = 𝜒1 and any character 𝜒 can be written as ∏𝑖=1 𝜒𝑖 𝑖 with 0 ≤ 𝑒𝑖 < 𝑝. A character 𝜒
acts on the vector space scheme by
                                                1
                                    [𝜒] ≔           ∑ 𝜒−1 (𝜆)[𝜆].
                                             𝑞 − 1 𝜆∈𝔽×
                                                       𝑞
    Let the base scheme 𝑆 be a ℤ𝑞 -scheme. Raynaud shows that there is an anti-equivalence
of categories
  ⎧n triples (ℒ , 𝑢 , 𝑣 ), where ℒ ’s are line bun-⎫          ⎧𝐺/𝑆, finite flat 𝔽 -vector space⎫
  {              𝑖   𝑖 𝑖             𝑖                  }     {                    𝑞               }
  {                                                     }     {                                    }
                                    ⊗𝑝            ⊗𝑝       ⟶
  ⎨  dles over 𝑆,  𝑢 𝑖  ∶ ℒ 𝑖+1 → ℒ 𝑖   , 𝑣𝑖  ∶ ℒ 𝑖  →  ⎬     ⎨   schemes whose   eigenspaces   of ⎬
  {                                                     }     {                                    }
  {ℒ       so that 𝑢   ⊗  𝑣 = 𝑤                         }     {𝜒 are of rank 1                     }
  ⎩ 𝑖+1                                                 ⎭     ⎩ 𝑖                                  ⎭
                                                    12


Here 𝑤 ∈ ℤ𝑞 is a constant.
    As in the Oort–Tate case, when 𝑆 = Spec 𝑅 with 𝑅 a local ring, the line bundles ℒ𝑖
are trivial and 𝑢𝑖 , 𝑣𝑖 are given by elements in 𝑅. Given 𝑛 pairs (𝑢𝑖 , 𝑣𝑖 ) with 𝑢𝑖 𝑣𝑖 = 𝑤, the
associated 𝔽𝑞 -vector space scheme is given by
                     Spec 𝑅[𝑥1 , … , 𝑥𝑛 ]/ (𝑥𝑝1 − 𝑢1 𝑥2 , 𝑥𝑝2 − 𝑢2 𝑥3 , … , 𝑥𝑝𝑛 − 𝑢𝑛 𝑥1 )
with coalgebra operations
                                                                         𝑟               𝑟
             ∗                                         𝑣𝑖−ℎ ⋯ 𝑣𝑖−1           𝑒′𝑗           𝑒′′
           𝑚 (𝑥𝑖 ) = 𝑥𝑖 ⊗ 1 + 1 ⊗ 𝑥𝑖 +         ∑                    (∏ 𝑥𝑗 ) ⊗ (∏ 𝑥𝑗 𝑗 ) ,
                                            𝜒′ 𝜒′′ =𝜒𝑖
                                                         𝑤𝜒′ 𝑤𝜒′′       𝑗=1            𝑗=1
          inv∗ (𝑥𝑖 ) = −𝑥𝑖 ,
            𝜖∗ (𝑥𝑖 ) = 0.
                          𝑟     𝑒′                  𝑟     𝑒′′
Here we write 𝜒′ = ∏𝑖=1 𝜒𝑖 𝑖 and 𝜒′′ = ∏𝑖=1 𝜒𝑖 𝑖 and 𝑤𝜒 ∈ ℤ𝑞 are constants. The index
0 < ℎ ≤ 𝑛 is uniquely characterized by 𝑒′𝑖−ℎ + 𝑒′′       𝑖−ℎ = 𝑝.
    Oort–Tate group schemes and Raynaud group schemes have explicit descriptions that are
locally quotients of polynomials rings as complete intersections. This fact holds for group
schemes more generally: for any group scheme 𝐺/𝑆 with 𝑆 Noetherian, the morphism 𝐺 → 𝑆
is a local complete intersection morphism, i.e. all fibers are locally complete intersections
(see [23, Lemma 31.14]). When further assuming that 𝑆 = Spec 𝑅 with 𝑅 a local complete
Noetherian ring, the group schemes 𝐺/𝑆 have the following form:
Proposition 2.1.2. ([41, Page 28, Corollary]) Let (𝑅, 𝑚) be a complete local noetherian
ring with the residue field 𝑅/𝑚 perfect of characteristic 𝑝. Let 𝐺 = Spec 𝐴 be a local group
scheme over Spec 𝑅. Then
                                   𝐴 ≅ 𝑅J𝑥1 , … , 𝑥𝑛 K/(𝑓1 , … , 𝑓𝑛 )
                𝑒𝑖
where 𝑓𝑖 = 𝑥𝑝𝑖 + 𝑔𝑖 and 𝑔𝑖 ’s are polynomials with coefficients in 𝑚 and degree < 𝑝𝑒𝑖 . In
particular, when 𝑅 is a perfect field in characteristic 𝑝, then
                                                             𝑒1
                                 𝐴 ≅ 𝑅[𝑥1 , … , 𝑥𝑛 ]/(𝑥𝑝1 , … , 𝑥𝑝𝑛 𝑛 ).
                                                                       𝑒
                                                     13


2.2      Siegel Modular Varieties and Ekedahl–Oort Stratification
Siegel modular varieties are generalizations of modular curves. In particular, over the com-
plex numbers ℂ, the Siegel modular variety 𝒜𝑔,1,𝑁 (ℂ) is the moduli spaces of isomorphism
classes of triples (𝐴, 𝜆, 𝜂), where 𝐴 is an abelian variety over ℂ, 𝜆 ∶ 𝐴 → 𝐴∨ is a principal
polarization on 𝐴, and 𝜂 ∶ 𝐴[𝑁 ] → (ℤ/𝑁 ℤ)2𝑔 is an isomorphism compatible with the Weil
pairing on 𝐴[𝑁 ] and the symplectic pairing on (ℤ/𝑁 ℤ)2𝑔 . In [28], Mumford shows that
𝒜𝑔,1,𝑁 is a fine moduli space when 𝑁 ≥ 3. Similar to the case of modular curves, the Siegel
modular variety 𝒜𝑔,1,𝑁 also has a “model” over ℤ[1/𝑁 ]. This is in fact a smooth scheme of
relative dimension 𝑔(𝑔 + 1)/2 over ℤ[1/𝑁 ] (see [28]).
     We are particularly interested in the geometry of Siegel modular variety in characteristic
𝑝, i.e. the scheme 𝒜 ≔ 𝒜𝑔,1,𝑁 ×Spec ℤ[1/𝑁] Spec 𝔽𝑝 where 𝑝 is prime to 𝑁 . In [30], Oort defines
a stratification on 𝒜 by the isomorphism class of the 𝑝-torsion (𝐴, 𝜆)[𝑝]. This stratification
is now called the “Ekedahl–Oort stratification”. The stratification is constructed as follows.
     Let (𝐴, 𝜆) be an principally polarized abelian variety of dimension 𝑔 over a base scheme 𝑆
of characteristic 𝑝 > 0. Consider the 𝑝-torsion group scheme 𝐺 ≔ 𝐴[𝑝]. Let 𝐹𝐺 ∶ 𝐺 → 𝐺(𝑝)
and 𝑉𝐺 ∶ 𝐺(𝑝) → 𝐺 be the Frobenius and Verschiebung respectively. Note that 𝐺 is a
truncated 𝑝-divisible group. Therefore ker 𝐹𝐺 = Im 𝑉𝐺 . The principal polarization on 𝐴
gives a group scheme isomorphism 𝜁 ∶ 𝐺 → 𝐺𝐷 of 𝐺 with its Cartier dual. For any subgroup
scheme 𝐻 of 𝐺, we define 𝑉 (𝐻) ≔ Im 𝑉𝐻 and 𝐹 −1 (𝐻) as the fiber product:
                                      𝐹 −1 (𝐻)         𝐻 (𝑝)
                                                  □
                                          𝐺     𝐹𝐺
                                                       𝐺(𝑝)
where the bottom arrow is the Frobenius. We also define
                                                𝜁      𝑖𝐷
                                 𝐻 ⟂ ≔ ker(𝐺 − → 𝐺𝐷 −→ 𝐻 𝐷 ).
                                               14


Note that by identifying 𝐺 with 𝐺𝐷 via 𝜁, we get a non-degenerate pairing ⟨⋅, ⋅⟩𝐺 ∶ 𝐺 × 𝐺 →
𝜇𝑝 . The notation “⟂” comes from the fact that
                                   𝑥 ∈ 𝐻 ⟂ (𝑆) ⟺ ⟨𝑥, 𝑦⟩𝐺 = 0
for all 𝑦 ∈ 𝐻.
     Consider the points (𝐴, 𝜆, 𝜂) of 𝒜. The Ekedahl–Oort stratification of 𝒜 is defined by the
isomorphism classes of (𝐴, 𝜆)[𝑝]. For each isomorphism class of (𝐴, 𝜆)[𝑝], one can associate it
with an “elementary sequence” 𝜑, which is an increasing sequence of integers of length 𝑔 + 1
with initial term 0 and increments less than or equal to 1, i.e. 𝜑(𝑖) ≤ 𝜑(𝑖 + 1) ≤ 𝜑(𝑖) + 1.
These sequences are called elementary sequences and it is easy to see that there are 2𝑔
elementary sequences. The Ekedahl–Oort strata are parametrized by elementary sequences;
we denote the stratum corresponding to 𝜑 by 𝑆𝜑 . The stratification has the following
properties:
   (i) Every stratum 𝑆𝜑 is non-empty, smooth, quasi-affine and equi-dimensional of dimension
        𝑔
       ∑ 𝜑(𝑖).
       𝑖=0
  (ii) Let 𝒳 be the universal abelian surface over 𝒜 and let 𝒳𝜑 ≔ 𝒳 ×𝒜 𝑆𝜑 . Over each
       strata 𝑆𝜑 , there is a so-called “canonical filtration” of group schemes
                              0 = 𝐺0 ⊂ ⋯ ⊂ 𝐺𝑟 = 𝑉 (𝐺) ⊂ ⋯ ⊂ 𝐺2𝑟 = 𝐺,
       of 𝐺 = 𝒳𝜑 [𝑝] satisfying
                                          Rank 𝐺𝑖 = 𝑝𝜌(𝑖)                                 (2.5)
                                            𝑉 (𝐺𝑖 ) = 𝐺𝑣(𝑖)                               (2.6)
                                         𝐹 −1 (𝐺𝑖 ) = 𝐺𝑓(𝑖)                               (2.7)
                                               𝐺⟂𝑖 = 𝐺2𝑟−𝑖                                (2.8)
                                        (𝐺𝑖 /𝐺𝑗 )𝐷 ≅ 𝐺2𝑟−𝑗 /𝐺2𝑟−𝑖                         (2.9)
                                                  15


    Here the index 𝑟 and maps 𝜌 ∶ {0, … , 2𝑟} → ℤ, 𝑣 ∶ {0, … , 2𝑟} → {0, … , 𝑟}, 𝑓 ∶
{0, … , 2𝑟} → {𝑟, … , 2𝑟} are determined and can be easily calculated from 𝜑. For a given 𝜑,
set 𝜌(0) = 0 and define 𝜌(𝑖) to be the maximum index > 𝜌(𝑖−1) so that 𝜑(𝜌(𝑖)) = 𝜑(𝜌(𝑖−1))
or 𝜑(𝜌(𝑖)) − 𝜑(𝜌(𝑖 − 1)) = 𝜌(𝑖) − 𝜌(𝑖 − 1). In other words, 𝜌(𝑖) is the maximal index after
𝜌(𝑖 − 1) so that 𝜑 keeps increasing or stationary from 𝜌(𝑖 − 1) to 𝜌(𝑖). In this way, we form
a 𝜌 ∶ {0, … , 𝑟} → ℤ (note that 𝑟 is the highest index in 𝜌). We define 𝑣 ∶ {0, … , 𝑟} → ℤ and
𝑓 ∶ {0, … , 𝑟} → ℤ by setting 𝑣(0) = 0, 𝑓(0) = 𝑟 and define 𝑣(𝑖) and 𝑓(𝑖) by
      𝑣(𝑖) = 𝑣(𝑖 − 1) ⇔ 𝑓(𝑖) = 𝑓(𝑖 − 1) + 1 ⇔ 𝜑(𝜌(𝑖)) = 𝜑(𝜌(𝑖 − 1)),
      𝑣(𝑖) = 𝑣(𝑖 − 1) + 1 ⇔ 𝑓(𝑖) = 𝑓(𝑖 − 1) ⇔ 𝜑(𝜌(𝑖)) − 𝜑(𝜌(𝑖 − 1)) = 𝜌(𝑖) − 𝜌(𝑖 − 1).
We then expand 𝜌, 𝑣, 𝑓 to {0, … , 2𝑟} by defining
                                   𝜌(𝑟 + 𝑖) = 2𝜌(𝑟) − 𝜌(𝑟 − 𝑖)
                                   𝑣(𝑟 + 𝑖) = 2𝑟 − 𝑓(𝑟 − 𝑖)
                                   𝑓(𝑟 + 𝑖) = 2𝑟 − 𝑣(𝑟 − 𝑖)
for all 1 ≤ 𝑖 ≤ 𝑟.
Example 2.2.1. Let 𝑔 = 2. We consider pricipally polarized abelian surfaces. We have 4
elementary sequences. For each 𝜑, let 𝒳𝜑 be the universal abelian surface over the associated
stratum 𝑆𝜑 and 𝐺 = 𝒳𝜑 [𝑝] as before. For any geometric point (𝐴, 𝜆, 𝜂) of 𝑆𝜑 , we define the
𝑝-rank 𝑘 and 𝑎-number of 𝐴 by
                                 𝐴[𝑝](𝔽̄𝑝 ) = (ℤ/𝑝ℤ)𝑘                                    (2.10)
                                     𝑎(𝐴) ≔ dim𝔽̄𝑝 Hom(𝛼𝑝 , 𝐴)                           (2.11)
For each elementary sequence 𝜑, the stratum 𝑆𝜑 has the following data:
  (i) Let 𝜑 = (0, 0, 0). In this case, the canonical filtration is
                                       0 = 𝐺0 ⊂ 𝐺1 ⊂ 𝐺2 = 𝐺.
                                                16


      The corresponding canonical type is 𝜌 = (0, 2, 4), 𝑣 = (0, 0, 1) and 𝑓 = (1, 2, 2). In this
      case, 𝐴 is of 𝑝-rank 0 and 𝑎-number 2, corresponding to superspecial abelian surfaces.
 (ii) Let 𝜑 = (0, 0, 1). In this case, the canonical filtration is
                                   0 = 𝐺0 ⊂ 𝐺1 ⊂ 𝐺2 ⊂ 𝐺3 ⊂ 𝐺4 = 𝐺.
      The corresponding canonical type is 𝜌 = (0, 1, 2, 3, 4), 𝑣 = (0, 0, 1, 1, 2) and 𝑓 =
      (2, 3, 3, 4, 4). In this case, 𝐴 is of 𝑝-rank 0 and 𝑎-number 1, corresponding to su-
      persingular but not superspecial (sometimes called supergeneral) abelian surfaces.
(iii) Let 𝜑 = (0, 1, 1). In this case, the canonical filtration is
                                   0 = 𝐺0 ⊂ 𝐺1 ⊂ 𝐺2 ⊂ 𝐺3 ⊂ 𝐺4 = 𝐺.
      The corresponding canonical type is 𝜌 = (0, 1, 2, 3, 4), 𝑣 = (0, 1, 1, 2, 2) and 𝑓 =
      (2, 2, 3, 3, 4). In this case 𝐴 is of 𝑝-rank 1 and 𝑎-number 1.
 (iv) Let 𝜑 = (0, 1, 2). In this case, the canonical filtration is
                                         0 = 𝐺0 ⊂ 𝐺1 ⊂ 𝐺2 = 𝐺.
      The corresponding canonical type is 𝜌 = (0, 2, 4), 𝑣 = (0, 1, 1) and 𝑓 = (1, 1, 2). In this
      case 𝐴 is of 𝑝-rank 2 and 𝑎-number 0. This corresponds to ordinary abelian surfaces.
2.3    Kottwitz-Wake Primitive Elements
Recall that all base schemes are assumed to be locally of finite presentation and all group
schemes are assumed to be commutative and flat over the base. Let 𝐺/𝑆 be a finite group
scheme. In [21], Kottwitz and Wake define a subscheme of “non-nullity” of 𝐺. This sub-
scheme is denoted by 𝐺× .
Definition 2.3.1 ([21]). Let 𝐺 be a finite group scheme over a base scheme 𝑆. Let ℐ ⊂ 𝒪𝐺
be the augmentation ideal of 𝐺. We define the scheme of “non-null” elements 𝐺× to be the
                                                 17


closed subscheme of 𝐺 with the defining ideal sheaf given by Ann(ℐ), the annihilator of the
augmentation ideal sheaf.
    A key property of “non-nullity” is that the scheme of “non-null” elements 𝐺× is locally
free of rank |𝐺| − 1 over 𝑆 (see [21]).
    This notion of “non-nullity” is used to describe the subscheme of “non-zero” points. But
one needs to be careful with this analogy. When 𝐺/𝑆 is not étale, the group scheme 𝐺 may
have no points other than the unit, but 𝐺× can still have points. Here is a concrete example:
Example 2.3.2. Consider the additive group 𝔾𝑎 /𝔽𝑝 . Let 𝛼𝑝 be the kernel of the Frobenius
of 𝔾𝑎 /𝔽𝑝 . In particular, the group scheme 𝛼𝑝 can be written as Spec 𝔽𝑝 [𝑥]/(𝑥𝑝 ) with aug-
mentation ideal (𝑥) and additive group operation. The annihilator of the augmentation ideal
(𝑥) is generated by 𝑥𝑝−1 and therefore 𝛼×  𝑝 = Spec 𝔽𝑝 [𝑥]/(𝑥
                                                              𝑝−1
                                                                  ).
    Note that for any reduced ring 𝑅 in characteristic 𝑝, we have 𝛼𝑝 (𝑅) = {0} and 𝛼×    𝑝 (𝑅) =
{0}. The unit section can still be non-null.
    Using “non-nullity”, Kottwitz and Wake define primitive elements for truncated 𝑝-divisible
groups. More precisely, let 𝒢/𝑆 be a truncated 𝑝-divisible group of height ℎ and level 𝑟 (i.e.
the smallest exponent such that 𝑝𝑟 kills 𝒢). This happens, for example, when 𝒢/𝑆 is the
𝑝𝑛 -torsion of a 𝑝-divisible group over 𝑆. In this case, the (scheme of) primitive elements 𝒢×
is defined to be the fiber product
                                       𝒢×          (𝒢[𝑝])×
                                              □
                                        𝒢           𝒢[𝑝]
                                             𝑝𝑟−1
where (𝒢[𝑝])× is the subscheme of “non-nullity”. The scheme of primitive elements 𝒢× /𝑆 is
flat of rank 𝑝𝑟−1 (𝑝ℎ −1) and describes the points “of exact order 𝑝𝑟 ”. Note that for truncated
𝑝-divisible groups of level 1, primitive elements are the same as “non-null” elements. For
simplicity, on general group schemes killed by 𝑝, we will abuse the name of primitive elements
for “non-null” elements.
                                                18


    In general, it may not be easy to calculate an annihilator ideal in a ring. However, by
Proposition 2.1.2, many group schemes are quotients of polynomial rings or power series
rings as complete intersections. In these cases, we have the following lemma:
Lemma 2.3.3. Let 𝑅 be a Noetherian ring and let (𝑅′ , 𝜖) be an augmented 𝑅-algebra, i.e.
an 𝑅-algebra 𝑅′ together with a ring homomorphism 𝜖 ∶ 𝑅′ → 𝑅 such that the composition
           𝜖
𝑅 → 𝑅′ −  → 𝑅 is identity. Assume that the augmentation ideal 𝐼 ≔ ker 𝜖 = (𝑥1 , … , 𝑥𝑛 ) is
generated by a regular sequence 𝑥1 , … , 𝑥𝑛 . Let 𝐽 = (𝑓1 , … , 𝑓𝑛 ) ⊂ 𝐼 be a subideal generated
by a regular sequence 𝑓1 , … , 𝑓𝑛 in 𝑅′ . Write
                            𝑓𝑖 = 𝑀𝑖1 𝑥1 + ⋯ + 𝑀𝑖𝑛 𝑥𝑛 , 𝑖 = 1, … , 𝑛
with 𝑀𝑖𝑗 ∈ 𝑅′ and set 𝑀 = (𝑀𝑖𝑗 ) ∈ Mat𝑛×𝑛 (𝑅′ ). Then we have
                                     (𝐽 ∶ 𝐼) = (det(𝑀 )) + 𝐽                                 (2.12)
where (𝐽 ∶ 𝐼) ≔ {𝑥 ∈ 𝑅′ |𝑥𝐼 ⊂ 𝐽 }. Let 𝐴 ≔ 𝑅′ /𝐽 . Let 𝐼𝐴 = 𝐼/𝐽 be the corresponding ideal
of 𝐴 and let 𝑑 be the image of det(𝑀 ) in 𝐴. Then we have
                                          Ann(𝐼𝐴 ) ≔ (𝑑)                                     (2.13)
Proof. The proof follows from [9, Proposition 2.1]. We sketch the argument here.
    The equivalence of Equation (2.12) and Equation (2.13) is immediate from the definition.
We will prove Equation (2.13).
    Consider the Koszul resolutions 𝐾((𝑥𝑖 ), 𝑅′ ) and 𝐾((𝑓𝑖 ), 𝑅′ ). They are complexes of
𝑅′ -modules defined by 𝐾𝑚 (−, 𝑅′ ) = Hom𝑅′ (∧𝑚            ′ 𝑛     ′                        ′
                                                     𝑅′ (𝑅 ) , 𝑅 ). For 𝜙 ∈ 𝐾𝑚 ((𝑥𝑖 ), 𝑅 ), the
boundary map is defined by 𝑑𝜙(𝑦) = 𝜙(𝑥 ∧ 𝑦) and 𝐾𝑚 ((𝑓𝑖 ), 𝑅′ ) is similarly defined. It is a
standard fact that 𝐾((𝑥𝑖 ), 𝑅′ ) and 𝐾((𝑓𝑖 ), 𝑅′ ) are exact from the regularity of the sequences
(see [24, Section 6]). There is a map 𝐾((𝑓𝑖 ), 𝑅′ ) → 𝐾((𝑥𝑖 ), 𝑅′ ) given as follows:
                                  𝑡
                                𝑓                               𝑓
      𝐾((𝑓𝑖 ), 𝑅′ ) ∶ 0    𝑅′       (𝑅′ )𝑛     ⋯       (𝑅′ )𝑛       𝑅′    𝐴 = 𝑅′ /𝐽         0
                            det(𝑀)                        (𝑀𝑖𝑗 )                 𝜋
                                  𝑡
                                𝑥                               𝑥
      𝐾((𝑥𝑖 ), 𝑅′ ) ∶ 0    𝑅′       (𝑅′ )𝑛     ⋯       (𝑅′ )𝑛       𝑅′    𝑅 = 𝑅′ /𝐼         0
                                                                                             (2.14)
                                                19


    Note that the exact rows are free resolutions of 𝐴 and 𝑅 respectively. By tensoring the
diagram with 𝐴 over 𝑅′ and consider the 𝑛-th homology, we get a commutative diagram
with exact rows:
                                        𝑛                     0
                           0        Tor𝑅′ (𝐴, 𝐴)         𝐴         𝐴𝑛
                                           𝜋∗             𝑑                               (2.15)
                                        𝑛                    𝑥
                           0        Tor𝑅′ (𝑅, 𝐴)         𝐴         𝐴𝑛
                                        𝑛
From the bottom row, we have Tor (𝑅, 𝐴) = Ann(𝐼𝐴 ). On the other hand, consider
                                                                   𝜋
𝐾((𝑓𝑖 ), 𝑅′ ) ⊗𝑅′ 𝐴 → 𝐾((𝑓𝑖 ), 𝑅′ ) ⊗𝑅′ 𝑅 given by 𝐴 = 𝑅′ /𝐽 −    → 𝑅 = 𝑅′ /𝐼 as in the right
colomn of Diagram (2.14). We get another commutative diagram with exact rows
                                        𝑛                    0
                            0       Tor (𝐴, 𝐴)          𝐴          𝐴𝑛
                                           𝜋∗             𝜋                               (2.16)
                                        𝑛                    0
                            0       Tor (𝑅, 𝐴)          𝑅          𝑅𝑛
This implies that 𝜋∗ is surjective and therefore by Equation (2.15), the image (𝑑) ⊂ 𝐴
                    𝑛
coincides with Tor (𝑅, 𝐴) = Ann(𝐼𝐴 ), as claimed.
    Lemma 2.3.3 is a very powerful tool to calculate primitive elements on finite group
schemes. In many cases, a group scheme killed by 𝑝 over Spec 𝑅 can be written in the
form
                                                                 𝑛
       𝐺 ≅ Spec 𝑅[𝑥]/(𝑥𝑝 − 𝐴 ⋅ 𝑥) ≔ 𝑅[𝑥1 , … , 𝑥𝑛 ]/ ({𝑥𝑝𝑖 − ∑𝑗=1 𝑎𝑖𝑗 𝑥𝑗 }         ),     (2.17)
                                                                             1≤𝑖≤𝑛
where 𝑥 = (𝑥1 , … , 𝑥𝑛 )𝑡 and 𝐴 is an 𝑛 × 𝑛 matrix over 𝑅. Pappas gives the following lemma
in [36]:
Lemma 2.3.4. Let 𝐺 be a group scheme over Spec 𝑅 killed by 𝑝 as given in Equation (2.17)
with augmentation ideal (𝑥) = (𝑥1 , … , 𝑥𝑛 ). Then the defining ideal of the primitive elements
𝐺× is generated by
                                                (−1)|𝐽 | det(𝐴𝐽 𝑐 ×𝐽 𝑐 ) ∏ 𝑥𝑝−1
                                                      𝑐
                  det(𝐷(𝑥𝑝−1 ) − 𝐴) =     ∑                                  𝑗  .         (2.18)
                                       𝐽⊂{1,…,𝑛}                         𝑗∈𝐽
Here the 𝐴𝐽 𝑐 ×𝐽 𝑐 is the minor of 𝐴 with rows and columns in 𝐽 𝑐 , the complement of 𝐽 in
{1, … , 𝑛}.
                                               20


Proof. By Lemma 2.3.3, the generator of the primitive elements in 𝐺 is generated by
det(𝐷(𝑥𝑝−1 ) − 𝐴). Expand the determinant det(𝐷(𝑥𝑝−1 ) − 𝐴) and consider the coefficient
of ∏𝑗∈𝐽 𝑥𝑝−1
          𝑗   . Note that the sign sgn(𝜎) of a permutation 𝜎 on {1, … , 𝑛} fixing a subset
𝐽 ⊂ {1, … , 𝑛} is the same as the sign sgn(𝜎)̄ of the induced permutation 𝜎̄ on 𝐽 𝑐 . Therefore
the coefficient of ∏𝑗∈𝐽 𝑥𝑝−1
                          𝑗    in det(𝐷(𝑥𝑝−1 ) − 𝐴) is the determinant det(−𝐴𝐽 𝑐 ×𝐽 𝑐 ), which is
                𝑐
equal to (−1)|𝐽 | det(𝐴𝐽 𝑐 ×𝐽 𝑐 ) as claimed in Equation (2.18).
    We can apply Lemma 2.3.4 to the Oort–Tate group schemes and Raynaud group schemes
in Section 2.1:
Example 2.3.5. Let 𝑆 = Spec 𝑅 where 𝑅 is a local ring. Consider an Oort–Tate group
scheme given by Spec 𝑅[𝑥]/(𝑥𝑝 − 𝑢𝑥) with augmentation ideal given generated by (𝑥). In this
case, the 1 × 1 matrix in Lemma 2.3.3 is given by 𝑥𝑝−1 − 𝑢, implying the primitive elements
are defined by the ideal (𝑥𝑝−1 − 𝑢).
    The Raynaud case is slightly more complicated. Using the notation in Equation (2.17),
the Raynaud group scheme over 𝑆 is given by
                                    𝐺 = Spec 𝑅[𝑥]/(𝑥𝑝 − 𝑈 ⋅ 𝑥)
where 𝑥 = (𝑥1 , … , 𝑥𝑛 )𝑡 and
                                            0 ⋯     0    𝑢𝑟
                                          ⎛
                                          ⎜𝑢1 ⋯     0     0⎞⎟
                                    𝑈 =⎜  ⎜                 ⎟ ,
                                          ⎜⋮ ⋱      ⋮     ⋮⎟⎟
                                          ⎝ 0 ⋯ 𝑢𝑟−1 0 ⎠
with augmentation ideal given by (𝑥1 , … , 𝑥𝑛 ). Note that det(𝑈 ) = (−1)𝑟−1 𝑢1 ⋯ 𝑢𝑟 and the
only principal minor of 𝑈 with nonzero determinant is the whole matrix 𝑈 . Therefore by
Lemma 2.3.4, the primitive elements 𝐺× ⊂ 𝐺 is defined by (𝑥𝑝−1   1  ⋯ 𝑥𝑝−1
                                                                        𝑟  − 𝑢1 ⋯ 𝑢𝑟 ).
    We will use the following special case of Lemma 2.3.4 later:
Example 2.3.6. Let 𝐺/ Spec 𝑅 be a group scheme as given in Equation (2.17). Suppose the
matrix 𝐴 is upper triangular, i.e. 𝑎𝑖𝑗 = 0 if 𝑖 > 𝑗. Then 𝐷(𝑥𝑝−1 ) − 𝐴 is an upper triangular
                                                21


matrix with diagonal elements 𝑥𝑝−1
                                𝑖   − 𝑎𝑖𝑖 . Therefore by Lemma 2.3.4, the defining ideal of
                           𝑛
𝐺× ⊂ 𝐺 is generated by ∏𝑖=1 (𝑥𝑝−1
                               𝑖   − 𝑎𝑖𝑖 ).
   In particular, suppose 𝐴 is strictly upper triangular, i.e. assume further that 𝑎𝑖𝑖 = 0.
Then the defining ideal of 𝐺× ⊂ 𝐺 is generated by 𝑥𝑝−1
                                                    1   ⋯ 𝑥𝑝−1
                                                            𝑛 .
                                             22


                                          CHAPTER 3
           FULL LEVEL STRUCTURE ON FINITE GROUP SCHEMES
In this chapter, we consider the Γ(𝑝)-level structure, also called “full level structure”, on
finite flat commutative group schemes.
     Suppose that 𝐻 is annihilated by 𝑝𝑟 and 𝐻[ 𝑝1 ] ≔ 𝐻 ×𝑆 𝑆[ 𝑝1 ] is étale-locally isomorphic
to the constant group scheme (ℤ/𝑝𝑟 ℤ)𝑔 for some 𝑟 and 𝑔. This happens, for example, when
𝐻 is the 𝑝𝑟 -torsion of some abelian variety of dimension 𝑔/2.
     Let Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) be the functor from the category of 𝑆-schemes Sch𝑆 to the
category of abelian groups Ab, defined by
                     Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻)(𝑇 ) ≔ Hom𝑔𝑝 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻(𝑇 )).
We will use Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) to denote the representing scheme. Since 𝐻 is annihilated
by 𝑝𝑟 , the representing scheme is just 𝐻 𝑔 . The general linear group GL𝑔 (ℤ/𝑝𝑟 ℤ) has a
natural right action on Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) by acting on (ℤ/𝑝𝑟 ℤ)𝑔 by precomposition.
     The problem we consider is to give a notion of full level structure on 𝐻. We expect it
                                                                                   ∗
to be a closed subscheme of Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻), which we denote by Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻),
satisfying:
             ∗
    1. Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) is flat over 𝑆 and of rank |GL𝑔 (ℤ/𝑝𝑟 ℤ)|.
             ∗
    2. Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) is GL𝑔 (ℤ/𝑝𝑟 ℤ)-invariant under the right GL𝑔 (ℤ/𝑝𝑟 ℤ)-action on
       Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻). When inverting 𝑝, we have an identification
                            ∗                   1                                1
                       Hom𝑆[ 1 ] ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻[ ]) = Isom𝑆[ 𝑝1 ] ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻[ ])
                              𝑝                 𝑝                                𝑝
       as closed subschemes of Hom𝑆[ 1 ] ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻[ 𝑝1 ]).
                                        𝑝
    3. When identifying Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) ×𝑆 𝑇 with Hom𝑇 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻𝑇 ) in the natural
                          ∗                              ∗
       way, we have Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻)×𝑆 𝑇 = Hom𝑇 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻𝑇 ) as closed subschemes,
       for any 𝑆-scheme 𝑇 .
                                                23


    We also expect our definition to coincide with the intuitive definition for some familiar
                                                                  ∗
group schemes. For example, for 𝐻 = 𝜇𝑝𝑟 , we expect Homℤ (ℤ/𝑝𝑟 ℤ, 𝐻) to be the closed
subscheme of 𝜇𝑝𝑟 defined by the cyclotomic polynomial
                                    𝑟
                                  𝑥𝑝 − 1              𝑟−1        𝑟−1
                     Φ𝑝𝑟 (𝑥) ≔    𝑝𝑟−1     = 𝑥(𝑝−1)𝑝 + 𝑥(𝑝−2)𝑝 + ⋯ + 1.
                                 𝑥     −1
When 𝐻 is the constant group scheme (ℤ/𝑝𝑟 ℤ)𝑔 , the resulting full level structure on 𝐻 given
         ∗
by Homℤ ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) should be GL𝑔 (ℤ/𝑝𝑟 ℤ) ⊂ Mat𝑔 (ℤ/𝑝𝑟 ℤ).
    The motivation for giving a well-behaved notion of full level structure comes from the
study of integral models of Shimura varieties. For example, for modular curves, finding an
integral model of the modular curve 𝑋(𝑝𝑟 ) essentially amounts to finding a flat model of full
level structure on the 𝑝𝑟 -torsion of elliptic curves. This is done by Katz and Mazur in their
book [19]: following an idea of Drinfeld in [10], Katz and Mazur consider the case when 𝐻
can be embedded into a curve. In this case a set of sections {𝑃1 , … , 𝑃𝑛 } of 𝐻 is defined to
be a “full set of sections”, if the points generate the group 𝐻 as Cartier divisors. Using this
notion, the full level structure on 𝐻 is defined to be the maps in Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) whose
                                                             ∗
image forms a full set of sections. As a scheme, Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) can be also described
as the closed subscheme of Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) cut out by the Cartier divisor equation
                                       𝐻=       ∑        [ℎ(𝑥)]
                                             𝑥∈(ℤ/𝑝𝑟 ℤ)𝑔
where ℎ is the universal homomorphism. Katz and Mazur’s construction, for example,
gives a definition of full level structure on ℤ/𝑝ℤ × 𝜇𝑝 , as it is the 𝑝-torsion of an ordinary
elliptic curve. They also suggest a natural generalization of their construction, given by
“×-homomorphisms” [19, Appendix of Chapter 1], that can be defined for general group
schemes. Unfortunately, the notion of ×-homomorphisms is deficient because the resulting
closed subscheme is generally not flat over the base. Such a negative result has been observed
by Chai and Norman in [3, Appendix 2]. For example, the nonflatness for ×-homomorphisms
even happens on 𝜇𝑝 × 𝜇𝑝 .
                                                 24


    As an improvement, Wake gives in [42] a good definition in the case of 𝐻 = 𝜇𝑝 × 𝜇𝑝
over Spec ℤ. By using a notion of “primitive elements”, he defines the full level structure,
called “scheme of full homomorphisms”, to be cut out by the condition that all nontrivial
linear combinations of rows and columns of the universal homomorphism are primitive.
Alternatively, Wake also gives another level structure, called “KM+D” level structure, short
for Katz-Mazur + Dual. The notion of KM+D level structure is defined by requiring both
universal homomorphism and its dual being ×-homomorphisms as defined by Katz and
Mazur. Wake proves that in the case 𝜇𝑝 × 𝜇𝑝 , the KM+D level structure coincides with
his original notion of full homomorphisms. Unfortunately, in general the “KM+D” level
structure does not give a flat scheme over the base. For example, it is observed in [42,
                         KM+D
Example 4.8] that Hom𝔽         ((ℤ/2ℤ)2 , (𝛼2 )2 ) has larger rank than expected.
                          2
    In this chapter, we give a definition of full level structure for 𝐻 of the form 𝐻 = 𝐺 × 𝐺,
where 𝐺 is a rank 𝑝 group scheme over a ℤ𝑝 -scheme 𝑆. When 𝐺 is 𝜇𝑝 , our definition
coincides with the one in [42]. The idea of our construction is to generalize Wake’s “rows-
and-columns” construction to a general group scheme 𝐺 using Kottwitz-Wake’s notion of
primitive elements [21]. In [21] the authors give a notion of primitive elements which is
well-behaved, even for general 𝑝-divisible groups. Using this notion, our full level structure
will be cut out by the condition that rows and columns of the universal homomorphism are
linearly independent, as in Wake’s construction. The precise description and properties are
discussed in Section 3. The main point is that this construction gives a flat model. We show
this by using Oort–Tate theory to reduce to Wake’s result.
    One might also expect the following naturality condition:
                                                 ∼
   4. For any group scheme isomorphism 𝐻 −       → 𝐻 ′ , the induced isomorphism
                            Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) → Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻 ′ )
                                         ∗                          ∗
      restricts to an isomorphism Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻) → Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐻 ′ ).
                                                25


This condition (4) can be interpreted as saying the notion of full level structure is defined
over the stack. Unfortunately, it turns out that in general there is no level structure on
𝐻 satisfying all conditions (1) − (4). Wake pointed out to us that the construction cannot
extend to the stack. We discuss this negative result in Section 3.3 and include their example
there. We thank Wake for the communication.
3.1     Full level structure on 𝐺 × 𝐺
The fundamental tool in defining the full level structure on 𝐺 × 𝐺 is the notion of “primitive
elements” (see Section 2.3). An important example of the primitive elements is the Oort–Tate
group scheme 𝐺 = Spec 𝐴[𝑥]/(𝑥𝑝 − 𝑡𝑥), where 𝐴 is a ℤ𝑝 -algebra. The augmentation ideal
is (𝑥). Thus 𝐺× is defined by the ideal (𝑥𝑝−1 − 𝑡), coinciding with the scheme of generators
defined in [16]. Another example is 𝐺× 𝐺. Its underlying algebra is 𝐴[𝑥, 𝑦]/(𝑥𝑝 −𝑡𝑥, 𝑦𝑝 −𝑡𝑦)
with the augmentation ideal (𝑥, 𝑦). By a direct calculation or using Lemma 2.3.3, we can
see that the scheme of primitive elements in 𝐺2 is
                         (𝐺2 )× = Spec 𝐴[𝑥, 𝑦]/ ((𝑥𝑝−1 − 𝑡)(𝑦𝑝−1 − 𝑡)) .
See also [21, Section 3.8].
    Now we consider the operation on the points of Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) = 𝐺4 (as functors).
We will identify 𝐺4 (𝑇 ) with Mat2 (𝐺(𝑇 )), the additive group of 2×2 matrices with entries in
𝐺(𝑇 ). On each entry Hom𝑆 (ℤ/𝑝ℤ, 𝐺)(𝑇 ) = 𝐺(𝑇 ), there is a natural addition arising from
the group structure of 𝐺. We denote this addition by +,̇ to distinguish it from the addition
on 𝒪𝐺 . For simplicity, for any 𝑓 ∈ 𝐺(𝑇 ), let [𝑚]𝑓 be 𝑓 +𝑓 ̇ +̇ ⋯ +𝑓,
                                                                    ̇ the sum of 𝑚 copies of
𝑓. Since the Oort–Tate Group is annihilated by 𝑝, the operation [𝑚] only depends on 𝑚
modulo 𝑝.
Example 3.1.1. Let 𝑆 = Spec ℤ𝑝 and 𝐺 = Spec ℤ𝑝 [𝑥]/(𝑥𝑝 − 𝑥) with comultiplication
                                                        𝑝−1
                                                    1       𝑤𝑝 𝑥𝑖 ⊗ 𝑥𝑝−𝑖
                      𝑚∗ (𝑥) = 1 ⊗ 𝑥 + 𝑥 ⊗ 1 +          ∑                .
                                                  1 − 𝑝 𝑖=1 𝑤𝑖 𝑤𝑝−𝑖
                                               26


This is obtained by taking 𝑢 = 1 and 𝑣 = 𝑤𝑝 from Section 2. Let 𝑇 = Spec ℤ𝑝 . In 𝐺(𝑇 ), let
𝜒(𝑗) ∈ Hom𝑆 (ℤ/𝑝ℤ, 𝐺)(𝑇 ) = 𝐺(𝑇 ) be the map sending 𝑥 to 𝜒(𝑗), where 𝜒 is the Teichmüller
character and let 𝜒(0) = 0. Since the elements in 𝐺(𝑇 ) are closed under the group action,
we have ([𝑗](1))𝑝 − [𝑗](1) = 0. On the other hand, by the definition of the comultiplication
of 𝐺, we have [𝑗](1) ≡ 𝑗 mod 𝑝. Therefore [𝑗](1) = 𝜒(𝑗). From [𝑗](1)+[𝑘](1) ̇    = [𝑗 + 𝑘](1),
we get a useful equation:
                                                    𝑝−1
                                                 1      𝑤𝑝 𝜒(𝑗𝑖 )𝜒(𝑘𝑝−𝑖 )
                     𝜒(𝑗 + 𝑘) = 𝜒(𝑗) + 𝜒(𝑘) +       ∑                     .              (3.1)
                                              1 − 𝑝 𝑖=1      𝑤𝑖 𝑤𝑝−𝑖
                                                                              ℤ/𝑝ℤ
    In fact, 𝐺 is isomorphic to the constant group scheme ℤ/𝑝ℤ = Spec ℤ𝑝           . The Hopf
                                                                  𝑆
                                                      ℤ/𝑝ℤ
algebra isomorphism between ℤ𝑝 [𝑥]/(𝑥𝑝 − 𝑥) and ℤ𝑝          is given by 𝑥 ↦ ∑ 𝜒(𝑖)𝑒𝑖 and
                                                              1
𝑒𝑖 ↦ 𝜆(𝑖) ∏𝑗≠𝑖 (𝑥 − 𝜒(𝑗)), where 𝜆(0) = −1 and 𝜆(𝑖) =        𝑝−1 otherwise. To see this, we
first easily observe that the maps give algebra isomorphisms. To see that it preserves the
comultiplication, we can check straightforwardly using Equation (3.1). We will skip the
detailed calculation here.
                         ∗
    Now we define Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) to be the subfunctor of Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 )� which
is given as follows:
                                ∗
Definition 3.1.2. Define Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) to be the functor whose 𝑇 -valued points are
the elements in Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 )(𝑇 ) = Mat2 (𝐺(𝑇 )) so that all nonzero 𝔽𝑝 -linear com-
binations of rows and columns are in (𝐺2 )× (𝑇 ). For nonzero 𝔽𝑝 -linear combinations, we
                            ̇
mean elements like [𝑚]𝑓 +[𝑛]𝑔   where 𝑚 and 𝑛 are not both zero in 𝔽𝑝 .
                                                        ∗
Remark 3.1.3. It is easy to see that the functor Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) we defined above is
representable. Indeed, each linear combination being primitive is a closed condition and thus
                              ∗                                               ∗
gives a subscheme of Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) = 𝐺4 . Therefore the functor Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 )
is represented by the scheme-theoretical intersection of those subschemes.       We will use
      ∗
Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) for the representing scheme.
                                                 ∗
    Here are some elementary properties of Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ):
                                             27


Proposition 3.1.4. Let 𝑆 be a ℤ𝑝 -scheme and let 𝐺, 𝐺′ be finite flat commutative group
schemes of rank 𝑝 over S. Let GL2 (𝔽𝑝 ) act on Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) by acting on (ℤ/𝑝ℤ)2 by
                              ∗
precomposition. Then Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) satisfies:
  (i) By identifying Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) ×𝑆 𝑇 = Hom𝑇 ((ℤ/𝑝ℤ)2 , 𝐺2𝑇 ) for any 𝑆-scheme 𝑇 ,
      we have
                                ∗                                  ∗
                          Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) ×𝑆 𝑇 = Hom𝑇 ((ℤ/𝑝ℤ)2 , 𝐺2𝑇 )
      as closed subschemes.
                                     ∗
 (ii) The full level structure Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) is GL2 (𝔽𝑝 )-invariant. Away from charac-
      teristic 𝑝, we have
                             ∗
                        Hom𝑆[ 1 ] ((ℤ/𝑝ℤ)2 , 𝐺[ 𝑝1 ]2 ) = Isom𝑆[ 𝑝1 ] ((ℤ/𝑝ℤ)2 , 𝐺[ 𝑝1 ]2 )
                                𝑝
      as closed subschemes of Hom𝑆[ 1 ] ((ℤ/𝑝ℤ)2 , 𝐺[ 𝑝1 ]2 ).
                                       𝑝
(iii) Let 𝜙 ∶ 𝐺 → 𝐺′ be an isomorphism and let Φ ∶ 𝐺2 → (𝐺′ )2 be the isomorphism given
            𝜙 0
      by (       ) . Then the isomorphism Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) → Hom𝑆 ((ℤ/𝑝ℤ)2 , (𝐺′ )2 ) in-
            0 𝜙
                                                                                         ∗
      duced by Φ restricts to an isomorphism on the full level structures Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) →
            ∗
      Hom𝑆 ((ℤ/𝑝ℤ)2 , (𝐺′ )2 ).
Proof.
  (i) It follows straightforwardly from Definition 3.1.2 and the fact that the notion of prim-
      itive elements is compatible with base change [21, 3.5].
                     ∗
 (ii) Let 𝑓 ∈ Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ), regarded as a 2 × 2 matrix in 𝐺(𝑆). Let 𝑔 ∈ GL2 (𝔽𝑝 ).
      Then 𝑔 acts on 𝑓 by 𝑓 ↦ 𝑔𝑡 𝑓, where the scalar multiplication is [⋅] and the addition is +.̇
      By an elementary calculation, one can see that it suffices to show that if (𝑢, 𝑣) ∈ (𝐺2 )×
      then (𝑢, 𝑣)𝑔 ∈ (𝐺2 )× for all 𝑔 ∈ GL2 (𝔽𝑝 ). Note that since 𝐺 is annihilated by 𝑝, every
      𝑚 ∈ 𝔽𝑝 defines an endomorphism of 𝐺 and therefore every 2 × 2 matrix over 𝔽𝑝 defines
      an endomorphism of 𝐺2 and invertible matrices induce automorphisms of 𝐺2 . In fact,
                                                   28


       (𝑢, 𝑣)𝑔 is the image of (𝑢, 𝑣) under the automorphism induced by 𝑔. Since group scheme
       automorphisms preserve the augmentation ideal sheaf, they also preserve the primitive
       elements by Definition 2.3.1. Therefore (𝑢, 𝑣)𝑔 ∈ (𝐺2 )× and we are done.
       For the second half of (ii), note that 𝐺[ 𝑝1 ] is étale locally isomorphic to ℤ/𝑝ℤ and by
                        ∗
       definition Hom ((ℤ/𝑝ℤ)2 , (ℤ/𝑝ℤ)2 ) = Isom((ℤ/𝑝ℤ)2 , (ℤ/𝑝ℤ)2 ). Then the statement is
       an immediate result of (i).
 (iii) As in (ii), since every group scheme isomorphism preserves the augmentation ideal
       sheaf, by Definition 2.3.1 it also preserves the primitive elements. Then it is straight-
       forward to check that (iii) holds by Definition 3.1.2.
     Now here is the main theorem in this chapter:
Theorem 3.1.5. Let 𝑆 be a ℤ𝑝 -scheme and let 𝐺 be a finite flat commutative group scheme
                               ∗
of rank 𝑝 over S. Let Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) be as defined in Definition 3.1.2. Then the full
                       ∗
level structure Hom𝑆 ((ℤ/𝑝ℤ)2 , 𝐺2 ) is flat over 𝑆 of rank |GL2 (𝔽𝑝 )|.
3.2      Proof of Theorem 3.1.5
By Proposition 3.1.4 (i), since being flat is a local property, we can reduce to the case where
𝑆 = Spec 𝐴 with 𝐴 being a local ℤ𝑝 -algebra. Recall from Section 2 that the group scheme
𝐺/𝑆 is determined by a triple (ℒ, 𝑢, 𝑣). Since 𝐴 is local, the line bundle ℒ on 𝑆 is trivial.
Let 𝒜 = ℤ𝑝 [𝑠, 𝑡]/(𝑠𝑡 − 𝑤𝑝 ) and 𝒮 = Spec 𝒜. Let 𝒢 = Spec 𝒜[𝑥]/(𝑥𝑝 − 𝑡𝑥) be the group
scheme over 𝒮 with comultiplication
                                                            𝑝−1
                           ∗                            1        𝑠𝑥𝑖 ⊗ 𝑥𝑝−𝑖
                          𝑚 (𝑥) = 1 ⊗ 𝑥 + 𝑥 ⊗ 1 +           ∑               .               (3.2)
                                                      1 − 𝑝 𝑖=1 𝑤𝑖 𝑤𝑝−𝑖
Then 𝐺/𝑆 will be the pull back of 𝒢/𝒮 through a morphism 𝑆 → 𝒮 determined by 𝑢 and 𝑣.
Applying Proposition 3.1.4 (i) again, we can see that it suffices to show the flatness of the
full level structure for 𝒢2 /𝒮.
                                                 29


                                 ∗
    We first look at Hom𝒮 ((ℤ/𝑝ℤ)2 , 𝒢2 ) over the two open subschemes Spec ℤ𝑝 [𝑠, 𝑠−1 ] and
Spec ℤ𝑝 [𝑡, 𝑡−1 ] of 𝒮. It is easy to check that after applying étale base changes by adding the 𝑝−
                                                       1       1                                      1        1
1th root of 𝑠, 𝑠−1 , 𝑡, 𝑡−1 , we get 𝒢×𝒮 Spec ℤ𝑝 [𝑠 𝑝−1 , 𝑠− 𝑝−1 ] ≅ 𝜇𝑝 and 𝒢×𝒮 Spec ℤ𝑝 [𝑡 𝑝−1 , 𝑡− 𝑝−1 ] ≅
ℤ/𝑝ℤ. In these cases, the following lemma is as expected:
Lemma 3.2.1. We have the following two isomorphisms of group schemes:
            ∗                                    1      1              full
  (i) Hom𝒮 ((ℤ/𝑝ℤ)2 , 𝒢2 ) ×𝒮 Spec ℤ𝑝 [𝑠 𝑝−1 , 𝑠− 𝑝−1 ] ≅ Hom                      1       1
                                                                                        − 𝑝−1   ((ℤ/𝑝ℤ)2 , 𝜇2𝑝 ).
                                                                       Spec ℤ𝑝 [𝑠 𝑝−1 ,𝑠      ]
                        full
      Here the Hom           is the full level structure for 𝜇𝑝 × 𝜇𝑝 defined by Wake in [42].
            ∗                                    1     1
 (ii) Hom𝒮 ((ℤ/𝑝ℤ)2 , 𝒢2 ) ×𝒮 Spec ℤ𝑝 [𝑡 𝑝−1 , 𝑡− 𝑝−1 ] ≅ GL2 (ℤ/𝑝ℤ).
                                                     ∗
Proof. Note that from the definition of Hom , we have
                                    ∗                         full
                              Hom ((ℤ/𝑝ℤ)2 , 𝜇2𝑝 ) = Hom           ((ℤ/𝑝ℤ)2 , 𝜇2𝑝 )
as they are defined in the same way. For the étale part, note that sections of constant group
                                                                              ∗
schemes being primitive exactly means being nonzero. So Hom ((ℤ/𝑝ℤ)2 , (ℤ/𝑝ℤ)2 ) consists
of the matrices satisfying that nonzero linear combinations of rows and columns are nonzero,
                                               ∗
thus invertible matrices. Hence Hom ((ℤ/𝑝ℤ)2 , (ℤ/𝑝ℤ)2 ) = GL2 (ℤ/𝑝ℤ) and the claim is
immediate from Proposition 3.1.4 (i).
    To make the full level structure explicit for 𝒢2 /𝒮, it is helpful to use the universal ho-
momorphism for description. Consider the universal base 𝒮univ = Spec 𝒜univ where 𝒜univ =
𝒜[𝑎, 𝑏, 𝑐, 𝑑]/(𝑎𝑝 − 𝑡𝑎, 𝑏𝑝 − 𝑡𝑏, 𝑐𝑝 − 𝑡𝑐, 𝑑𝑝 − 𝑡𝑑). Then we have 𝒮univ = Hom𝒮 ((ℤ/𝑝ℤ)2 , 𝒢2 ).
Let ℎ ∈ Hom𝒮univ ((ℤ/𝑝ℤ)2 , 𝒢2𝒮univ ) be the universal homomorphism defined over 𝒮univ , given
                                                           ∗
by (1, 0) ↦ (𝑎, 𝑏), (0, 1) ↦ (𝑐, 𝑑). Then Hom𝒮 ((ℤ/𝑝ℤ)2 , 𝒢2 ), as a subscheme of the uni-
                                                                     ∗
versal base 𝒮univ , is cut out by the condition ℎ ∈ Hom𝒮univ ((ℤ/𝑝ℤ)2 , 𝒢2𝒮univ ). Therefore, by
                   ∗
definition, Hom𝒮 ((ℤ/𝑝ℤ)2 , 𝒢2 ) is given by the ideal 𝐼 ⊂ 𝒜univ generated by
                                      𝑝−1                        𝑝−1
                   {(([𝑚]𝑎+[𝑛]𝑏)  ̇        − 𝑡) (([𝑚]𝑐+[𝑛]𝑑)̇          − 𝑡) ,
                                     𝑝−1                       𝑝−1
                   (([𝑚]𝑎+[𝑛]𝑐)̇                         ̇
                                         − 𝑡) (([𝑚]𝑏+[𝑛]𝑑)           − 𝑡)}                        .
                                                                             (𝑚,𝑛)∈𝔽2𝑝 {(0,0)}
                                                     30


    Recall that in the notion [𝑚]𝑎+[𝑛]𝑏,̇     we are regarding 𝑎, 𝑏, 𝑐, 𝑑 as elements in 𝒢(𝒮univ ),
corresponding to the homomorphisms
              𝐴[𝑥]/(𝑥𝑝 − 𝑡𝑥) → 𝒜[𝑎, 𝑏, 𝑐, 𝑑]/(𝑎𝑝 − 𝑡𝑎, 𝑏𝑝 − 𝑡𝑏, 𝑐𝑝 − 𝑡𝑐, 𝑑𝑝 − 𝑡𝑑)
sending 𝑥 to 𝑎, 𝑏, 𝑐, 𝑑. As an abstract group, 𝒢(𝒮univ ) is given by
                {𝑥 ∈ 𝒜[𝑎, 𝑏, 𝑐, 𝑑]/(𝑎𝑝 − 𝑡𝑎, 𝑏𝑝 − 𝑡𝑏, 𝑐𝑝 − 𝑡𝑐, 𝑑𝑝 − 𝑡𝑑)∣𝑥𝑝 = 𝑡𝑥}
                                                            1 𝑝−1 𝑠𝑥𝑖 𝑦𝑝−𝑖
with the group structure given by 𝑥+𝑦     ̇ =𝑥+𝑦+                 ∑               . Therefore
                                                          1 − 𝑝 𝑖=1 𝑤𝑖 𝑤𝑝−𝑖
                       𝑝−1                           𝑝−1                                𝑝−1
                  1          𝑠𝑎𝑝                1           𝑠𝑡𝑎                     1           𝑤𝑝
   [2]𝑎 = 2𝑎 +         ∑              = 2𝑎 +         ∑              = (2 +              ∑           ) 𝑎.
               1 − 𝑝 𝑖=1 𝑤𝑖 𝑤𝑝−𝑖             1 − 𝑝 𝑖=1 𝑤𝑖 𝑤𝑝−𝑖                    1 − 𝑝 𝑖=1 𝑤𝑖 𝑤𝑝−𝑖
Using Equation (3.1), we get [2]𝑎 = 𝜒(2)𝑎 and in general by induction we have [𝑚]𝑎 =
𝜒(𝑚)𝑎. Therefore the full level structure on 𝒢2 /𝒮 has the following expression:
               ∗
           Hom𝒮 ((ℤ/𝑝ℤ)2 , 𝒢2 )
                                                   𝑠𝑡−𝑤𝑝 ,𝑎𝑝 −𝑡𝑎,𝑏𝑝 −𝑡𝑏,𝑐𝑝 −𝑡𝑐,𝑑𝑝 −𝑡𝑑,               (3.3)
                                                             𝑝−1                       𝑝−1
           ≅ Spec ℤ𝑝 [𝑠, 𝑡, 𝑎, 𝑏, 𝑐, 𝑑]/ ⎛              ̇
                                          ⎜{((𝜒(𝑚)𝑎+𝜒(𝑛)𝑏)                      ̇
                                                                 −𝑡)((𝜒(𝑚)𝑐+𝜒(𝑛)𝑑)         −𝑡), ⎞
                                                                                                ⎟.
                                                            𝑝−1                       𝑝−1
                                                      ̇                       ̇
                                          ⎝ ((𝜒(𝑚)𝑎+𝜒(𝑛)𝑐) −𝑡)((𝜒(𝑚)𝑏+𝜒(𝑛)𝑑) −𝑡)} ⎠
                                                                          ∗
    Having all these set up, we will prove the flatness of Hom𝒮 ((ℤ/𝑝ℤ)2 , 𝒢2 ) over 𝒮 using
the lemma below:
Lemma 3.2.2 ([27] Page 51 Lemma 1). Let 𝑌 be a reduced scheme and ℱ a coherent sheaf
on 𝑌 such that dim𝑘(𝑦) ℱ ⊗𝒪𝑦 𝑘(𝑦) = 𝑟, for all 𝑦 ∈ 𝑌 . Then ℱ is a locally free of rank 𝑟 on
𝑌.
    Apply Lemma 3.2.2 to 𝑌 = 𝒮. Note that for 𝑦 ∈ Spec ℤ𝑝 [𝑡, 𝑡−1 ], we know that
                             dim𝑘(𝑦) (𝒪Hom∗𝒮 ⊗𝒪𝑦 𝑘(𝑦)) = |GL2 (𝔽𝑝 )|
from Lemma 3.2.1 (ii) and étale descent. For 𝑦 ∈ Spec ℤ𝑝 [𝑠, 𝑠−1 ], we can get
                             dim𝑘(𝑦) (𝒪Hom∗𝒮 ⊗𝒪𝑦 𝑘(𝑦)) = |GL2 (𝔽𝑝 )|
                                                  31


                                                                                        full
by combining Lemma 3.2.1 (i) together with Wake’s result on Hom                              and étale descent. The
only remaining point is 𝑦0 for 𝑠 = 𝑡 = 𝑝 = 0.
    Consider 𝒢/𝒮 modulo 𝑝, denoted by 𝒢/̄ 𝒮.̄ The underlying base scheme, which is given
by 𝒮 ̄ = Spec 𝔽𝑝 [𝑠, 𝑡]/(𝑠𝑡), is a union of two affine lines and the concerning point 𝑦0 is the
origin of 𝒮.̄ Note that 𝜒(𝑚) ≡ 𝑚 mod 𝑝. Therefore, by setting 𝑝 = 0 from Equation (1.6),
we get
                         ∗
                   Hom𝒮 ((ℤ/𝑝ℤ)
                           ̄
                                        2
                                          , 𝒢2̄ )
                                                           𝑎𝑝 −𝑡𝑎,𝑏𝑝 −𝑡𝑏,𝑐𝑝 −𝑡𝑐,𝑑𝑝 −𝑡𝑑,                          (3.4)
                                                                   𝑝−1                𝑝−1
                                                       {((𝑚𝑎+𝑛𝑏)̇      −𝑡)((𝑚𝑐+𝑛𝑑)̇        −𝑡),
                   ≅ Spec 𝒪𝒮 [𝑎,  ̄     𝑏, 𝑐, 𝑑]/ (                                             ).
                                                                  𝑝−1               𝑝−1
                                                              ̇
                                                        ((𝑚𝑎+𝑛𝑐)                ̇
                                                                      −𝑡)((𝑚𝑏+𝑛𝑑)         −𝑡)}
                                                                  𝑝−1
                                                                         𝑠𝑥𝑖 𝑦𝑝−𝑖
Here the “+”  ̇ operation is given as 𝑥+𝑦          ̇ = 𝑥+𝑦+ ∑                        (recall that 𝑤𝑖 ≡ 𝑖! mod 𝑝
                                                                  𝑖=1 𝑖! (𝑝 − 𝑖)!
from Section 2). Now we have a key observation on Equation (3.4).
Theorem 3.2.3. Let 𝒢/̄ 𝒮 ̄ be the “universal” Oort–Tate group scheme in characteristic 𝑝
                                                                                    ∗
as above. Then the ideal defining the full level structure Hom𝒮 ((ℤ/𝑝ℤ)               ̄
                                                                                                 2
                                                                                                   , 𝒢2̄ ) as a closed
subscheme of 𝒢4̄ is generated by elements which do not involve the parameter 𝑠.
Proof. We claim that in the coordinate ring (3.4), we have
                                      ̇
                                (𝑚𝑎+𝑛𝑏)      𝑝−1
                                                    − 𝑡 = 𝑢 ((𝑚𝑎 + 𝑛𝑏)𝑝−1 − 𝑡)
for some unit 𝑢. Then it follows that
                          ∗
                    Hom𝒮 ((ℤ/𝑝ℤ)
                              ̄
                                         2
                                           , 𝒢2̄ )
                                                           𝑎𝑝 −𝑡𝑎,𝑏𝑝 −𝑡𝑏,𝑐𝑝 −𝑡𝑐,𝑑𝑝 −𝑡𝑑,
                                                                  𝑝−1               𝑝−1
                                                                                                                 (3.5)
                    ≅   Spec 𝒪𝒮 [𝑎, ̄   𝑏, 𝑐, 𝑑]/ ({((𝑚𝑎+𝑛𝑏)𝑝−1 −𝑡)((𝑚𝑐+𝑛𝑑)𝑝−1 −𝑡), ) .
                                                        ((𝑚𝑎+𝑛𝑐)     −𝑡)((𝑚𝑏+𝑛𝑑)         −𝑡)}
and we are done.
    When 𝑝 = 2, since 𝑠𝑡 = 0 and 𝑎2 = 𝑡𝑎, we simply have
                      𝑚𝑎+𝑛𝑏  ̇ = 𝑚𝑎 + 𝑛𝑏 + 𝑠𝑚𝑛𝑎𝑏 = (𝑚𝑎 + 𝑛𝑏)(1 + 𝑠𝑚𝑎).
Here 1 + 𝑠𝑚𝑎 is a unit as (1 + 𝑠𝑚𝑎)2 = 1 + 𝑠2 𝑚2 𝑎2 = 1 + 𝑠2 𝑚2 𝑎𝑡 = 1.
                                                          32


                                                𝑝−1
                                                      𝑥𝑖 𝑦𝑝−𝑖
    Now suppose that 𝑝 > 2. Let 𝑔(𝑥, 𝑦) = ∑                     be a polynomial in 𝔽𝑝 [𝑥, 𝑦]. Note
                                                𝑖=1 𝑖! (𝑝 − 𝑖)!
this polynomial 𝑔(𝑥, 𝑦) is divisible by 𝑥 + 𝑦 as 𝑔(𝑥, −𝑥) = 0 (note that 𝑝 is odd). Assume
𝑔(𝑥, 𝑦) = (𝑥 + 𝑦)𝑔′ (𝑥, 𝑦). Then 𝑚𝑎+𝑛𝑏   ̇ = (𝑚𝑎 + 𝑛𝑏)(1 + 𝑠𝑔′ (𝑚𝑎, 𝑛𝑏)). Note that 𝑔′ has no
constant term and 𝑠𝑡 = 0. So we have
                                            𝑝
                         (1 + 𝑠𝑔′ (𝑚𝑎, 𝑛𝑏)) = 1 + 𝑠𝑝 𝑔′ (𝑚𝑝 𝑎𝑡, 𝑛𝑝 𝑏𝑡) = 1.
Therefore 1 + 𝑠𝑔′ (𝑚𝑎, 𝑛𝑏) is a unit and we have
                                     𝑝−1
                 (1 + 𝑠𝑔′ (𝑚𝑎, 𝑛𝑏))       ((𝑚𝑎 + 𝑛𝑏)𝑝−1 − 𝑡) = (𝑚𝑎+𝑛𝑏) ̇  𝑝−1
                                                                              −𝑡
as claimed.
    As a consequence of Theorem 3.2.3, for any point 𝑦 ∈ 𝒮 ̄ away from 𝑦0 , we have
              dim𝑘(𝑦0 ) 𝒪Hom∗𝒮 ⊗𝒪𝑦 𝑘(𝑦0 ) = dim𝑘(𝑦) 𝒪Hom∗𝒮 ⊗𝒪𝑦 𝑘(𝑦) = |GL2 (𝔽𝑝 )|.
                                   0
Applying Lemma 3.2.2, we finish proving the flatness.
3.3     Nonexistence of full level structure over the stack
Let C be a stack of group schemes of certain type over Schℤp . (By a stack here we simply
mean a category fibered in groupoids over Schℤp as in [6].) So, we assume that the objects
in C are group schemes 𝐺/𝑆 of certain fixed type (for example, finite flat commutative and
of certain rank) and the morphisms are Cartesian squares.
Definition 3.3.1. Let C be a stack of group schemes as above. By a full level structure over
C, we mean a fibered functor ℱ ∶ C → Sch, such that
(1) For any 𝐺/𝑆, the scheme ℱ(𝐺/𝑆) is a closed subscheme of Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐺).
(2) For any 𝑓 ∶ 𝐺/𝑆 → 𝐺′ /𝑆 ′ , the morphism ℱ(𝑓) ∶ ℱ(𝐺/𝑆) → ℱ(𝐺′ /𝑆 ′ ) is the restriction
     of the induced morphism Hom𝑆 ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐺) → Hom𝑆′ ((ℤ/𝑝𝑟 ℤ)𝑔 , 𝐺′ ).
(3) The scheme ℱ(𝐺/𝑆) satisfies the conditions (1)-(3) in the beginning of this chapter.
                                                 33


    Note that the condition (4) is automatically satisfied since ℱ is a functor.
    Using this terminology of full level structure over the stack, we may briefly summarize
the results in Section 3.1 as that we define a well-behaved notion of full level structure on
the stack 𝑂𝑇 , whose objects are group schemes of the form 𝐺 × 𝐺 where 𝐺 is an Oort–Tate
scheme over a ℤ𝑝 -scheme 𝑆, and morphisms are group scheme isomorphisms 𝐺×𝐺 → 𝐺′ ×𝐺′
                      𝜙 0
of the form Φ = (         ) as in Proposition 3.1.4.
                      0 𝜙
    However, this full level structure on 𝑂𝑇 cannot be extended to the stack of finite flat
commutative group schemes. In fact, consider the substack 𝑂𝑇 × 𝑂𝑇 , whose objects are
𝐺 × 𝐺′ where 𝐺, 𝐺′ are Oort–Tate schemes with morphisms be arbitrary group scheme
isomorphisms. We will see that even on 𝑂𝑇 × 𝑂𝑇 , there is no good notion of full level
structure:
Theorem 3.3.2. There is no notion of full level structure over the stack 𝑂𝑇 × 𝑂𝑇 in the
sense of Definition 3.3.1�
Proof. Let 𝒢/𝒮 be as in Section 3. Assume there is a full level structure on 𝑂𝑇 × 𝑂𝑇
satisfying (1)-(4). Then the full level structure on 𝒢2 /𝒮 must be the one we defined. In fact
over the generic fiber of 𝒮, the full level structure is given by the condition (2). Therefore
the only way to satisfy condition (1) is defining the full level structure over 𝑆 as the Zariski
closure of the corresponding scheme over the generic fiber. Note that any group scheme of
rank 𝑝 over a local ring can be obtained from 𝒢/𝒮 by base change. Because of condition (3),
the full level structure on 𝐺×𝐺 over a local base must be the one we defined above. However,
this only possible structure is not preserved under all group scheme automorphisms. Here is
one example communicated to the author by Wake:
    Consider the full level structure on 𝛼𝑝 × 𝛼𝑝 over 𝔽̅𝑝 with 𝑝 > 2. By our definition and
Theorem 3.2.3, we have
                                                                   𝑎𝑝 ,𝑏𝑝 ,𝑐𝑝 ,𝑑𝑝
                  ∗                                                    𝑝−1         𝑝−1
            Hom𝔽̅ ((ℤ/𝑝ℤ)2 , 𝛼2𝑝 ) ≅ Spec 𝔽̅𝑝 [𝑎, 𝑏, 𝑐, 𝑑]/ ({(𝑚𝑎+𝑛𝑏) (𝑚𝑐+𝑛𝑑) , ) .
                                                                      𝑝−1         𝑝−1
                    𝑝
                                                              (𝑚𝑎+𝑛𝑐)     (𝑚𝑏+𝑛𝑑)     }
                                                 34


                                                                            𝑎  𝑏
Note that Aut𝔽̅𝑝 (𝛼2𝑝 ) = GL2 (𝔽̅𝑝 ), with the action given by multiplying (     ) by elements
                                                                             𝑐 𝑑
           ̅                                     2
of GL2 (𝔽𝑝 ) from the right. Since (𝑚, 𝑛) ∈ 𝔽𝑝 {(0, 0)}, it is not hard to see that the ideal is
not invariant under the action of GL2 (𝔽̅𝑝 ).
Remark 3.3.3. Although as shown, a good notion of full level structure on the stack of all
finite group schemes does not exist, one might still hope to define a full level structure on
truncated 𝑝-divisible groups. However, some new idea is needed.
3.4     Full level structure on 𝐺 × 𝐺 × 𝐺
A natural question we may ask is whether we can have some similar results for group schemes
of the form 𝐺𝑛 , where 𝐺 is an Oort–Tate group scheme. We record some partial results here.
However, a full answer to this question requires some new idea.
     Let us take 𝐺 = 𝜇𝑝 over Spec ℤ. One intermediate step towards defining a full level struc-
ture on 𝐺3 is defining a “partial level structure” as a subscheme of Homℤ ((ℤ/𝑝ℤ)2 , (𝜇𝑝 )3 ).
We will still require that the resulting scheme is flat over the base and when inverting 𝑝
                ∗                           ∗                   ∗
we want Homℤ ((ℤ/𝑝ℤ)2 , (𝜇𝑝 )3 ) ≅ Mat2×3 (𝔽𝑝 ), where Mat2×3 denote the set of all 2 × 3
matrices of rank 2. It turns out that this can be done using our result in this paper. Let
                                                    ∗
ℎ be the universal homomorphism. Then Homℤ ((ℤ/𝑝ℤ)2 , (𝜇𝑝 )3 ) is cut out by the following
conditions:
   (i) All nonzero linear combinations of rows and columns are primitive.
  (ii) After applying any left GL2 (𝔽𝑝 )-action and right GL3 (𝔽𝑝 )-action to ℎ, one of the
       three 2 × 2 blocks of the resulting homomorphism lies in the full level structure
             ∗
       Homℤ ((ℤ/𝑝ℤ)2 , (𝜇𝑝 )2 ).
     Let us make (ii) clear here. Let
                                              𝑎11 𝑎12 𝑎13
                                       ℎ=(                 )
                                              𝑎21 𝑎22 𝑎23
                                                  35


be the universal homomorphism. Let 𝐼1 , resp. 𝐼2 , 𝐼3 , be the ideal defined by requiring that
                           𝑎     𝑎                𝑎    𝑎          𝑎    𝑎
                          ( 11 12 ) ,      resp. ( 11 13 ) , ( 12 13 ) ,
                           𝑎21 𝑎22                𝑎21 𝑎23         𝑎22 𝑎23
                                                  ∗
lies in the full level structure subscheme Hom ((ℤ/𝑝ℤ)2 , (𝜇𝑝 )2 ). Then the ideal defining “one
of the three 2 × 2 blocks lies the full level structure” is the ideal 𝐼1 𝐼2 ∩ 𝐼1 𝐼3 ∩ 𝐼2 𝐼3 . The closed
                    ∗                                                                           ∗
subscheme Homℤ ((ℤ/𝑝ℤ)2 , (𝜇𝑝 )3 ) cut out by these conditions is flat of rank |Mat2×3 (𝔽𝑝 )|
                                                                ∗
over the base. This result of “partial level structure” Homℤ ((ℤ/𝑝ℤ)2 , (𝜇𝑝 )3 ) can be extended
         ∗
to Hom ((ℤ/𝑝ℤ)2 , 𝐺3 ).
                                      ∗
     One might hope to define Homℤ ((ℤ/𝑝ℤ)3 , (𝜇𝑝 )3 ) using the “partial level structure” above,
by requiring that after applying the left and right GL3 (𝔽𝑝 )-action and possibly Cartier dual
to the universal homomorphism, the resulting homomorphism is such that any 2 × 3 block
is giving a “partial level structure”. It turns out that this condition is very close to what we
want, but still not enough. Here are some numerical results. Consider 𝜇𝑝 over 𝔽𝑝 . For 𝑝 = 2,
the above condition will give a closed subscheme of rank 169 over 𝔽𝑝 , while |GL3 (𝔽2 )| = 168.
For 𝑝 = 3, the obtained subscheme has rank 11473 over 𝔽𝑝 , while |GL3 (𝔽3 )| = 11232
(comparing with 39 = 19683). So, some further conditions need to be discovered.
                                                  36


                                       CHAPTER 4
       CONSTRUCTIONS OF GROUP SCHEMES USING DIEUDONNÉ
                                        MODULES
Dieudonné theory is a powerful tool for studying group schemes, 𝑝-divisible groups and
abelian varieties. The classical Dieudonné theory works over perfect fields in characteristic
𝑝 > 0. There has been many variations of Dieudonné theory over difference bases. In
this chapter, we will use a version of Dieudonné theory due to de Jong to construct group
schemes.
    Let 𝐺/𝑆 be a finite group scheme. (Recall that all group schemes are assumed to be
commutative and flat over the base.) We define its Cartier dual 𝐺𝐷 by
                                 𝐺𝐷 (𝑇 ) = Hom(𝐺𝑇 , 𝔾𝑚,𝑇 ).
This is a priori a functor but can be shown to be representable. This makes 𝐺𝐷 a finite
                                                                 ∼
group scheme over 𝑆 and there is a canonical isomorphism 𝐺 −     → (𝐺𝐷 )𝐷 . Furthermore,
for any group scheme homomorphism 𝑓 ∶ 𝐺 → 𝐻, we have an induced homomorphism
𝑓 𝐷 ∶ 𝐻 𝐷 → 𝐺𝐷 such that (𝑓 𝐷 )𝐷 = 𝑓 under the canonical isomorphism 𝐺 = (𝐺𝐷 )𝐷 .
    Let 𝐺/𝑆 be a group scheme (not necessarily finite) with the 𝑝𝒪𝒮 = 0. Then there are
Frobenius morphisms 𝜎𝐺 ∶ 𝐺 → 𝐺 and 𝜎𝑆 ∶ 𝑆 → 𝑆 that are defined by Frobenius maps
𝑓𝐺 ∶ 𝒪𝐺 → 𝒪𝐺 and 𝑓𝑆 ∶ 𝒪𝑆 → 𝒪𝑆 . This induces a morphism 𝐹𝐺 ∶ 𝐺 → 𝐺(𝑝) ≔ 𝐺 ×𝑆,𝜎𝑆 𝑆,
which turns out to be a homomorphism of group schemes:
                                                𝜎𝐺
                                    𝐺    𝐹𝐺
                                           𝐺(𝑝)       𝐺
                                                  □
                                                  𝜎𝑆
                                            𝑆         𝑆
    We can define another group scheme homomorphism 𝑉𝐺 ∶ 𝐺(𝑝) → 𝐺. Consider the
                                             37


following diagram:
                              𝒪𝐺         (𝒪⊗𝑝
                                           𝐺 )
                                               𝑆𝑝
                                                         𝒪𝐺 ⊗𝒪𝑆 ,𝑓𝑆 𝒪𝑆
                               𝑐𝑜𝑚𝑢𝑙𝑡
                                           𝒪⊗𝑝
                                             𝐺
Here (𝒪⊗𝑝𝐺 )
             𝑆𝑝
                is the 𝑆𝑝 -invariant elements of 𝒪⊗𝑝
                                                   𝐺 . Since 𝐺 is commutative, the comultiplica-
tion factors through (𝒪⊗𝑝      𝑆𝑝
                            𝐺 ) . The map (𝒪𝐺 )
                                                ⊗𝑝 𝑆𝑝
                                                      → 𝒪𝐺 ⊗𝒪𝑆 ,𝑓𝑆 𝒪𝑆 is the unique homomor-
phism such that 𝑎(𝑥 ⊗ ⋯ ⊗ 𝑥) ↦ 𝑥 ⊗ 𝑎. In this way, we get a morphism 𝑉𝐺 ∶ 𝐺 → 𝐺(𝑝) which
also turns out to be a homomorphism of group schemes. This group scheme homomorphism
is called the Verschiebung of 𝐺. When the group scheme 𝐺/𝑆 is finite, the Verschiebung 𝑉𝐺
can also be defined as (𝐹𝐺𝐷 )𝐷 ∶ ((𝐺𝐷 )(𝑝) )𝐷 = 𝐺(𝑝) → (𝐺𝐷 )𝐷 = 𝐺. It is a basic fact that
𝐹𝐺 ∘ 𝑉𝐺 = 𝑝 ⋅ Id𝐺(𝑝) and 𝑉𝐺 ∘ 𝐹𝐺 = 𝑝 ⋅ Id𝐺 (see [5, II]).
4.1 A Version of Dieudonné Theory by De Jong
Let 𝑊 be the Witt ring (over ℤ) as in [5, III]. It is a ring scheme over Spec ℤ. Let 𝑘 be a
perfect field in characteristic 𝑝 > 0. The 𝑘-points of 𝑊 are called Witt vectors and 𝑊 (𝑘)
is the ring of Witt vectors. The ring 𝑊 (𝑘) is a complete discrete valuation ring in mixed
characteristics (0, 𝑝) with uniformizer 𝑝 and residue field 𝑘 (see [5, III. 3]).
    Moreover, there is a ring homomorphism 𝜎 ∶ 𝑊 (𝑘) → 𝑊 (𝑘) which lifts the Frobenius
map on the residue field 𝑘. Let 𝐷𝑘 ≔ 𝑊 (𝑘){𝐹 , 𝑉 }/(𝐹 𝑉 − 𝑝) be the Dieudonné ring, where
𝑊 (𝑘){𝐹 , 𝑉 } is the non-commutative polynomial ring in variables 𝐹 , 𝑉 with relations
                                          𝐹𝑉 = 𝑉 𝐹,
                                           𝐹 𝑎 = 𝜎(𝑎)𝐹 ,
                                           𝑉 𝑎 = 𝜎−1 (𝑎)𝑉 .
The left 𝐷𝑘 -modules are called Dieudonné modules. Equivalently, one can think of Dieudonné
modules as 𝑊 (𝑘)-modules with actions of 𝐹 and 𝑉 that are subject to the relations above.
    Dieudonné modules are powerful tools to study group schemes, 𝑝-divisible groups and
abelian varieties. (Recall that all group schemes are assumed to be commutative and flat over
                                                  38


the base.) The classical (contravariant) Dieudonné theory gives an anti-equivalence between
the category of finite group schemes over a perfect field 𝑘 with positive characteristics and
the category of Dieudonné modules over 𝑊 (𝑘) with finite length (see [5, III. 7]):
                   finite group schemes                 Dieudonné modules with
                {                         }⟶{                                    }.
                         𝐺/ Spec 𝑘                         finite 𝑊 (𝑘)-length
    The classical Dieudonné theory also gives an similar anti-equivalence for 𝑝-divisible groups
over a perfect field (see [5, III. 8]):
                                               ⎧ Dieudonné modules that are       ⎫
                  𝑝-divisible groups           {
                                               {                                  }
                                                                                  }
                                                         finitely generated free
             {
                       𝒢/ Spec 𝑘
                                        }⟶⎨                                       ⎬.
                                               {
                                               {                                  }
                                                                                  }
                                               ⎩              𝑊 (𝑘)-modules       ⎭
    Starting from the classical Dieudonné theory, there has been variations of Dieudonné
theory for different base schemes. Let 𝑆 = Spec 𝑅 with 𝑅 a ring of characteristic 𝑝 > 0
and let Σ ≔ Spec ℤ𝑝 . In [2], using the notion of crystals over the crystalline site, Berthelot–
Breen–Messing define the crystalline Dieudonné functor 𝔻 from the category of 𝑝-divisible
groups over 𝑆 to the category of 𝐹 -crystals over the crystalline site CRIS(𝑆/Σ), given by
                                    𝔻(𝒢) ≔ ℰ𝑥𝑡1𝑆/Σ (𝒢, 𝒪𝑆/Σ ) .                             (4.1)
Here 𝒪𝑆/Σ is the structure sheaf defined by
                                        𝒪𝑆/Σ (𝑈 , 𝑇 , 𝛿) ≔ 𝒪𝑇 .                             (4.2)
and 𝒢 is regarded as an abelian sheaf on CRIS(𝑆/Σ). It is shown in [1] and [2] that when 𝑅
is a perfect field or more generally a perfect valuation ring, the Dieudonné functor 𝔻 defines
an anti-equivalence of categories. Moreover, let 𝑓 ∶ 𝐴 → Spec 𝑅 be an abelian scheme, then
we have
                                     𝔻(𝐴[𝑝∞ ])𝑆 = 𝑅1 𝑓∗ 𝒪𝐴/Σ .
    In [4], de Jong established a version of Dieudonné theory for group schemes 𝐺/𝑆 satisfying
𝑝𝒪𝒮 = 0 and 𝑆 and its Frobenius 𝑓𝑆 lift modulo 𝑝2 . More precisely, let 𝑆 = Spec 𝑅 be as
above. Assume that 𝑆 admits a lift to ℤ/𝑝2 ℤ, i.e. a scheme 𝑆 ′ flat over Spec ℤ/𝑝2 ℤ with
𝑆 ′ ×Spec ℤ/𝑝2 ℤ Spec 𝔽𝑝 ≅ 𝑆:
                                                  39


                                        𝑆                 𝑆′
                                                 □
                                     Spec 𝔽𝑝          Spec ℤ/𝑝2 ℤ
Then the divided power on ℤ𝑝 extends to a unique divided power 𝛾 on 𝑝𝒪𝑆′ . In this way,
the triple (𝑆, 𝑆 ′ , 𝛾) forms an object in the Crystalline site CRIS(𝑆/Σ).
    Assume further that there is a morphism 𝑓𝑆′ ∶ 𝑆 ′ → 𝑆 ′ which lifts the Frobenius map on
𝑆. Let 𝐶(1) denote the category of group schemes over 𝑆 killed by 𝑝 and let 𝑀 (1) denote the
category of triples (𝑀 , 𝐹 , 𝑉 ), where 𝑀 is a finite locally free 𝒪𝑆 -module and 𝐹 ∶ 𝑓 ∗ 𝑀 → 𝑀 ,
𝑉 ∶ 𝑀 → 𝑓 ∗ 𝑀 are 𝒪𝑆 -linear maps such that 𝑉 ∘ 𝐹 = 𝑝 ⋅ Id𝑓 ∗ 𝑀 and 𝐹 ∘ 𝑉 = 𝑝 ⋅ Id𝑀 .
    Consider the functor 𝑀𝑆 ∶ 𝐶(1) → 𝑀 (1) defined by
                                 𝑀𝑆 (𝐺) ≔ ℰ𝑥𝑡1𝑆/Σ (𝐺, ℐ𝑆/Σ )                                 (4.3)
                                                               (𝑆,𝑆 ′ ,𝛾)
where 𝐺 is regarded as an abelian sheaf on CRIS(𝑆/Σ) and ℐ𝑆/Σ is the abelian sheaf on
CRIS(𝑆/Σ) defined by
                                  ℐ𝑆/Σ (𝑈 , 𝑇 , 𝛿) ≔ ker(𝒪𝑇 → 𝒪𝑈 ).
Proposition 4.1.1. (de Jong [4, Proposition 8.6]) The functor 𝑀𝑆 ∶ 𝐶(1) → 𝑀 (1) induces
the following isomorphisms when 𝑉𝐺 = 0 or 𝐹𝐻 = 0:
                                             ∼
                           Hom𝐶(1) (𝐻, 𝐺) −  → Hom𝑀(1) (𝑀𝑆 (𝐺), 𝑀𝑆 (𝐻))
and
                               1             ∼      1
                            Ext𝐶(1) (𝐻, 𝐺) −→ Ext𝑀(1) (𝑀𝑆 (𝐺), 𝑀𝑆 (𝐻)).
Remark 4.1.2. The conditions in Proposition 4.1.1 are satisfied when 𝑆 is affine and smooth.
In fact, when 𝑆 is smooth, the obstruction class lifting 𝑆 together with the Frobenius lies in
certain cohomology 𝐻 1 (𝑆, 𝑇𝑆 ⊗ 𝐵𝑆1 ) (see [26, Appendix]). When 𝑆 is affine, the cohomology
vanishes. In this case, the lift of 𝑆 together with the Frobenius exists automatically.
    Note that this version of Dieudonné theory by de Jong is slightly different from the
crystalline Dieudonné theory of finite group schemes or abelian schemes as in the definitions
                                                   40


(4.1) and (4.3). In the rest of this section, we will compare these two versions of Dieudonné
theory.
    Let 𝔾𝑎 be the abelian sheaf on CRIS(𝑆/Σ) defined by
                                        𝔾𝑎 (𝑈 , 𝑇 , 𝛿) ≔ 𝒪𝑈 .
Therefore there is a exact sequence of abelian sheaves
                                 0 → 𝒥𝑆/Σ → 𝒪𝑆/Σ → 𝔾𝑎 → 0.
    Let 𝐴/𝑆 (resp. 𝐺/𝑆) be an abelian scheme (resp. a finite group scheme). We define
the crystalline Dieudonné crystal of 𝐴/𝑆 (resp. 𝐺/𝑆) as 𝔻(−) ≔ ℰ𝑥𝑡1𝑆/Σ (−, 𝒪𝑆/Σ ) . As a
standard result of crystalline Dieudonné theory, we have the following proposition:
Proposition 4.1.3. ([2, Proposition 2.5.8]) Let 𝐴/𝑆 be an abelian scheme. Then we have
                           ℰ𝑥𝑡𝑖𝑆/Σ (𝐴, 𝒥𝑆/Σ ) = ℰ𝑥𝑡𝑖𝑆/Σ (𝐴, 𝔾𝑎 ) = 0
for 𝑖 = 0 or 𝑖 = 2. In particular, when evaluating at 𝑆, we have the following commutative
diagram that connects with De Rham cohomology:
           0      ℰ𝑥𝑡1𝑆/Σ (𝐴, 𝒥𝑆/Σ )             𝔻(𝐴)𝑆         ℰ𝑥𝑡1𝑆/Σ (𝐴, 𝔾𝑎 )        0
                                      𝑆                                         𝑆
                                                      ≅                   ≅
                             ≅
                                                 1
           0               𝜔𝐴                𝐻𝐷𝑅     (𝐴/𝑆)         𝑅1 𝑓∗ (𝒪𝐴 )         0
Here 𝑤𝐴 is the pullback of Ω1𝐴/𝑆 along the unit section, and 𝐻𝐷𝑅   1
                                                                       (𝐴/𝑆) is the first De Rham
cohomology.
Remark 4.1.4. The identification in Proposition 4.1.3 can also be realized as following:
               0      𝑉 (𝔻(𝐴)𝑆 )          𝔻(𝐴)𝑆           𝔻(𝐴)𝑆 /𝑉 (𝔻(𝐴)𝑆 )       0
                            ≅                  ≅                     ≅
                                         1
               0          𝜔𝐴            𝐻𝐷𝑅  (𝐴/𝑆)            𝑅1 𝑓∗ (𝒪𝐴 )         0
                                                  41


where 𝑉 is the induced action of Verschibung on 𝔻(𝐴)𝑆 . In particular, ℰ𝑥𝑡1𝑆/Σ (𝐴, 𝒥𝑆/Σ ) ⊂
                                                                                                         𝑆
𝔻(𝐴)𝑆 identifies with 𝑉 𝔻(𝐴)𝑆 . Similarly,           ℰ𝑥𝑡1𝑆/Σ (𝐴, 𝒥𝑆/Σ )             ⊂ 𝔻(𝐴)(𝑆,𝑆′ ,𝛾) identifies
                                                                         (𝑆,𝑆 ′ ,𝛾)
with 𝑉 𝔻(𝐴)(𝑆,𝑆′ ,𝛾) . This follows from the observation over a field in [29, Corollary 5.11]
and the fact that the Dieudonné crystal functor commutes with base change.
    Now we give the following comparison lemma:
Lemma 4.1.5. Let 0 → 𝐺 → 𝐴 → 𝐵 → 0 be an exact sequence, where 𝐺 is a finite group
scheme killed by 𝑝 and 𝐴, 𝐵 are abelian varieties over 𝑆. Then we have
                              𝑀𝑆 (𝐺) ≅ Coker (𝑉 𝔻(𝐵)𝑆 → 𝑉 𝔻(𝐴)𝑆 ) .                                     (4.4)
In particular, we have
                             𝑀𝑆 (𝐴[𝑝]) ≅ Coker (𝑉 𝔻(𝐴)𝑆 /𝑝𝑉 𝔻(𝐴)𝑆 ) .                                   (4.5)
Proof. By Proposition 4.1.3, we can get a commutative diagram
            ℰ𝑥𝑡1𝑆/Σ (𝐵, 𝒥𝑆/Σ )              → ℰ𝑥𝑡1𝑆/Σ (𝐴, 𝒥𝑆/Σ )                → 𝑀𝑆 (𝐺) → 0.
                                 (𝑆,𝑆 ′ ,𝛾)                          (𝑆,𝑆 ′ ,𝛾)
From Remark 4.1.4, we have
                        𝑀𝑆 (𝐺) ≅ Coker (𝑉 𝔻(𝐵)(𝑆,𝑆′ ,𝛾) → 𝑉 𝔻(𝐴)(𝑆,𝑆′ ,𝛾) ) .                           (4.6)
Note that 𝐺 is killed by 𝑝. Therefore 𝑀𝑆 (𝐺) is also annihilated by 𝑝. Therefore we may
modulo 𝑝 before taking the cokernel in Equation (4.6). Note that the base 𝑆 ⊂ 𝑆 ′ is defined
by the ideal (𝑝). Therefore we have 𝔻(𝐴)(𝑆,𝑆′ ,𝛾) ⊗𝔽𝑝 = 𝔻(𝐴)𝑆 and the statement follows.
    Assume that 𝑆, 𝑆 ′ are spectra of local rings. From Proposition 4.1.3, by evaluating the
first row at (𝑆, 𝑆 ′ , 𝛾), we get that
             0 → ℰ𝑥𝑡1𝑆/Σ (𝐴, 𝒥𝑆/Σ )                  → 𝔻(𝐴)(𝑆,𝑆′ ,𝛾) → 𝑖∗ (𝑅1 𝑓∗ (𝒪𝐴 )) → 0.
                                          (𝑆,𝑆 ′ ,𝛾)
where 𝑖 ∶ 𝑆 → 𝑆 ′ is the embedding. Let 𝑠1 , … , 𝑠2𝑔 ∈ 𝔻(𝐴)(𝑆,𝑆′ ,𝛾) be a basis so that
the images 𝑠1̄ , … , 𝑠𝑔̄ ∈ 𝔻(𝐴)𝑆 generate 𝜔𝐴 and 𝑠𝑔+1                  ̄ ∈ 𝔻(𝐴)𝑆 generates 𝑅1 𝑓∗ (𝒪𝐴 ).
                                                             ̄ , … , 𝑠2𝑔
                                                       42


Then ℰ𝑥𝑡1𝑆/Σ (𝐴, 𝒥𝑆/Σ )                = 𝑉 𝔻(𝐴)(𝑆,𝑆′ ,𝛾) is generated by 𝑠1 , … , 𝑠𝑔 , 𝑝𝑠𝑔+1 , … , 𝑝𝑠2𝑔 .
                            (𝑆,𝑆 ′ ,𝛾)
The Dieudonné module 𝑀𝑆 (𝐴[𝑝]), by Equation (4.5), is generated by
               𝑠1 , … , 𝑠𝑔 , 𝑝𝑠𝑔+1 , … , 𝑝𝑠2𝑔 mod ⟨𝑝𝑠1 , … , 𝑝𝑠𝑔 , 𝑝2 𝑠𝑔+1 , … , 𝑝2 𝑠2𝑔 ⟩.
4.2     Group Schemes Annihilated by 𝑉
Let 𝐺/𝑆 be a group scheme as in Section 4.1. Suppose the rank of 𝐺/𝑆 is 𝑟. From Section 4.1,
the Dieudonné module 𝑀 ≔ 𝑀𝑆 (𝐺) is a locally free 𝒪𝑆 -module of rank 𝑟. Let 𝑓 ∶ 𝒪𝑆 → 𝒪𝑆
be the Frobenius map. Then 𝑓 ∗ 𝑀 = 𝑀 ⊗𝒪𝑆 ,𝑓 𝒪𝑆 is also a locally free 𝒪𝑆 -module of rank
𝑟. Let 𝐹𝐺 , 𝑉𝐺 be the Frobenius and Verschiebung on 𝐺 respectively. The Frobenius 𝐹𝐺 and
Verschiebung 𝑉𝐺 induce two linear maps 𝐹𝑀 ∶ 𝑓 ∗ 𝑀 → 𝑀 and 𝑉𝑀 ∶ 𝑀 → 𝑓 ∗ 𝑀 , so that
𝐹𝑀 ∘ 𝑉𝑀 = 0 and 𝑉𝑀 ∘ 𝐹𝑀 = 0.
    The goal of the rest of this chapter is to determine the group scheme 𝐺 when we are given
its Dieudonné module 𝑀𝑆 (𝐺). Upon a choice of bases of 𝑀 and 𝑓 ∗ 𝑀 , this is equivalent to
the two matrices 𝐹 , 𝑉 with 𝐹 𝑉 = 𝑉 𝐹 = 0. The first theorem of this type is the following
result of Grothendieck in [12, Exposé VII, Theorem 7.4]:
Proposition 4.2.1. Let 𝑆 be a base scheme over 𝔽𝑝 . We have the following anti-equivalence
of categories:
 ⎧ 𝐺/𝑆, finite (flat commutative)           ⎫     ⎧ pairs (𝑀 , 𝐹 ), where 𝑀 is a locally free        ⎫
 {
 {                                          }
                                            }     {
                                                  {                                                  }
                                                                                                     }
      group schemes killed by 𝑝 with          ⟶            𝒪𝑆 -module, 𝐹𝑀 ∶ 𝑓 ∗ 𝑀 → 𝑀 is a
 ⎨                                          ⎬     ⎨                                                  ⎬
 {
 {                                          }
                                            }     {
                                                  {                                                  }
                                                                                                     }
 ⎩                𝑉𝐺 = 0                    ⎭     ⎩         homomorphism of 𝒪𝑆 -modules              ⎭
    In particular, when 𝑆 = Spec 𝑅 where 𝑅 is a local ring, the Dieudonné module 𝑀 is a
free 𝑅-module. Let 𝑥1 , … , 𝑥𝑟 be a basis of 𝑀 . Then 𝑥1 ⊗ 1, … , 𝑥𝑟 ⊗ 1 form a basis of 𝑓 ∗ 𝑀 .
We write the linear map 𝐹𝑀 ∶ 𝑓 ∗ 𝑀 → 𝑀 as
                                                               𝑎11 𝑎12     ⋯ 𝑎1𝑟
                                                             ⎛
                                                             ⎜𝑎21 𝑎22      ⋯ 𝑎2𝑟 ⎞  ⎟
                 𝐹𝑀 (𝑥1 ⊗ 1, … , 𝑥𝑟 ⊗ 1) = (𝑥1 , … , 𝑥𝑟 ) ⎜  ⎜                      ⎟.
                                                             ⎜ ⋮      ⋮    ⋱ ⋮ ⎟    ⎟
                                                             ⎝𝑎𝑟1 𝑎𝑟2      ⋯ 𝑎𝑟𝑟 ⎠
                                                  43


Then the corresponding group scheme 𝐺/𝑆 is given by
                                                       𝑟
                    𝐺 = Spec 𝑅[𝑥1 , … , 𝑥𝑟 ]/ ({𝑥𝑝𝑖 − ∑𝑗=1 𝑎𝑗𝑖 𝑥𝑗 }      ),
                                                                   1≤𝑖≤𝑟
with additive coalgebra operations
                                 𝑚∗ (𝑥𝑖 ) = 1 ⊗ 𝑥𝑖 + 𝑥𝑖 ⊗ 1,
                                inv∗ (𝑥𝑖 ) = −𝑥𝑖 ,
                                  𝜖∗ (𝑥𝑖 ) = 0.
4.3    Group Schemes Annihilated by 𝑉 2
Let 𝐺/𝑆 be a group scheme as in Section 4.1. Suppose that 𝑉𝐺2 = 0 and all images and
coimages of Verschiebung are flat over 𝑆. Let 𝐺1 ≔ 𝐺/ Im 𝑉𝐺 and 𝐺2 ≔ Im 𝑉𝐺 . Then we
have an exact sequence of group schemes:
                                 0 → 𝐺2 → 𝐺 → 𝐺1 → 0.
    By our assumption, 𝐺1 and 𝐺2 are both annihilated by the Verschiebung, i.e. 𝑉𝐺1 = 0
and 𝑉𝐺2 = 0. According to Proposition 4.2.1, we may write
                                                         𝑛
                   𝐺1 = Spec 𝑅[𝑥1 , … , 𝑥𝑛 ]/ ({𝑥𝑝𝑖 − ∑𝑗=1 𝑎𝑗𝑖 𝑥𝑗 }        )
                                                                     1≤𝑖≤𝑛
and
                                                         𝑚
                   𝐺2 = Spec 𝑅[𝑦1 , … , 𝑦𝑚 ]/ ({𝑦𝑖𝑝 − ∑𝑗=1 𝑏𝑗𝑖 𝑦𝑗 }        )
                                                                    1≤𝑖≤𝑚
                                                                  𝑛              𝑚
    The Dieudonné module of 𝐺1 (resp. 𝐺2 ) is given by 𝑀1 = ⨁ 𝑅𝑥𝑖 (resp. 𝑀2 = ⨁ 𝑅𝑦𝑖 )
                                                                𝑖=1             𝑖=1
with Frobenius matrix 𝐴 = (𝑎𝑖𝑗 )𝑛×𝑛 (resp. 𝐵 = (𝑏𝑖𝑗 )𝑚×𝑚 ) and Verschibung matrix 0𝑛×𝑛
(resp. 0𝑚×𝑚 ). Let 𝑀 be a Dieudonné module that is an extension of 𝑀2 by 𝑀1 :
                                0 → 𝑀1 → 𝑀 → 𝑀2 → 0.
                                               44


Therefore 𝑀 is a free 𝑅-module of rank 𝑚 + 𝑛 with basis 𝑥1 , … , 𝑥𝑛 , 𝑦1 , … , 𝑦𝑛 and the
Frobenius and Verschiebung of 𝑀 has the description
                                                                               𝐴𝑛×𝑛 𝐶𝑛×𝑚
  𝐹 (𝑥1 ⊗ 1, … , 𝑥𝑛 ⊗ 1, 𝑦1 ⊗ 1, … , 𝑦𝑚 ⊗ 1) = (𝑥1 , … , 𝑥𝑟 , 𝑦1 , … , 𝑦𝑚 ) (                  ),
                                                                               0𝑚×𝑛 𝐵𝑚×𝑚
                                                                                                   (4.7)
                                                                                0        𝐷𝑛×𝑚
  𝑉 (𝑥1 , … , 𝑥𝑟 , 𝑦1 , … , 𝑦𝑚 ) = (𝑥1 ⊗ 1, … , 𝑥𝑛 ⊗ 1, 𝑦1 ⊗ 1, … , 𝑦𝑚 ⊗ 1) ( 𝑛×𝑛              ).
                                                                               0𝑚×𝑛 0𝑚×𝑚
so that 𝐹 ∘ 𝑉 = 𝑉 ∘ 𝐹 = 0. This is equivalent to 𝐴𝐷 = 𝐷𝐵 = 0.
    By Proposition 4.1.1, there is a bijection between the extensions 𝐺 of 𝐺1 by 𝐺2 and the
extensions 𝑀 of 𝑀1 by 𝑀2 . Therefore, to get all group schemes 𝐺 that are extensions of
𝐺1 by 𝐺2 , we only need to construct a group scheme of 𝐺1 by 𝐺2 that has 𝐹 and 𝑉 as in
Equation (4.7). Here is the theorem:
Theorem 4.3.1. Let 𝐺/𝑆 be a group scheme as in Section 4.1. Suppose that 𝑆 = Spec 𝑅
where 𝑅 is a local ring. Assume that Im 𝑉 and 𝐺/ Im 𝑉𝐺 are flat over 𝑆. Then 𝐺 can be
written as
                                                     𝑛                    𝑚               𝑛
    Spec 𝑅[𝑥1 , … , 𝑥𝑛 , 𝑦1 , … , 𝑦𝑚 ]/ ({𝑥𝑝𝑖 − ∑𝑗=1 𝑎𝑗𝑖 𝑥𝑗 } , {𝑦𝑖𝑝 − ∑𝑗=1 𝑏𝑗𝑖 𝑦𝑗 − ∑𝑗=1 𝑐𝑗𝑖 𝑥𝑗 })
with coalgebra structure given by
                                                                          𝑝−1 𝑛   𝑑𝑗𝑖 𝑥𝑘𝑗 ⊗ 𝑥𝑝−𝑘
                                                                                             𝑗
       𝑚∗𝐺 (𝑥𝑖 )   = 1 ⊗ 𝑥𝑖 + 𝑥𝑖 ⊗ 1,     𝑚∗𝐺 (𝑦𝑖 )  = 1 ⊗ 𝑦𝑖 + 𝑦 𝑖 ⊗ 1 + ∑ ∑                    ,
                                                                          𝑘=1 𝑗=1 𝑘! (𝑝 − 𝑘)!
       𝜖∗𝐺 (𝑥𝑖 ) = 0,                     𝜖∗𝐺 (𝑦𝑖 ) = 0,
       inv∗𝐺 (𝑥𝑖 ) = −𝑥𝑖 ,                inv∗𝐺 (𝑦𝑖 ) = −𝑦𝑖 .
Proof. Note that the coordinate ring of 𝐺 is a free module of rank 𝑝𝑚+𝑛 generated by
 𝑒     𝑒    𝑓       𝑓
𝑥11 ⋯ 𝑥𝑛𝑛 𝑦11 ⋯ 𝑦𝑚𝑚 for 0 ≤ 𝑒𝑖 , 𝑓𝑖 < 𝑝. Therefore 𝐺 → 𝑆 is flat. The only thing to check is
that these operations give a group scheme. This is similar to the proof of Theorem 4.4.1 in
the next section but simpler. We will skip the direct calculations here.
4.4    Group Schemes Annihilated by 𝑉 3
Now we consider the next case. Let 𝐺/𝑆 be a finite group scheme that is killed by 𝑝 and
such that 𝑉𝐺3 = 0. We will also assume that all kernels and cokernels of 𝑉𝐺 are flat group
                                                      45


schemes over the base 𝑆. In this case, we have exact sequences
                       0 → Im 𝑉𝐺 / Im 𝑉𝐺2 → 𝐺/ Im 𝑉𝐺2 → 𝐺/ Im 𝑉𝐺 → 0
and
                               0 → Im 𝑉𝐺2 → 𝐺 → 𝐺/ Im 𝑉𝐺2 → 0.
Both left terms in these two exact sequences are annihilated by 𝑉𝐺 to satisfy the condition
in Proposition 4.1.1. The idea to obtain the explicit expression of 𝐺 is to use the first exact
sequence to get a description of 𝐺/ Im 𝑉𝐺2 using the result in Section 5.1. Then we will use
the second exact sequence to construct the group scheme 𝐺.
    Since the notation and calculation get very complicated immediately, we will only state
the result in a specific case that we need in Section 6.2. However, note that the theorem can
be stated with more generality.
Theorem 4.4.1. Let 𝐺/𝑆 be a group scheme as in Section 4.1. Assume 𝑆 = Spec 𝑅 with
𝑅 a local ring. Assume the rank of 𝐺 is 𝑝4 and suppose that 𝐺 admits a filtration
                                 0 = 𝐺0 ⊂ 𝐺1 ⊂ 𝐺3 ⊂ 𝐺4 = 𝐺
such that
                                       Rank 𝐺𝑖 = 𝑝𝑖 ,                                             (4.8)
                                          𝑉 (𝐺𝑖 ) = 𝐺𝑣(𝑖) ,                                       (4.9)
                                       𝐹 −1 (𝐺𝑖 ) = 𝐺𝑓(𝑖) ,                                      (4.10)
                                    (𝐺𝑖 /𝐺𝑗 )𝐷 ≅ 𝐺4−𝑗 /𝐺4−𝑖 ,                                    (4.11)
where 𝑣 = (0, 0, 1, 2) and 𝑓 = (2, 3, 4, 4). Then there exists a 10-tuple
                                 (𝑎1 , 𝑎2 , 𝑏1 , 𝑏2 , 𝑐, 𝑑, 𝑒1 , 𝑒2 , 𝑓1 , 𝑓2 )
over 𝑅 such that
       𝐺 = 𝐺4 ≅ Spec 𝑅[𝑥, 𝑦1 , 𝑦2 , 𝑧]/ (𝑥𝑝 , 𝑦1𝑝 − 𝑎1 𝑥, 𝑦2𝑝 − 𝑎2 𝑥, 𝑧 𝑝 − 𝑐𝑥 − 𝑒1 𝑦1 − 𝑒2 𝑦2 )
                                                      46


where coalgebra operations given by
   𝜖∗ (𝑥) = 0; inv∗ (𝑥) = −𝑥;          𝑚∗ (𝑥) = 1 ⊗ 𝑥 + 𝑥 ⊗ 1,
                                                                     𝑝−1
   ∗              ∗                      ∗                               𝑏𝑖 𝑥𝑘 ⊗ 𝑥𝑝−𝑘
  𝜖 (𝑦𝑖 ) = 0; inv (𝑦𝑖 ) = −𝑦𝑖 ;       𝑚 (𝑦𝑖 ) = 1 ⊗ 𝑦𝑖 + 𝑦𝑖 ⊗ 1 + ∑                   ,        (4.12)
                                                                     𝑘=1
                                                                          𝑘! (𝑝 − 𝑘)!
                                                                  𝑝−1
                                                                       𝑑𝑥 ⊗ 𝑥𝑝−𝑘
                                                                         𝑘
   𝜖∗ (𝑧) = 0; inv∗ (𝑧) = −𝑧;          𝑚∗ (𝑧) = 1 ⊗ 𝑧 + 𝑧 ⊗ 1 + ∑
                                                                  𝑘=1
                                                                       𝑘! (𝑝 − 𝑘)!
                           𝑝−1                         𝑝−1                         𝑝−1
                        2
                                 𝑓𝑖 𝑦𝑖𝑘 ⊗ 𝑦𝑖𝑝−𝑘     2
                                                           𝑓 𝑏 (1 ⊗ 𝑦𝑖 + 𝑦𝑖 ⊗ 1)       (𝑥𝑘 ⊗ 𝑥𝑝−𝑘 )
                    + ∑∑                        − ∑∑ 𝑖 𝑖                                            .
                       𝑖=1 𝑘=1
                                  𝑘! (𝑝 − 𝑘)!      𝑖=1 𝑘=1
                                                                          𝑘! (𝑝  − 𝑘)!
    �
Proof. We start by analyzing 𝐺3 /𝐺1 . Note that we have 𝐹 −1 (𝐺1 ) = 𝐺3 and 𝑉 (𝐺3 ) = 𝐺1 .
Therefore we have 𝑉𝐺3 /𝐺1 = 0 and 𝐹𝐺3 /𝐺1 = 0. By Proposition 4.2.1, we have 𝐺3 /𝐺1 ≅
𝛼𝑝 × 𝛼𝑝 = Spec 𝑅[𝑦1 , 𝑦2 ]/(𝑦1𝑝 , 𝑦2𝑝 ).
    Now consider the exact sequence
                            0 → 𝐺3 /𝐺1 → 𝐺4 /𝐺1 → 𝐺4 /𝐺3 → 0.
Similarly, we have 𝐹𝐺4 /𝐺3 = 0 and 𝑉𝐺4 /𝐺3 = 0. Thereofore we can write 𝐺4 /𝐺3 ≅ 𝛼𝑝 =
Spec 𝑅[𝑥]/(𝑥𝑝 ). By Theorem 4.3.1, we have
                     𝐺4 /𝐺1 = Spec 𝑅[𝑥, 𝑦1 , 𝑦2 ]/ (𝑥𝑝 , 𝑦1𝑝 − 𝑎1 𝑥, 𝑦2𝑝 − 𝑎2 𝑥)
with
                       𝑚∗𝐺4 /𝐺1 (𝑥) = 1 ⊗ 𝑥 + 𝑥 ⊗ 1
                                                             𝑝−1
                                                                 𝑏𝑖 𝑥𝑘 ⊗ 𝑥𝑝−𝑘
                      𝑚∗𝐺4 /𝐺1 (𝑦𝑖 )   = 1 ⊗ 𝑦𝑖 + 𝑦 𝑖 ⊗ 1 + ∑                   .
                                                             𝑘=1
                                                                  𝑘! (𝑝 − 𝑘)!
The Frobenius and Verschebung acts on 𝑀𝑆 (𝐺4 /𝐺1 ) by
                                                                   0 𝑎1     𝑎2
                                                                ⎛
                     𝐹 (𝑥 ⊗ 1, 𝑦1 ⊗ 1, 𝑦2 ⊗ 1) = (𝑥, 𝑦1 , 𝑦2 ) ⎜0 0         0⎞ ⎟,
                                                                ⎝0 0        0⎠
                                                                                                (4.13)
                                                                   0 𝑏1    𝑏2
                                                                ⎛
                     𝑉 (𝑥, 𝑦1 , 𝑦2 ) = (𝑥 ⊗ 1, 𝑦1 ⊗ 1, 𝑦2 ⊗ 1) ⎜0 0         0⎞⎟.
                                                                ⎝0 0        0⎠
                                                   47


    With the Hopf algebra of 𝐺4 /𝐺1 , now we consider the exact sequence
                                    0 → 𝐺1 → 𝐺4 → 𝐺4 /𝐺1 → 0.
Similarly, we have 𝐺1 ≅ 𝛼𝑝 = Spec 𝑅[𝑧]/(𝑧 𝑝 ). Consider the extension of Dieudonné modules
                          0 → 𝑀𝑆 (𝐺4 /𝐺1 ) → 𝑀𝑆 (𝐺4 ) → 𝑀𝑆 (𝐺1 ) → 0.
As an 𝑅 module, 𝑀𝑆 (𝐺) ≅ 𝑅4 . Using the matrices of the Dieudonné modules in Equations
(4.13), we can calculate that the Frobenius of 𝑀𝑆 (𝐺4 ) acts by
                                                                    0 𝑎1 𝑎2 𝑐
                                                                   ⎛
                                                                   ⎜0 0 0 𝑒1 ⎞ ⎟
               𝐹 (𝑥 ⊗ 1, 𝑦1 ⊗ 1, 𝑦2 ⊗ 1, 𝑧 ⊗ 1) = (𝑥, 𝑦1 , 𝑦2 , 𝑧) ⎜
                                                                   ⎜           ⎟,       (4.14)
                                                                   ⎜0 0 0 𝑒2 ⎟ ⎟
                                                                   ⎝0 0 0 0 ⎠
and the Verschiebung acts by
                                                                     0 𝑏1 𝑏2 𝑑
                                                                   ⎛
                                                                   ⎜0 0 0 𝑓 1 ⎞⎟
               𝑉 (𝑥, 𝑦1 , 𝑦2 , 𝑧) = (𝑥 ⊗ 1, 𝑦1 ⊗ 1, 𝑦2 ⊗ 1, 𝑧 ⊗ 1) ⎜
                                                                   ⎜           ⎟,       (4.15)
                                                                   ⎜0 0 0 𝑓 2 ⎟⎟
                                                                   ⎝0 0 0 0 ⎠
so that 𝐹 𝑉 = 𝑉 𝐹 = 0. Equivalently, we have the following equations:
                                            𝑎1 𝑓1 + 𝑎2 𝑓2 = 0                           (4.16)
                                            𝑏1 𝑒1 + 𝑏2 𝑒2 = 0                           (4.17)
By Proposition 4.1.1, it suffices to construct a group scheme with Frobenius and Verschiebung
in forms of (4.14) and (4.15). We construct this group scheme explicitly as stated in Theorem
4.4.1. The rest of this proof is devoted to checking that the scheme 𝐺 with operations above
defines a group scheme. We will check the following conditions:
   (I) The comultiplication, counit, coinverse defined in Equation (4.12) give well-defined
       algebra homomorphisms.
  (II) The Hopf algebra axioms are satisfied, i.e.
                                      (Id ⊗𝑚∗ ) ∘ 𝑚∗ = (𝑚∗ ⊗ Id) ∘ 𝑚∗ ,
                                       (Id ⊗𝜖∗ ) ∘ 𝑚∗ = Id,                             (4.18)
                                      (Id, inv∗ ) ∘ 𝑚∗ = 𝜖∗ .
                                                    48


    We first check condition (I). We will check that the comultiplication gives a well-defined
algebra homomorphism 𝑚∗ ∶ 𝒪𝐺 → 𝒪𝐺 ⊗ 𝒪𝐺 . The well-definedness of the counit and coin-
verse are easy to check.
    The well-definedness of the comultiplication amounts to checking that
                                𝑝
                       (𝑚∗ (𝑧)) − 𝑐𝑚∗ (𝑥) − 𝑒1 𝑚∗ (𝑦1 ) − 𝑒2 𝑚∗ (𝑦2 ) = 0.                    (4.19)
Note that
                                                       𝑝−1
                     ∗    𝑝          𝑝     𝑝
                                                    2
                                                           𝑓𝑖𝑝 (𝑎𝑖 𝑥)𝑘 ⊗ (𝑎𝑖 𝑥)𝑝−𝑘
                 (𝑚 (𝑧)) = 1 ⊗ 𝑧 + 𝑧 ⊗ 1 + ∑ ∑
                                                   𝑖=1 𝑘=1
                                                                  𝑘! (𝑝 − 𝑘)!
                                                   𝑝−1
                                                                                              (4.20)
                                                          2
                                                                       𝑥𝑘 ⊗ 𝑥𝑝−𝑘
                            = 1 ⊗ 𝑧 𝑝 + 𝑧 𝑝 ⊗ 1 + ∑ (∑ 𝑓𝑖𝑝 𝑎𝑝𝑖 )                   .
                                                   𝑘=1   𝑖=1
                                                                       𝑘! (𝑝 − 𝑘)!
Note that we have 𝑎1 𝑓1 + 𝑎2 𝑓2 = 0 from Equation (4.16). So we have 𝑎𝑝1 𝑓1𝑝 + 𝑎𝑝2 𝑓2𝑝 = 0 and
therefore
              𝑝
      (𝑚∗ (𝑧)) = 1 ⊗ 𝑧 𝑝 + 𝑧 𝑝 ⊗ 1
                                                                                              (4.21)
                = 𝑐 (1 ⊗ 𝑥 + 𝑥 ⊗ 1) + 𝑒1 (1 ⊗ 𝑦1 + 𝑦1 ⊗ 1) + 𝑒2 (1 ⊗ 𝑦2 + 𝑦2 ⊗ 1) .
    On the other hand, we have
    𝑐𝑚∗ (𝑥)+𝑒1 𝑚∗ (𝑦1 ) + 𝑒2 𝑚∗ (𝑦2 ) = 𝑐 (1 ⊗ 𝑥 + 𝑥 ⊗ 1)                                     (4.22)
                                                                   𝑝−1     2
                                                                                     𝑥𝑘 ⊗ 𝑥𝑝−𝑘
              + 𝑒1 (1 ⊗ 𝑦1 + 𝑦1 ⊗ 1) + 𝑒2 (1 ⊗ 𝑦2 + 𝑦2 ⊗ 1) +      ∑ (∑ 𝑒𝑝𝑖 𝑏𝑖𝑝 )                .
                                                                   𝑘=1    𝑖=1
                                                                                     𝑘! (𝑝 − 𝑘)!
Similarly, note that 𝑒1 𝑏1 + 𝑒2 𝑏2 = 0 from Equation (4.16) and therefore 𝑒𝑝1 𝑏1𝑝 + 𝑒𝑝2 𝑏2𝑝 = 0.
Hence we have
                𝑐𝑚∗ (𝑥) + 𝑒1 𝑚∗ (𝑦1 + 𝑒2 𝑚∗ (𝑦2 )
                                                                                              (4.23)
              =𝑐 (1 ⊗ 𝑥 + 𝑥 ⊗ 1) + 𝑒1 (1 ⊗ 𝑦1 + 𝑦1 ⊗ 1) + 𝑒2 (1 ⊗ 𝑦2 + 𝑦2 ⊗ 1) .
    By combining Equation (4.21) and (4.23), Equation (4.19) holds. It follows that the
algebra homomorphism 𝑚∗ ∶ 𝒪𝐺 → 𝒪𝐺 ⊗ 𝒪𝐺 is well-defined.
    Now we check condition (II). We will check the equation
                            (Id ⊗𝑚∗ ) ∘ 𝑚∗ (𝑧) = (𝑚∗ ⊗ Id) ∘ 𝑚∗ (𝑧).                          (4.24)
                                                49


The calculation for the other two equations of (4.18) is straightforward and therefore omitted.
     We first fix some notations. Let 𝑅′ be an 𝑅-algebra and let 𝑥, 𝑦, 𝑧 be points of 𝐺(𝑅′ ). A
                                                (1)  (2)
point 𝑥 ∈ 𝐺(𝑅′ ) is given by 𝑥 = (𝑥0 , 𝑥1 , 𝑥1 , 𝑥2 ), a quadruple of elements in 𝑅′ satisfying
              (𝑖)                                       (1)         (2)
𝑥𝑝0 = 0, (𝑥1 )𝑝 = 𝑎𝑖 𝑥0 and 𝑥𝑝2 = 𝑐𝑥0 + 𝑒1 𝑥1 + 𝑒2 𝑥1 . Similar conditions hold for 𝑦 =
        (1)  (2)                     (1)   (2)
(𝑦0 , 𝑦1 , 𝑦1 , 𝑦2 ) and 𝑧 = (𝑧0 , 𝑧1 , 𝑧1 , 𝑧2 ). For any 𝑎, 𝑏, 𝑐 ∈ 𝑅′ , we define 𝑃 (𝑎, 𝑏) ∈ 𝑅′ as
                                                      𝑝−1
                                                               𝑎𝑘 𝑏𝑝−𝑘
                                         𝑃 (𝑎, 𝑏) ≔ ∑                                                (4.25)
                                                      𝑘=1
                                                            𝑘! (𝑝 − 𝑘)!
and define
                                                                    𝑎𝑘 𝑏𝑙 𝑐𝑚
                                    𝑃 (𝑎, 𝑏, 𝑐) ≔        ∑                    .                      (4.26)
                                                    0≤𝑘,𝑙,𝑚≤𝑝−1
                                                                    𝑘! 𝑙! 𝑚!
                                                      𝑘+𝑙+𝑚=𝑝
One basic property of this expression is
                     𝑃 (𝑎, 𝑏, 𝑐) = 𝑃 (𝑎, 𝑏) + 𝑃 (𝑎 + 𝑏, 𝑐) = 𝑃 (𝑎, 𝑏 + 𝑐) + 𝑃 (𝑏, 𝑐).                (4.27)
     Now we define 𝑠(𝑥, 𝑦) to be the sum of 𝑥 and 𝑦 under the group operation defined by
the coalgebra operators in Equation (4.12). More explicitly, using the notation 𝑃 (𝑎, 𝑏), we
                          (1)   (2)
define 𝑠(𝑥, 𝑦) ≔ (𝑠0 , 𝑠1 , 𝑠1 , 𝑠2 ), where
     𝑠0 (𝑥, 𝑦) ≔ 𝑥0 + 𝑦0
     (𝑖)           (𝑖)    (𝑖)
   𝑠1 (𝑥, 𝑦) ≔ 𝑥1 + 𝑦1 + 𝑏𝑖 𝑃 (𝑥0 , 𝑦0 )
                                                 2                         2
                                                            (𝑖) (𝑖)               (𝑖)  (𝑖)
     𝑠2 (𝑥, 𝑦) ≔ 𝑥2 + 𝑦2 + 𝑑𝑃 (𝑥0 , 𝑦0 ) +     ∑ 𝑓𝑖 𝑃 (𝑥1 , 𝑦1 )      − ∑ 𝑓𝑖 𝑏𝑖 (𝑥1 + 𝑦1 )𝑝−1 𝑃 (𝑥0 , 𝑦0 )
                                               𝑖=1                        𝑖=1
     By Yoneda’s Lemma, Equation (4.24) is equivalent to the associativity of 𝑠, i.e.
                                       𝑠(𝑥, 𝑠(𝑦, 𝑧)) = 𝑠(𝑠(𝑥, 𝑦), 𝑧)
                                                                                           (𝑖)
for any 𝑥, 𝑦, 𝑧 ∈ 𝐺(𝑅′ ). Note that 𝑠0 (𝑥, 𝑠(𝑦, 𝑧)) = 𝑠0 (𝑠(𝑥, 𝑦), 𝑧) is trivial and 𝑠1 (𝑥, 𝑠(𝑦, 𝑧)) =
 (𝑖)
𝑠1 (𝑠(𝑥, 𝑦), 𝑧) follows from Equation (4.27). The difficult part is checking 𝑠2 (𝑥, 𝑠(𝑦, 𝑧)) =
𝑠2 (𝑠(𝑥, 𝑦), 𝑧).
                                                       50


                                                                         (1)     (2)
   We start with the left side. Denote 𝑤 = (𝑤0 , 𝑤1 , 𝑤1 , 𝑤2 ) = 𝑠(𝑥, 𝑦). Therefore
      𝑠2 (𝑠(𝑥, 𝑦), 𝑧)
                                       2                                  2
                                                      (𝑖) (𝑖)                            (𝑖)        (𝑖)
   =𝑤2 + 𝑧2 + 𝑑𝑃 (𝑤0 , 𝑧0 ) + ∑ 𝑓𝑖 𝑃 (𝑤1 , 𝑧1 ) − ∑ 𝑓𝑖 𝑏𝑖 (𝑤1 + 𝑧1 )𝑝−1 𝑃 (𝑤0 , 𝑧0 )
                                     𝑖=1                                𝑖=1
                                      2                                 2
                                                    (𝑖) (𝑖)                            (𝑖)        (𝑖)
   =𝑥2 + 𝑦2 + 𝑑𝑃 (𝑥0 , 𝑦0 ) + ∑ 𝑓𝑖 𝑃 (𝑥1 , 𝑦1 ) − ∑ 𝑓𝑖 𝑏𝑖 (𝑥1 + 𝑦1 )𝑝−1 𝑃 (𝑥0 , 𝑦0 )                                   (4.28)
                                    𝑖=1                                𝑖=1
                                           2
                                                         (𝑖)        (𝑖)                            (𝑖)
      + 𝑧2 + 𝑑𝑃 (𝑥0 + 𝑦0 , 𝑧0 ) + ∑ 𝑓𝑖 𝑃 (𝑥1 + 𝑦1 + 𝑏𝑖 𝑃 (𝑥0 , 𝑦0 ), 𝑧1 )
                                         𝑖=1
            2
                     (𝑖)    (𝑖)                               (𝑖)
      − ∑ 𝑓𝑖 𝑏𝑖 (𝑥1 + 𝑦1 + 𝑏𝑖 𝑃 (𝑥0 , 𝑦0 ) + 𝑧1 )𝑝−1 𝑃 (𝑥0 + 𝑦0 , 𝑧0 )
          𝑖=1
Note that 𝑥𝑝0 = 𝑦0𝑝 = 0. By definition of 𝑃 in (4.25), we have (𝑃 (𝑥0 , 𝑦0 ))2 = 0. Therefore
                                                 𝑝−1        (𝑖)        (𝑖)                              (𝑖)
        (𝑖)    (𝑖)                     (𝑖)              (𝑥1 + 𝑦1 + 𝑏𝑖 𝑃 (𝑥0 , 𝑦0 ))𝑘 (𝑧1 )𝑝−𝑘
   𝑃 (𝑥1    + 𝑦1   + 𝑏𝑖 𝑃 (𝑥0 , 𝑦0 ),𝑧1 )    =∑
                                                 𝑘=1
                                                                              𝑘! (𝑝 − 𝑘)!
                                     𝑝−1          (𝑖)        (𝑖)                  (𝑖)        (𝑖)                  (𝑖)
                                             ((𝑥1 +        𝑦1 )𝑘 + 𝑏𝑖 𝑘(𝑥1 + 𝑦1 )𝑘−1 𝑃 (𝑥0 , 𝑦0 )) (𝑧1 )𝑝−𝑘
                                  =∑
                                     𝑘=1
                                                                                  𝑘! (𝑝 − 𝑘)!
                                                                        𝑝−1           (𝑖)        (𝑖)         (𝑖)
                                            (𝑖)       (𝑖) (𝑖)                  𝑏 (𝑥 + 𝑦1 )𝑘−1 (𝑧1 )𝑝−𝑘 𝑃 (𝑥0 , 𝑦0 )
                                  =𝑃 (𝑥1        +   𝑦1 , 𝑧 1 )      +∑ 𝑖 1
                                                                        𝑘=1
                                                                                               (𝑘 − 1)! (𝑝 − 𝑘)!
Plugging this into (4.28), we get
    𝑠2 (𝑠(𝑥, 𝑦), 𝑧) =
                                                 2                                  2
                                                                 (𝑖) (𝑖)                           (𝑖)      (𝑖)
    𝑥2 + 𝑦2 + 𝑧2 + 𝑑𝑃 (𝑥0 , 𝑦0 , 𝑧0 ) + ∑ 𝑓𝑖 𝑃 (𝑥1 , 𝑦1 ) − ∑ 𝑓𝑖 𝑏𝑖 (𝑥1 + 𝑦1 )𝑝−1 𝑃 (𝑥0 , 𝑦0 )
                                                𝑖=1                               𝑖=1
          2                                    𝑝−1          (𝑖)        (𝑖)           (𝑖)
                       (𝑖)     (𝑖) (𝑖)               𝑏𝑖 (𝑥1 + 𝑦1 )𝑘−1 (𝑧1 )𝑝−𝑘 𝑃 (𝑥0 , 𝑦0 )
    + ∑ 𝑓𝑖 (𝑃 (𝑥1 + 𝑦1 , 𝑧1 ) + ∑                                                                               )
        𝑖=1                                   𝑘=1
                                                                     (𝑘 − 1)! (𝑝 − 𝑘)!
          2
                      (𝑖)    (𝑖)      (𝑖)                                    (𝑖)       (𝑖)        (𝑖)
    − ∑ 𝑓𝑖 𝑏𝑖 ((𝑥1 + 𝑦1 + 𝑧1 )𝑝−1 − 𝑏𝑖 𝑃 (𝑥0 , 𝑦0 )(𝑥1 + 𝑦1 + 𝑧1 )𝑝−2 )
        𝑖=1
                                                                                  ⋅ (𝑃 (𝑥0 , 𝑦0 , 𝑧0 ) − 𝑃 (𝑥0 , 𝑦0 ))
                                                              51


After rearranging and simplifying, we get
   𝑠2 (𝑠(𝑥, 𝑦), 𝑧) =
                                                  2
                                                              (𝑖)  (𝑖)     (𝑖)
   𝑥2 + 𝑦2 + 𝑧2 + 𝑑𝑃 (𝑥0 , 𝑦0 , 𝑧0 ) + ∑ 𝑓𝑖 𝑃 (𝑥1 , 𝑦1 , 𝑧1 )                                                            (4.29)
                                                𝑖=1
         2                                              2   𝑝−1             (𝑖)       (𝑖)           (𝑖)
                     (𝑖)      (𝑖)                                 𝑓𝑖 𝑏𝑖 (𝑥1 + 𝑦1 )𝑘−1 (𝑧1 )𝑝−𝑘 𝑃 (𝑥0 , 𝑦0 )
    −  ∑ 𝑓𝑖 𝑏𝑖 (𝑥1        +  𝑦1 )𝑝−1 𝑃 (𝑥0 , 𝑦0 )   + ∑∑
        𝑖=1                                            𝑖=1 𝑘=1
                                                                                   (𝑘 − 1)! (𝑝 − 𝑘)!
         2                                                               2
                       (𝑖)      (𝑖)    (𝑖) 𝑝−1                                          (𝑖)       (𝑖)    (𝑖) 𝑝−1
    − ∑ 𝑓𝑖 𝑏𝑖 (𝑥1 + 𝑦1 + 𝑧1 )                    𝑃 (𝑥0 , 𝑦0 , 𝑧0 ) + ∑ 𝑓𝑖 𝑏𝑖 (𝑥1 + 𝑦1 + 𝑧1 )                     𝑃 (𝑥0 , 𝑦0 )
        𝑖=1                                                            𝑖=1
         2
                      (𝑖)      (𝑖)    (𝑖)
    + ∑ 𝑓𝑖 𝑏𝑖2 (𝑥1 + 𝑦1 + 𝑧1 )𝑝−2 𝑃 (𝑥0 , 𝑦0 , 𝑧0 )𝑃 (𝑥0 , 𝑦0 )
        𝑖=1
Now we calculate the term 𝑃 (𝑥0 , 𝑦0 , 𝑧0 )𝑃 (𝑥0 , 𝑦0 ). Note that 𝑥𝑝0 = 𝑦0𝑝 = 𝑧0𝑝 = 0. Therefore
we have
                                                          𝑘     𝑙  𝑚                            𝑘     𝑙
                                                         𝑥01 𝑦01 𝑧0 1                         𝑥02 𝑦02
   𝑃 (𝑥0 , 𝑦0 , 𝑧0 )𝑃 (𝑥0 ,𝑦0 ) =            ∑                           ⋅       ∑
                                           ,𝑙 ,𝑚 ≤𝑝−1 1 1
                                     0≤𝑘1 1 1
                                                         𝑘 ! 𝑙 ! 𝑚1 ! 1≤𝑘 ,𝑙 ≤𝑝−1 𝑘2 ! 𝑙2 !
                                                                                2   2
                                       𝑘1 +𝑙1 +𝑚1 =𝑝                           𝑘2 +𝑙2 =𝑝
                                                𝑝−𝑚
                                                                                    1
                           =        ∑        (∑                                                                ) 𝑥𝑘0 𝑦0𝑙 𝑧0𝑚 .
                              2≤𝑘,𝑙,𝑚≤𝑝−1       𝑙1 =0
                                                      (𝑝 − 𝑙1 − 𝑚)! 𝑙1 ! 𝑚! (𝑝 − 𝑙 + 𝑙1 )! (𝑙 − 𝑙1 )!
                                𝑘+𝑙+𝑚=2𝑝
Here 𝑚1 = 𝑚, 𝑙2 = 𝑙 − 𝑙1 , 𝑘2 = 𝑝 − 𝑙2 = 𝑝 − 𝑙 + 𝑙1 , 𝑘1 = 𝑝 − 𝑙1 − 𝑚. Note that all 𝑘, 𝑙, 𝑚 are
at least 2. Otherwise at least one of the other 2 variables will be greater than or equal to 𝑝.
The bounds for 𝑙1 is from 0 to 𝑝 − 𝑚 since 𝑙 − 𝑝 − 𝑚 > 0 from 𝑘 ≤ 𝑝 − 1 and 𝑘 + 𝑙 + 𝑚 = 2𝑝.
So 𝑙2 = 𝑙 − 𝑙1 > 0, as we want. Now we want to calculate the constant in the parenthesis,
which is given by the following lemma:
Lemma 4.4.2. For any 2 ≤ 𝑘, 𝑙, 𝑚 ≤ 𝑝 − 1 such that 𝑘 + 𝑙 + 𝑚 = 2𝑝, we have
                   𝑝−𝑚
                                                    1                                         1
                   ∑                                                                 ≡                  mod 𝑝.
                   𝑙1 =0
                           (𝑝 − 𝑙1 − 𝑚)! 𝑙1 ! 𝑚! (𝑝 − 𝑙 + 𝑙1 )! (𝑙 − 𝑙1 )!                𝑘! 𝑙! 𝑚!
Proof. Note that
                                                                      𝑝−𝑚
                                                                                 𝑙        2𝑝 − 𝑙 − 𝑚             2𝑝 − 𝑚
  𝑝−𝑚                                                                  ∑ ( )(                            )    (           )
                                    1                                 𝑙1 =0     𝑙1        𝑝 − 𝑚 − 𝑙1              𝑝−𝑚
   ∑                                                              =                                         =                .
   𝑙1 =0
         (𝑝 − 𝑙1 − 𝑚)! 𝑙1 ! 𝑚! (𝑝 − 𝑙 + 𝑙1 )! (𝑙 − 𝑙1 )!                   𝑚! 𝑙! (2𝑝 − 𝑙 − 𝑚)!                   𝑘! 𝑙! 𝑚!
                                                               52


Now observe that modulo 𝑝, we have
     2𝑝 − 𝑚            (𝑝 + 𝑝 − 𝑚) ⋅ (𝑝 + 𝑝 − 𝑚 − 1) ⋯ (𝑝 + 1)                      (𝑝 − 𝑚)(𝑝 − 𝑚 − 1) ⋯ 1
   (          )=                                                                ≡                          = 1.
       𝑝−𝑚                               1 ⋅ 2 ⋯ (𝑝 − 𝑚)                                  1 ⋅ 2 ⋯ (𝑝 − 𝑚)
This finishes the proof of the lemma.
    From Lemma 4.4.2, we get
                                                                                   1
                           𝑃 (𝑥0 , 𝑦0 , 𝑧0 )𝑃 (𝑥0 , 𝑦0 ) =          ∑                    𝑥𝑘0 𝑦0𝑙 𝑧0𝑚 .      (4.30)
                                                               2≤𝑘,𝑙,𝑚≤𝑝−1
                                                                               𝑘! 𝑙! 𝑚!
                                                                 𝑘+𝑙+𝑚=2𝑝
    Plug it into Equation (4.29), we get
   𝑠2 (𝑠(𝑥, 𝑦), 𝑧) =
                                                 2
                                                              (𝑖)  (𝑖)  (𝑖)
   𝑥2 + 𝑦2 + 𝑧2 + 𝑑𝑃 (𝑥0 , 𝑦0 , 𝑧0 ) + ∑ 𝑓𝑖 𝑃 (𝑥1 , 𝑦1 , 𝑧1 )
                                                𝑖=1
        2
                     (𝑖)       (𝑖)     (𝑖) 𝑝−1
   − ∑ 𝑓𝑖 𝑏𝑖 (𝑥1 + 𝑦1 + 𝑧1 )                    𝑃 (𝑥0 , 𝑦0 , 𝑧0 )
       𝑖=1
        2
                               𝑓𝑖 𝑏𝑖2     (𝑖)     (𝑖)      (𝑖)                                              (4.31)
   +∑           ∑                     (𝑥1 + 𝑦1 + 𝑧1 )𝑝−2 𝑥𝑘0 𝑦0𝑙 𝑧0𝑚
       𝑖=1 2≤𝑘,𝑙,𝑚≤𝑝−1
                            𝑘! 𝑙! 𝑚!
            𝑘+𝑙+𝑚=2𝑝
        2                                               2    𝑝−1         (𝑖)       (𝑖)          (𝑖)
                   (𝑖)       (𝑖)                                  𝑓 𝑏 (𝑥 + 𝑦1 )𝑘−1 (𝑧1 )𝑝−𝑘 𝑃 (𝑥0 , 𝑦0 )
   −   ∑ 𝑓𝑖 𝑏𝑖 (𝑥1      +  𝑦1 )𝑝−1 𝑃 (𝑥0 , 𝑦0 )     + ∑∑ 𝑖 𝑖 1
       𝑖=1                                             𝑖=1 𝑘=1
                                                                                (𝑘 − 1)! (𝑝 − 𝑘)!
        2
                     (𝑖)       (𝑖)     (𝑖) 𝑝−1
   + ∑ 𝑓𝑖 𝑏𝑖 (𝑥1 + 𝑦1 + 𝑧1 )                    𝑃 (𝑥0 , 𝑦0 )
       𝑖=1
    Now we observe that the last three terms cancel. To see this, we just need to expand the
last term using the binomial expansion:
        2
                    (𝑖)       (𝑖)     (𝑖) 𝑝−1
      ∑ 𝑓𝑖 𝑏𝑖 (𝑥1 + 𝑦1 + 𝑧1 )                   𝑃 (𝑥0 , 𝑦0 )
       𝑖=1
        2                                                 2    𝑝−2         (𝑖)       (𝑖)      (𝑖)
                    (𝑖)       (𝑖) 𝑝−1                              𝑓 𝑏 (𝑥 + 𝑦1 )𝑘 (𝑧1 )𝑝−𝑘 𝑃 (𝑥0 , 𝑦0 )
   = ∑ 𝑓𝑖 𝑏𝑖 (𝑥1         + 𝑦1 )         𝑃 (𝑥0 , 𝑦0 )  + ∑∑ 𝑖 𝑖 1                                            (4.32)
       𝑖=1                                               𝑖=1 𝑘=0
                                                                         (𝑝 − 1)! (𝑘 − 1)! (𝑝 − 𝑘)!
        2                                               2   𝑝−1         (𝑖)       (𝑖)          (𝑖)
                  (𝑖)       (𝑖)                                   𝑓 𝑏 (𝑥 + 𝑦1 )𝑘−1 (𝑧1 )𝑝−𝑘 𝑃 (𝑥0 , 𝑦0 )
   = ∑ 𝑓𝑖 𝑏𝑖 (𝑥1       +  𝑦1 )𝑝−1 𝑃 (𝑥0 , 𝑦0 )     − ∑∑ 𝑖 𝑖 1
       𝑖=1                                             𝑖=1 𝑘=1
                                                                               (𝑘 − 1)! (𝑝 − 𝑘)!
                                                               53


Here, the last step uses (𝑝 − 1)! ≡ −1 mod 𝑝. After canceling the last three terms, we get
                                                              2
                                                                          (𝑖)  (𝑖) (𝑖)
        𝑠2 (𝑠(𝑥, 𝑦), 𝑧) =𝑥2 + 𝑦2 + 𝑧2 + 𝑑𝑃 (𝑥0 , 𝑦0 , 𝑧0 ) + ∑ 𝑓𝑖 𝑃 (𝑥1 , 𝑦1 , 𝑧1 )
                                                             𝑖=1
                             2
                                       (𝑖)   (𝑖)     (𝑖) 𝑝−1
                         − ∑ 𝑓𝑖 𝑏𝑖 (𝑥1 + 𝑦1 + 𝑧1 )           𝑃 (𝑥0 , 𝑦0 , 𝑧0 )         (4.33)
                            𝑖=1
                             2
                                             𝑓𝑖 𝑏𝑖2     (𝑖)    (𝑖)    (𝑖)
                         +∑         ∑                (𝑥1 + 𝑦1 + 𝑧1 )𝑝−2 𝑥𝑘0 𝑦0𝑙 𝑧0𝑚
                            𝑖=1 2≤𝑘,𝑙,𝑚≤𝑝−1
                                            𝑘! 𝑙! 𝑚!
                                 𝑘+𝑙+𝑚=2𝑝
By the symmetry of this expression, we can conclude that 𝑠2 (𝑠(𝑥, 𝑦), 𝑧) = 𝑠2 (𝑥, 𝑠(𝑦, 𝑧)).
This completes the proof that the 𝐺 we constructed in Theorem 4.4.1 is a group scheme.
                                               54


                                         CHAPTER 5
                  Γ1 (𝑝) -COVER OVER THE SIEGEL THREEFOLD
     Let 𝒜 ≔ 𝒜2,1,𝑁 , 𝑁 ≥ 3 be the Siegel threefold, which is the fine moduli scheme of
principally polarized abelian surfaces with symplectic level-𝑁 structure as in Section 2.3. For
the existence of 𝒜, see [28]. In this chapter, we consider the special fiber 𝒜 ̄ ≔ 𝒜 × Spec 𝔽𝑝 .
As in Example 2.2.1, there are 4 Ekedahl–Oort strata of 𝒜,̄ corresponding to the superspecial
locus, supersingular (but not superspecial) locus, 𝑝-rank-1 locus and ordinary locus. The loci
have dimensions 0,1,2,3 respectively. On each stratum, there is a canonical group scheme
filtration of the 𝑝-torsion of the universal abelian surface as in Example 2.2.1.
     Let 𝒳 be the universal abelian surface over 𝒜 and 𝒳̄ ≔ 𝒳 ×𝒜 𝒜.̄ Let 𝒳[𝑝]      ̄  be the 𝑝-
torsion group scheme of 𝒳̄ and let 𝒳̄ × [𝑝] ≔ (𝒳[𝑝])
                                                  ̄   ×
                                                        be its subscheme of primitive elements.
We will call the morphism 𝒳̄ × [𝑝] → 𝒜 ̄ “the Γ1 (𝑝)-cover” of 𝒜.̄ (Note that the name “Γ1 (𝑝)-
cover” has different meaning in various papers. For example, Haines and Rapoport use
“Γ1 (𝑝)-cover” for the pro-𝑝 Iwahori structure in [16].)
     Let 𝑆𝜑 be an Ekedahl–Oort stratum of 𝒜.̄ Let 𝒳̄ 𝜑 be the restriction of 𝒳̄ over 𝑆𝜑 and
let 𝒳̄ ×
       𝜑 [𝑝] → 𝑆𝜑 be the restriction of the Γ1 (𝑝)-cover. We want to study the geometry of
the Γ1 (𝑝)-cover 𝒳/𝑆̄
                       𝜑 on each Ekedahl–Oort stratum by calculating the (local) eqautions.
The main tool in this chapter is the machinery of constructing group schemes from their
Dieudonné modules built in Chapter 4 and Lemma 2.3.3 (particularly Example 2.3.6) for
calculating primitive elements.
5.1 Γ1 (𝑝) -cover over the Superspecial Locus
The superspecial locus consists of discrete points. Over each point Spec 𝔽̄𝑝 , the universal
abelian surface 𝒳̄ 𝜑 is a product of supersingular elliptic curves. Let 𝐸 be a supersingular
elliptic curve over Spec 𝔽̄𝑝 . Then 𝒳̄ 𝜑 ≅ 𝐸 × 𝐸 by a result due to Deligne ([40, Theorem
3.5]). Therefore 𝒳̄ 𝜑 [𝑝] ≅ 𝐸[𝑝] × 𝐸[𝑝].
                                               55


     Note that 𝐸[𝑝] is a self-dual group scheme of rank 𝑝2 , killed by 𝑝, of local-local type and
has nonzero Frobenius and Verschiebung. By classical Dieudonné theory over perfect fields,
there is a unique group scheme with these properties, given by
                                              Spec 𝔽̄𝑝 [𝑥]/(𝑥𝑝 )
                                                                  2
with coalgebra operations
                                                                𝑝−1
                               ∗                                     𝑥𝑝𝑘 ⊗ 𝑥𝑝(𝑝−𝑘)
                             𝑚 (𝑥) = 1 ⊗ 𝑥 + 𝑥 ⊗ 1 + ∑                              ,
                                                                𝑘=1
                                                                        𝑘! (𝑝 − 𝑘)!
                              𝜖∗ (𝑥) = 0,
                           inv∗ (𝑥) = −𝑥.
     Therefore, we have 𝒳̄ 𝜑 [𝑝] ≅ 𝐸[𝑝] × 𝐸[𝑝] ≅ 𝔽̄𝑝 [𝑥, 𝑦]/(𝑥𝑝 , 𝑦𝑝 ) with coalgebra operations
                                                                            2   2
as above. The augmentation ideal is given by (𝑥, 𝑦). By Lemma 2.3.3, we can see that
𝒳̄ ×        ̄
   𝜑 [𝑝] ⊂ 𝒳𝜑 [𝑝] is defined by ideal (𝑥
                                              𝑝2 −1 𝑝2 −1
                                                   𝑦       ). To conclude, we have the following result:
for the Γ1 (𝑝)-cover 𝒳̄ ×𝜑 [𝑝]/𝑆𝜑 over the superspecial locus:
Theorem 5.1.1. Let 𝑆𝜑 be the superspecial locus of the Sigel threefold 𝒜.̄ Over each point
of 𝑆𝜑 , the Γ1 (𝑝)-cover 𝒳̄ ×𝜑 [𝑝]/𝑆𝜑 is given by
                                  Spec 𝔽̄𝑝 [𝑥, 𝑦]/(𝑥𝑝 , 𝑦𝑝 , 𝑥𝑝 −1 𝑦𝑝 −1 )
                                                        2     2     2       2
over Spec 𝔽̄𝑝 . In particular, the scheme 𝒳̄ ×      𝜑 is Cohen–Macaulay, but not Gorenstein.
Proof. It only remains to prove the last statement. Note that the expression of 𝒪𝒳̄ ×𝜑 [𝑝] in
Theorem 5.1.1 is an Artin ring. Therefore 𝒳̄ ×            𝜑 [𝑝] is automatically Cohen–Macaulay. Also
it is easy to see that the socle of 𝔽̄𝑝 [𝑥, 𝑦]/(𝑥𝑝 , 𝑦𝑝 , 𝑥𝑝 −1 𝑦𝑝 −1 ) has dimension 1 as an 𝔽̄𝑝 -
                                                          2     2     2       2
                                             , while 𝒪𝒳̄ ×𝜑 [𝑝] has dimension 0. Therefore 𝒳̄ ×
                                  2
                                    −1 𝑝2 −1
vector space, spanned by 𝑥𝑝           𝑦                                                       𝜑 [𝑝] is not
Gorenstein.
                                                       56


5.2 Γ1 (𝑝) -cover over the Supersingular Locus
In this section, let 𝑆𝜑 be the supersingular locus of the Siegel threefold and 𝒳̄ 𝜑 be the
restriction of the universal abelian surface on 𝑆𝜑 . Let 𝐺 ≔ 𝒳̄ 𝜑 [𝑝] be the 𝑝-torsion of 𝒳̄ 𝜑 .
Recall that from Example 2.2.1, the canonical filtration of 𝐺 has the form
                             0 = 𝐺0 ⊂ 𝐺1 ⊂ 𝐺2 ⊂ 𝐺3 ⊂ 𝐺4 = 𝐺.
The corresponding canonical type is 𝜌 = (0, 1, 2, 3, 4), 𝑣 = (0, 0, 1, 1, 2) and 𝑓 = (2, 3, 3, 4, 4).
Recall that this means the following:
                                     Rank 𝐺𝑖 = 𝑝𝜌(𝑖) ,                                                 (5.1)
                                         𝑉 (𝐺𝑖 ) = 𝐺𝑣(𝑖) ,                                             (5.2)
                                     𝐹 −1 (𝐺𝑖 ) = 𝐺𝑓(𝑖) ,                                              (5.3)
                                              𝐺⟂ 𝑖 = 𝐺4−𝑖 ,                                            (5.4)
                                   (𝐺𝑖 /𝐺𝑗 )𝐷 ≅ 𝐺4−𝑗 /𝐺4−𝑖 .                                           (5.5)
    We want to give an explicit description of 𝐺 = 𝐺4 . This is precisely the situation in
Theorem 4.4.1. From this, we have the following result:
Theorem 5.2.1. Let 𝒳̄ 𝜑 /𝑆𝜑 be the universal abelian surface over the supersingular locus of
the Siegel threefold. Let 𝑥 ∈ 𝑆𝜑 be a point and let 𝑅 ≔ 𝒪𝑥,𝑆𝜑 be the local ring at 𝑥 ∈ 𝑆𝜑 .
Let 𝑆 = Spec 𝑅 and let 𝒳̄ 𝑆 be the pullback of 𝒳̄ 𝜑 to 𝑆. Consider its 𝑝-torsion 𝐺 ≔ 𝒳̄ 𝑆 [𝑝].
Then there exists a 10-tuple (𝑎1 , 𝑎2 , 𝑏1 , 𝑏2 , 𝑐, 𝑑, 𝑒1 , 𝑒2 , 𝑓1 , 𝑓2 ) with entries in 𝑅, so that
             𝐺 = 𝐺4 ≅ Spec 𝑅[𝑥, 𝑦1 , 𝑦2 , 𝑧]/ (𝑥𝑝 , 𝑦𝑖𝑝 − 𝑎𝑖 𝑥, 𝑧 𝑝 − 𝑐𝑥 − 𝑒1 𝑦1 − 𝑒2 𝑦2 )
                                                     57


where coalgebra operations given by
                 𝑚∗ (𝑥) ≔1 ⊗ 𝑥 + 𝑥 ⊗ 1,
                                                𝑝−1
                  ∗                                  𝑏𝑖 𝑥𝑘 ⊗ 𝑥𝑝−𝑘
                 𝑚 (𝑦𝑖 ) ≔1 ⊗ 𝑦𝑖 + 𝑦𝑖 ⊗ 1 + ∑                      ,
                                                𝑘=1
                                                      𝑘! (𝑝 − 𝑘)!
                                              𝑝−1                        𝑝−1
                                                   𝑑𝑥𝑘 ⊗ 𝑥𝑝−𝑘         2
                                                                             𝑓 𝑦𝑘 ⊗ 𝑦𝑖𝑝−𝑘
                   ∗
                 𝑚 (𝑧) ≔1 ⊗ 𝑧 + 𝑧 ⊗ 1 + ∑                        + ∑∑ 𝑖 𝑖
                                              𝑘=1
                                                    𝑘! (𝑝 − 𝑘)!      𝑖=1 𝑘=1
                                                                              𝑘! (𝑝 − 𝑘)!
                               2  𝑝−1                           𝑝−1
                                      𝑓𝑖 𝑏𝑖 (1 ⊗ 𝑦𝑖 + 𝑦𝑖 ⊗ 1)        (𝑥𝑘 ⊗ 𝑥𝑝−𝑘 )
                           − ∑∑                                                    .
                              𝑖=1 𝑘=1
                                                        𝑘! (𝑝 − 𝑘)!
The primitive elements of 𝐺 is given by
      𝐺× ≅ Spec 𝑅[𝑥, 𝑦1 , 𝑦2 , 𝑧]/ (𝑥𝑝 , 𝑦𝑖𝑝 − 𝑎𝑖 𝑥, 𝑧 𝑝 − 𝑐𝑥 − 𝑒1 𝑦1 − 𝑒2 𝑦2 , 𝑥𝑝−1 𝑦1𝑝−1 𝑦2𝑝−1 𝑧 𝑝−1 )
Proof. This is an immediate corollary of Theorem 4.4.1 and Example 2.3.6.
    In Theorem 5.2.1, the only property we used about 𝒳̄ 𝑆 [𝑝]/𝑆 is that it allows a canoni-
cal filtration that satisfies properties (5.2)-(5.5). It does not use the properties of the base
scheme, i.e. the supersingular locus of the Siegel threefold. To make a more precise descrip-
tion of the group scheme 𝒳̄ 𝑆 [𝑝]/𝑆, we can use a construction of the supersingular locus by
Moret-Bailly [25] and Oort [32]. This construction is also studied in [18].
    Let 𝐸 be a supersingular elliptic curve over 𝔽̄𝑝 and consider 𝐸 × 𝐸. Note that the kernel
of Frobenius on 𝐸 × 𝐸 is 𝛼𝑝 × 𝛼𝑝 . For each 𝜇 ∈ 𝔽̄𝑝 , we define a group scheme morphism
𝜇∗ ∶ 𝛼𝑝 → 𝛼𝑝 by sending 𝑎 ↦ 𝜇𝑎. Let (𝜇, 𝜈) ⊂ 𝔽̄2𝑝 − {0, 0} and consider the embedding
           (𝜇,𝜈)
𝑖𝜇,𝜈 ∶ 𝛼𝑝 −−−→ 𝛼𝑝 × 𝛼𝑝 ⊂ 𝐸 × 𝐸. Note that for any 𝜆 ∈ 𝔽̄×             𝑝 , the image 𝑖𝜆𝜇,𝜆𝜈 (𝛼𝑝 ) is equal
to the image 𝑖𝜇,𝜈 (𝛼𝑝 ) (though the maps are not the same).
    Consider ℙ1 = Proj (𝔽̄𝑝 [𝜇, 𝜈]) and write 𝛼𝑝 × 𝛼𝑝 × ℙ1 = Spec 𝒪ℙ1 [𝑥, 𝑦]/(𝑥𝑝 , 𝑦𝑝 ). We
define 𝐻 ⊂ 𝛼𝑝 × 𝛼𝑝 × ℙ1 ⊂ 𝐸 × 𝐸 × ℙ1 to be the subgroup scheme defined by 𝜈𝑥 − 𝜇𝑦 = 0.
In particular, at each point [𝜇, 𝜈] ∈ ℙ1 , the restriction 𝐻𝜇,𝜈 ⊂ 𝛼𝑝 × 𝛼𝑝 is the image 𝑖𝜇,𝜈 (𝛼𝑝 ).
Over any affine chart 𝑈 ⊂ ℙ1 , the restriction of 𝐻 on 𝑈 satisfies that 𝐻𝑈 ≅ 𝛼𝑝,𝑈 .
    Now let 𝒴 ≔ (𝐸 × 𝐸 × ℙ1 )/𝐻 to be the quotient abelian surface over ℙ1 . This abelian
surface 𝒴/ℙ1 has the following property:
                                                    58


Proposition 5.2.2. Let 𝒜 ̄ = 𝒜2,1,𝑁 × Spec 𝔽𝑝 be the Siegel threefold in characteristic 𝑝 with
𝑁 ≥ 3. Consider the supersingular locus 𝒜𝑠𝑠        ̄ , which is the union of superspecial stratum
and supersingular (but not superspecial) stratum. Then we have
(1) The singular points of 𝒜𝑠𝑠   ̄ are exactly the points in the superspecial stratum.
(2) Each irreducible component of 𝒜𝑠𝑠      ̄ is isomorphic to ℙ1 and there are exactly (𝑝 + 1)
    branches of ℙ1 intersecting transversally at each superspecial point.
(3) Let 𝑉 ⊂ 𝒜𝑠𝑠  ̄ be an irreducible component and let 𝒳̄ 𝑉 be the restriction of the universal
    abelian surface on 𝑉 . Then there is an isomorphism 𝜙 ∶ ℙ1 → 𝑉 so that 𝒴 ≅ 𝒳̄ 𝑉 ×𝑉 ℙ1 .
Proof. See [20, Page 193] and [18, Section 2].
   Let 𝑆𝜑 be the supersingular stratum and 𝒳̄ 𝜑 be the universal abelian surface over 𝑆𝜑 .
By Proposition 5.2.2, we have
                                         𝒳̄ 𝜑 ≅ ⨆ 𝒴 ×ℙ1 𝑈𝑖
                                                 𝑖
               1
where 𝑈𝑖 ⊂ ℙ are open subschemes. Therefore, we would want to have a description of
𝒴[𝑝]× /ℙ1 to study the Γ1 (𝑝)-cover.
   Consider the exact sequence 0 → 𝐻 → 𝐸 × 𝐸 → 𝒴 → 0 and restrict it to 𝑈 =
                                                                   (𝜇,1)
Spec 𝔽̄𝑝 [𝜇] ⊂ Proj 𝔽̄𝑝 [𝜇, 𝜈]. In this case, we have 𝐻𝑈 ≅ 𝛼𝑝 −−−→ 𝛼𝑝 × 𝛼𝑝 ↪ 𝐸 × 𝐸. The
exact sequence above yields an exact sequence of Dieudonné modules
                               𝔻(𝒴)𝑈 → 𝔻(𝐸 × 𝐸)𝑈 → 𝔻(𝐻)𝑈 → 0.
Note that 𝐻 and 𝐸 × 𝐸 are base changed from Spec 𝔽̄𝑝 and 𝔻 is a crystal. Therefore the
Dieudonné module of 𝔻(𝐻)𝑈 and 𝔻(𝐸 × 𝐸)𝑈 can be directly obtained from the Dieudonné
modules of 𝐸 and 𝛼𝑝 over 𝔽̄𝑝 . In particular, we have that
                                   𝔻(𝐻)𝑈 = 𝔽̄𝑝 ⊗𝑊 (𝔽̄𝑝 ) 𝒪𝑈 = 𝒪𝑈
and
                                          2
                         𝔻(𝐸 × 𝐸)𝑈 = ⨁ 𝑊 (𝔽̄𝑝 ){𝐹 , 𝑉 }/(𝐹 − 𝑉 ) ⊗ 𝒪𝑈 .
                                         𝑖=1
                                                  59


From this, we write 𝔻(𝐸 × 𝐸)𝑈 as
         𝐿 ≔ 𝔻(𝐸 × 𝐸)𝑈 = 𝑊 (𝔽̄𝑝 )[𝜇]𝑒1 ⊕ 𝑊 (𝔽̄𝑝 )[𝜇]𝑒2 ⊕ 𝑊 (𝔽̄𝑝 )[𝜇]𝑒3 ⊕ 𝑊 (𝔽̄𝑝 )[𝜇]𝑒4 ,
where 𝐹 𝑒1 = 𝑒3 , 𝐹 𝑒2 = 𝑒4 , 𝐹 𝑒3 = 𝑝𝑒1 , 𝐹 𝑒4 = 𝑝𝑒2 , and same for 𝑉 .
    Now we want to identify 𝔻(𝒴)𝑈 ⊂ 𝔻(𝐸 × 𝐸)𝑈 , which is the kernel of the map 𝜙 ∶
𝔻(𝐸×𝐸)𝑈 → 𝔻(𝐻)𝑈 . Since 𝐻 is killed by 𝐹 , the kernel ker 𝜙 contains 𝐹 𝐿, and 𝔻(𝒴)𝑈 can be
identified with a line ℓ ⊂ 𝐿/𝐹 𝐿 which generated by 𝜇𝑒1 +𝑒2 . To sum up, 𝔻(𝒴)𝑈 ⊂ 𝔻(𝐸×𝐸)𝑈
is generated by 𝑒3 , 𝑒4 , 𝑝𝑒1 , 𝑝𝑒2 , 𝜇𝑒1 + 𝑒2 . Note that 𝑝𝑒2 = 𝑝(𝜇𝑒1 + 𝑒2 ) − 𝜇(𝑝𝑒1 ). Therefore,
we have
          𝔻(𝒴)𝑈 = 𝑊 (𝔽̄𝑝 )[𝜇]𝑝𝑒1 ⊕ 𝑊 (𝔽̄𝑝 )[𝜇]𝑒3 ⊕ 𝑊 (𝔽̄𝑝 )[𝜇]𝑒4 ⊕ 𝑊 (𝔽̄𝑝 )[𝜇](𝜇𝑒1 + 𝑒2 ),
with Frobenius given by
                                                                                  0 1 −𝜇 0
                                                                               ⎛
                                                                               ⎜ 0 0 0 𝜇⎞       ⎟
  𝐹 (𝑝𝑒1 ⊗ 1, 𝑒3 ⊗ 1, 𝑒4 ⊗ 1, (𝜇𝑒1 + 𝑒2 ) ⊗ 1) = (𝑝𝑒1 , 𝑒3 , 𝑒4 , 𝜇𝑒1 + 𝑒2 ) ⎜ ⎜                ⎟ , (5.6)
                                                                               ⎜𝑝 0 0 1 ⎟       ⎟
                                                                               ⎝0 0 𝑝 0 ⎠
and the Verschiebung given by
                                                                                  0 1 −𝜇 0
                                                                               ⎛
                                                                               ⎜ 0 0 0 𝜇⎞       ⎟
  𝑉 (𝑝𝑒1 , 𝑒3 , 𝑒4 , 𝜇𝑒1 + 𝑒2 ) = (𝑝𝑒1 ⊗ 1, 𝑒3 ⊗ 1, 𝑒4 ⊗ 1, (𝜇𝑒1 + 𝑒2 ) ⊗ 1) ⎜ ⎜                ⎟ . (5.7)
                                                                               ⎜𝑝 0 0 1 ⎟       ⎟
                                                                               ⎝0 0 𝑝 0 ⎠
Now we are ready for the following result:
Theorem 5.2.3. Let 𝑆𝜑 be the supersingular stratum and let 𝑊 be an irreducible component
of 𝑆𝜑 . The Γ1 (𝑝)-cover 𝒳̄ ×                                   ×        1
                                 𝑊 [𝑝]/𝑊 is the pullback of 𝒴 [𝑝]/ℙ𝔽̄ via some open immersion
                                                                            𝑝
𝑊 →     ℙ1𝔽̄ .   Over each affine chart of the standard cover         ℙ1𝔽̄     = 𝔸10 ∪ 𝔸1∞ , the restricted
            𝑝                                                             𝑝
Γ1 (𝑝)-cover 𝒴× [𝑝]|𝔸1̄ /𝔸1𝔽̄ is isomorphic to
                         𝔽𝑝    𝑝
                          Spec 𝔽̄𝑝 [𝜇, 𝑥, 𝑦]/(𝑥𝑝 , 𝑦𝑝 − (𝜇𝑝 − 𝜇)𝑥𝑝 , 𝑥𝑝 −1 𝑦𝑝 −1 )
                                                2    2                 2        2
                                                                                                     (5.8)
over Spec (𝔽̄𝑝 [𝜇]).
                                                       60


Proof. Without loss of generality, consider 𝑈 = Spec 𝔽̄𝑝 [𝜇] ⊂ ℙ1 as above. Other affine
charts are similar. Write 𝐺 = 𝒴[𝑝]. By Section 2.4, we have 𝑀𝑆 (𝐺) = 𝑉 𝔻(𝒴)𝑈 /𝑝𝑉 𝔻(𝒴)𝑈 .
Following the calculations in (5.6) and (5.7), we can see that 𝑉 𝔻(𝒴)𝑈 /𝑝𝑉 𝔻(𝒴)𝑈 is a free
𝒪𝑈 -module with basis 𝑝𝑒3 , 𝑝𝑒1 , 𝑝𝑒2 , 𝜇𝑒3 + 𝑒4 with Frobenius and Verschiebung given by
                                                                                           0 1 −𝜇 0
                                                                                        ⎛
                                                                                        ⎜0 0 0 𝜇 ⎞         ⎟
  𝐹 (𝑝𝑒3 ⊗ 1, 𝑝𝑒1 ⊗ 1, 𝑝𝑒2 ⊗ 1, (𝜇𝑒3 + 𝑒4 ) ⊗ 1) = (𝑝𝑒3 , 𝑝𝑒1 , 𝑝𝑒2 , 𝜇𝑒3 + 𝑒4 ) ⎜      ⎜                  ⎟,
                                                                                        ⎜0 0 0 1 ⎟         ⎟
                                                                                        ⎝0 0 0 0 ⎠
                                                                                           0 1 −𝜇 0
                                                                                        ⎛
                                                                                        ⎜  0 0 0 𝜇⎞        ⎟
  𝑉 (𝑝𝑒3 , 𝑝𝑒1 , 𝑝𝑒2 , 𝜇𝑒3 + 𝑒4 ) = (𝑝𝑒3 ⊗ 1, 𝑝𝑒1 ⊗ 1, 𝑝𝑒2 ⊗ 1, (𝜇𝑒3 + 𝑒4 ) ⊗ 1) ⎜      ⎜                  ⎟.
                                                                                        ⎜0 0 0 1 ⎟         ⎟
                                                                                        ⎝  0 0       0   0 ⎠
By Theorem 4.4.1, the group scheme 𝐺 can be written as
                     Spec 𝔽̄𝑝 [𝜇, 𝑥, 𝑦1 , 𝑦2 , 𝑧]/(𝑥𝑝 , 𝑦1𝑝 − 𝑥, 𝑦2𝑝 + 𝜇𝑥, 𝑧 𝑝 − 𝜇𝑦1 − 𝑦2 ).               (5.9)
After substituting 𝑥 = 𝑦1𝑝 and 𝑦2 = 𝑧 𝑝 − 𝜇𝑦1 and changing the variables, we have the
expression (5.8).
    On the other chart 𝑇 = Spec 𝔽̄𝑝 [𝜈] ⊂ ℙ1 , we have 𝑉 𝔻(𝒴)𝑇 /𝑝𝑉 𝔻(𝒴)𝑇 a free 𝒪𝑇 -module
with basis 𝑝𝑒4 , 𝑝𝑒1 , 𝑝𝑒2 , 𝑒3 +𝜈𝑒4 . A similar calculation follows and we have the group scheme
as
                     Spec 𝔽̄𝑝 [𝜈, 𝑥, 𝑦1 , 𝑦2 , 𝑧]/(𝑥𝑝 , 𝑦1𝑝 − 𝜈𝑥, 𝑦2𝑝 − 𝑥, 𝑧 𝑝 − 𝑦1 − 𝜈𝑦2 ).             (5.10)
Note that (5.9) and (5.10) are isomorphic by sending 𝜇 ↦ 𝜈 and swapping 𝑦1 and 𝑦2 .
    The calculation of primitive elements is immediate from Example 2.3.6.
5.3 Γ1 (𝑝) -cover over 𝑝-rank-1 Locus
Let 𝑆𝜑 be the 𝑝-rank-1 stratum and let 𝑥 be a point of 𝑆𝜑 . We set 𝑅 = 𝒪𝑠ℎ                   𝑆𝜑 ,𝑥̄ , the strict
henselization of the local coordinate ring 𝒪𝑆𝜑 ,𝑥̄ and set 𝑆 = Spec 𝑅. Recall that in this case,
the canonical filtration is
                                  0 = 𝐺0 ⊂ 𝐺1 ⊂ 𝐺2 ⊂ 𝐺3 ⊂ 𝐺4 = 𝐺.
                                                          61


The corresponding canonical type is 𝜌 = (0, 1, 2, 3, 4), 𝑣 = (0, 1, 1, 2, 2) and 𝑓 = (2, 2, 3, 3, 4).
    We will use the notion mixed extensions in this case, established by Grothendieck. First,
we will give a quick review of this theory. For more details, see [11, Exposé IX, 9.3] and [3,
4.2].
5.3.1   Mixed extensions
Let 𝒞 be an abelian category. Suppose that there are two extensions in 𝒞:
                                  (𝐹 ) ∶ 0 → 𝑃 → 𝐹 → 𝑅 → 0,                                  (5.11)
                                  (𝐸) ∶ 0 → 𝑅 → 𝐸 → 𝑄 → 0.                                   (5.12)
Definition 5.3.1. A mixed extension (extension panachée) of 𝐸 by 𝐹 is an object 𝑥 in 𝒞
together with a filtration
                                        0 ⊂ 𝑋2 ⊂ 𝑋1 ⊂ 𝑋
such that
                       0 → 𝑋 2 → 𝑋 2 → 𝑋 1 /𝑋 2 → 0 ≅ 0 → 𝑃 → 𝐹 → 𝑅 → 0,                     (5.13)
               0 → 𝑋 1 /𝑋 2 → 𝑋/𝑋 2 → 𝑋/𝑋 1 → 0 ≅ 0 → 𝑅 → 𝐸 → 𝑄 → 0.                         (5.14)
    Let Extpan(𝐸, 𝐹 ) be the category of all mixed extensions of 𝐸 by 𝐹 and let Ext(𝑄, 𝑃 ) be
the category of all extensions of 𝑄 by 𝑃 . We define a functor 𝑤 ∶ Ext(𝑄, 𝑃 )×Extpan(𝐸, 𝐹 ) →
Extpan(𝐸, 𝐹 ) as follows: Let 𝑋 be a mixed extension of 𝐸 by 𝐹 . Regard 𝑋 as an ordinary
extension of 𝑄 by 𝐹 . Let 𝐺 be an extension of 𝑄 by 𝑃 . Let 𝐺 ̄ be the induced extension of
𝑄 by 𝐹 via the injection 𝑃 ↪ 𝐹 :
                            0          𝑃       𝐺          𝑄         0
                            0          𝐹       𝐺̄         𝑄         0
                                               62


Then we define 𝑤(𝐺, 𝑋) ≔ 𝐺 ̄ ∧ 𝑋. Here ∧ denotes the Baer sum in the Ext group. Let
𝑌 = 𝑤(𝐺, 𝑋). We need to define the filtration on 𝑌 to get an element in Extpan(𝐸, 𝐹 ). By
the definition of 𝑌 , we have an extension in 𝒞:
                                    0 → 𝐹 → 𝑌 → 𝑄 → 0.
Let 𝑌 1 be the image of 𝐹 under the inclusion. Push this exact sequence out along 𝐹 ↠ 𝑅:
                            0        𝐹        𝑌        𝑄          0
                            0        𝑅        𝑌̃       𝑄          0
By 𝑌 = 𝑤(𝐺, 𝑋), one can get 𝑌 ̃ ≅ 𝐸. We take 𝑌 2 = ker(𝑌 → 𝑌 ̃ ). In this way we form the
filtration 0 ⊂ 𝑌 2 ⊂ 𝑌 1 ⊂ 𝑌 .
     The main result of the theory of mixed extensions is the following proposition:
Proposition 5.3.2. Let 𝐸 and 𝐹 be two extensions as above.            Consider the category
Extpan(𝐸, 𝐹 ) of mixed extensions of 𝐸 by 𝐹 . Let Ext(𝑄, 𝑃 ) be the category of all ex-
tensions of 𝑄 by 𝑃 . Then the set of all isomorphism classes of objects in Extpan(𝐸, 𝐹 )
                                               1
is either empty, or it is a torsor under Ext (𝑄, 𝑃 ) (the group of isomorphism classes of
extensions) via the functor
                       𝑤 ∶ Ext(𝑄, 𝑃 ) × Extpan(𝐸, 𝐹 ) → Extpan(𝐸, 𝐹 ).
defined above.
5.3.2    Calculations on extensions
Apply the theory of mixed extensions to the filtration 0 ⊂ 𝐺1 ⊂ 𝐺3 ⊂ 𝐺4 . We need to
understand the extensions
                          (𝐹 ) ∶ 0 → 𝐺1 → 𝐺3 → 𝐺3 /𝐺1 → 0
                                                                                     (5.15)
                          (𝐸) ∶ 0 → 𝐺3 /𝐺1 → 𝐺4 /𝐺1 → 𝐺4 /𝐺3 → 0
                                              63


and the Ext group Ext(𝐺4 /𝐺3 , 𝐺1 ). We will analyze these groups one by one. We will
analyze (𝐸) first.
    First, we consider 𝐺3 /𝐺1 . Consider the 𝑝-divisible group 𝒳̄ 𝑆 [𝑝∞ ]. There is a short exact
sequence
                           0 → 𝒳̄ 𝑆 [𝑝∞ ]0 → 𝒳̄ 𝑆 [𝑝∞ ] → 𝒳̄ 𝑆 [𝑝∞ ]𝑒𝑡 → 0,                  (5.16)
where 0 denotes the identity component and 𝑒𝑡 denotes the maximal étale quotient. Let ∨
denote the Cartier dual of 𝑝-divisible groups, defined by
                                      𝒳̄ 𝑆 [𝑝∞ ]∨ ≔ (𝒳̄ 𝑆 [𝑝𝑛 ]𝐷 )𝑛 ,
where 𝐷 denotes the Cartier dual of finite group schemes. Apply the same exact sequence
to (𝒳̄ 𝑆 [𝑝∞ ]0 )∨ and take Cartier dual again: We get
                       0 → 𝒳̄ 𝑆 [𝑝∞ ]𝑚𝑢𝑙 → 𝒳̄ 𝑆 [𝑝∞ ]0 → (𝒳̄ 𝑆 [𝑝∞ ]0 )𝑢𝑛𝑖 → 0.              (5.17)
    Consider 𝒢 = (𝒳̄ 𝑆 [𝑝∞ ]0 )𝑢𝑛𝑖 . By the construction above, we have 𝒢[𝑝] = 𝐺3 /𝐺1 and the
Newton polygon of each point of 𝒢 is of slope (1/2, 1/2). Now we need the following result
by Oort and Zink from [34]:
Proposition 5.3.3. ([4, Proposition 8.6]) Let 𝑅 be a strictly henselian reduced local ring
over 𝔽̄𝑝 . Let 𝒢 be an isoclinic 𝑝-divisible group over 𝑆 = Spec 𝑅. Then there is a 𝑝-divisible
group 𝒢0 over 𝔽𝑝 with an isogeny 𝒢0 ×Spec 𝔽̄𝑝 𝑆 → 𝒢.
    Applying Proposition 5.3.3 to 𝒢 = (𝒳̄ 𝑆 [𝑝∞ ]0 )𝑢𝑛𝑖 , we obtain an isogeny 𝒢0 ×Spec 𝔽̄𝑝 𝑆 → 𝒢.
By classical Dieudonné theory ([5, III. 8]), there is a unique 𝑝-divisible group of dimension
1 and height 2 over an algebraically closed field, which is the 𝑝-divisible group associated
to a supersingular elliptic curve. Therefore we assume 𝒢0 = 𝐸[𝑝∞ ] for some supersingular
elliptic curve over 𝔽̄𝑝 and we get an isogeny
                                       𝐸[𝑝∞ ] ×Spec 𝔽̄𝑝 𝑆 → 𝒢.                               (5.18)
                                                   64


Lemma 5.3.4. The isogeny 𝐸[𝑝∞ ] ×Spec 𝔽̄𝑝 𝑆 → 𝒢 is an isomorphism.
Proof. Here we use the tool of Rapoport–Zink space ℳ of 𝐸[𝑝∞ ]. For an 𝔽̄𝑝 -scheme 𝑇 ,
a 𝑇 -point of ℳ is given by a pair (𝒢, 𝜌), where 𝒢 is a 𝑝-divisible group over 𝑇 and 𝜌 ∶
𝐸[𝑝∞ ] ×Spec 𝔽̄𝑝 𝑇 → 𝒢 is a quasi-isogeny of height 0. Two points (𝒢1 , 𝜌1 ) and (𝒢2 , 𝜌2 ) are
identified if they are isomorphic. The fundamental result of ℳ is that it is represented by
the formal scheme Spf 𝑊 (𝔽̄𝑝 )J𝑡K. For more details of this Rapoport–Zink space, see [37, 3.78,
3.79].
    The isogeny in (5.18) gives an 𝑆-point of ℳ, which corresponds to a morphism 𝑆 →
Spf 𝑊 (𝔽̄𝑝 )J𝑡K. Note that the only possible map 𝑆 → Spf 𝑊 (𝔽̄𝑝 )J𝑡K is by sending 𝑡 ↦ 0
since 𝑆 is smooth (thus integral). So ℳ(𝑆) has only one point and therefore the isogeny
𝐸[𝑝∞ ] ×Spec 𝔽̄𝑝 𝑆 → 𝒢 is in fact an isomorphism.
    By Lemma 5.3.4 above, we see that 𝐺3 /𝐺1 is isomorphic to the 𝑝-torsion of a supersin-
gular elliptic curve base changed from a field. Therefore we have
                          𝐺3 /𝐺1 ≅ 𝐸[𝑝] ≅ Spec 𝑅[𝑦1 , 𝑦2 ]/(𝑦1𝑝 , 𝑦2𝑝 − 𝑦1 )                  (5.19)
with
                        𝑚∗𝐺3 /𝐺1 (𝑦1 ) = 1 ⊗ 𝑦1 + 𝑦1 ⊗ 1,
                                                             𝑝−1
                                                                  𝑦1𝑘 ⊗ 𝑦1𝑝−𝑘
                        𝑚∗𝐺3 /𝐺1 (𝑦2 ) = 1 ⊗ 𝑦2 + 𝑦 2 ⊗ 1 + ∑
                                                             𝑘=1
                                                                  𝑘! (𝑝 − 𝑘)!
    Now consider 𝐺4 /𝐺3 . From the canonical type of 𝐺, we get that 𝐹𝐺4 /𝐺3 is an isomor-
phism and 𝑉𝐺4 /𝐺3 = 0. Since the base Spec 𝑅 is strictly henselian, we have 𝐺4 /𝐺3 ≅ ℤ/𝑝ℤ.
Dually, we also have 𝐺1 ≅ 𝜇𝑝 . We write them as
     𝐺4 /𝐺3 ≅ ℤ/𝑝ℤ ≅ Spec 𝑅[𝑥]/(𝑥𝑝 − 𝑥),             𝑚∗𝐺4 /𝐺3 (𝑥) = 1 ⊗ 𝑥 + 𝑥 ⊗ 1,
                                                                            𝑝−1
                                                                                𝑧 𝑖 ⊗ 𝑧 𝑝−𝑖   (5.20)
                               𝑝
     𝐺1 ≅ 𝜇𝑝 ≅ Spec 𝑅[𝑧]/(𝑧 ),              𝑚∗𝐺1 (𝑧)  =1⊗𝑧+𝑧⊗1+∑                            .
                                                                            𝑖=1
                                                                                𝑖! (𝑝 − 𝑖)!
                                                65


   From the discussion above, the two extensions in (5.15) are now given by
                          (𝐹 ) ∶ 0 → 𝜇𝑝 → 𝐺3 → 𝐸[𝑝] → 0
                                                                                    (5.21)
                          (𝐸) ∶ 0 → 𝐸[𝑝] → 𝐺4 /𝐺1 → ℤ/𝑝ℤ → 0
   Now we want to give the expression of 𝐺4 /𝐺1 in (𝐸). Consider 𝐺3 /𝐺2 . By the canonical
type, we can see that 𝑉𝐺3 /𝐺2 = 0 and 𝐹𝐺3 /𝐺2 = 0. Then 𝐺3 /𝐺2 ≅ 𝛼𝑝 and we write 𝐺3 /𝐺2
as
                                  𝐺3 /𝐺2 ≅ Spec 𝑅[𝑦1 ]/(𝑦1𝑝 )
with
                               𝑚∗𝐺3 /𝐺2 (𝑦1 ) = 1 ⊗ 𝑦1 + 𝑦1 ⊗ 1.
   Note that we have 𝐺4 /𝐺3 from (5.20). Consider the extension
                            0 → 𝐺3 /𝐺2 → 𝐺4 /𝐺2 → 𝐺4 /𝐺3 → 0
By Theorem 4.3.1, 𝐺4 /𝐺2 is of the form
                          𝐺4 /𝐺2 ≅ Spec 𝑅[𝑥, 𝑦1 ]/(𝑥𝑝 − 𝑥, 𝑦1𝑝 − 𝑎𝑥)                (5.22)
with
               𝑚∗𝐺4 /𝐺2 (𝑥) = 1 ⊗ 𝑥 + 𝑥 ⊗ 1, 𝑚∗𝐺4 /𝐺2 (𝑦1 ) = 1 ⊗ 𝑦1 + 𝑦1 ⊗ 1.
   Then we consider the extension
                            0 → 𝐺2 /𝐺1 → 𝐺4 /𝐺1 → 𝐺4 /𝐺2 → 0
Here again 𝐺2 /𝐺1 ≅ 𝛼𝑝 . We write it as
                                  𝐺2 /𝐺1 ≅ Spec 𝑅[𝑦2 ]/(𝑦2𝑝 )
with
                               𝑚∗𝐺2 /𝐺1 (𝑦2 ) = 1 ⊗ 𝑦2 + 𝑦2 ⊗ 1.
Again by Theorem 4.3.1, 𝐺4 /𝐺1 is of the form
                𝐺4 /𝐺1 ≅ Spec 𝑅[𝑥, 𝑦1 , 𝑦2 ]/(𝑥𝑝 − 𝑥, 𝑦1𝑝 − 𝑎𝑥, 𝑦2𝑝 − 𝑏𝑥 − 𝑐𝑦1 )    (5.23)
                                               66


with
             𝑚∗𝐺4 /𝐺1 (𝑥) = 1 ⊗ 𝑥 + 𝑥 ⊗ 1,
           𝑚∗𝐺4 /𝐺1 (𝑦1 ) = 1 ⊗ 𝑦1 + 𝑦1 ⊗ 1,
                                                𝑝−1                     𝑝−1
                                                    −𝑎𝑑𝑥𝑘 ⊗ 𝑥𝑝−𝑘            𝑑𝑦𝑘 ⊗ 𝑦1𝑝−𝑘
           𝑚∗𝐺4 /𝐺1 (𝑦2 ) = 1 ⊗ 𝑦2 + 𝑦 2 ⊗ 1 + ∑                     +∑ 1               .
                                                𝑘=1
                                                     𝑘! (𝑝  − 𝑘)!       𝑘=1
                                                                            𝑘! (𝑝 − 𝑘)!
Note that 𝐺3 /𝐺1 = ker 𝐹𝐺2 4 /𝐺1 . Comparing with (5.19), we get 𝑐 = 𝑑 = 1 and therefore we
have the expression of 𝐺4 /𝐺1 as
                   𝐺4 /𝐺1 ≅ Spec 𝑅[𝑥, 𝑦1 , 𝑦2 ]/(𝑥𝑝 − 𝑥, 𝑦1𝑝 − 𝑎𝑥, 𝑦2𝑝 − 𝑏𝑥 − 𝑦1 )        (5.24)
with
              𝑚∗𝐺4 /𝐺1 (𝑥) = 1 ⊗ 𝑥 + 𝑥 ⊗ 1,
             𝑚∗𝐺4 /𝐺1 (𝑦1 ) = 1 ⊗ 𝑦1 + 𝑦1 ⊗ 1,
                                                 𝑝−1                    𝑝−1
                                                     −𝑎𝑥𝑘 ⊗ 𝑥𝑝−𝑘            𝑦𝑘 ⊗ 𝑦1𝑝−𝑘
             𝑚∗𝐺4 /𝐺1 (𝑦2 ) = 1 ⊗ 𝑦2 + 𝑦2 ⊗ 1 + ∑                    +∑ 1
                                                 𝑘=1
                                                      𝑘! (𝑝 − 𝑘)!       𝑘=1
                                                                            𝑘! (𝑝 − 𝑘)!
    This expression of 𝐺4 /𝐺1 gives the description of all terms in (𝐸). Now we want to
analyze (𝐹 ). Note that (𝐹 ) is in fact the Cartier dual of 𝐸 from the information of the
canonical type. However, the explicit expressions of the group schemes are complicated and
difficult to directly write down the algebras and coalgebras. Therefore, we will also analyse
(𝐹 ) using another approach based on the fact that 𝐺3 is an extension of the 𝑝-torsion of
supersingular elliptic curves by 𝜇𝑝 . The result is as follows:
                                             1
Lemma 5.3.5. The extension group Ext𝑓𝑝𝑝𝑓,𝑆 (𝐸[𝑝], 𝜇𝑝 ) is isomorphic to 𝐸(𝑆)/𝑝𝐸(𝑆).
Proof. We will be using fppf topology throughout this proof. It is a classical result that
                                                                                 1
for any group scheme 𝐺/𝑆, we have an isomorphism 𝐻 1 (𝑆, 𝐺) ≅ Ext𝑆 (𝐺𝐷 , 𝔾𝑚 ). This
isomorphism is explained in many places, for example [43, Theorem 2, Theorem 3]. Apply
this isomorphism to 𝐸[𝑝]. Note that 𝐸[𝑝] is self-dual. Therefore we get an isomorphism
                     1
𝐻 1 (𝑆, 𝐸[𝑝]) ≅ Ext𝑆 (𝐸[𝑝], 𝔾𝑚 )
                                                 67


    For 𝐻 1 (𝑆, 𝐸[𝑝]), consider the long exact sequence associated to
                                                    𝑝
                                   0 → 𝐸[𝑝] → 𝐸 −  → 𝐸 → 0.                             (5.25)
We get
                                 𝑝
                          𝐸(𝑆) −→ 𝐸(𝑆) → 𝐻 1 (𝑆, 𝐸[𝑝]) → 𝐻 1 (𝑆, 𝐸).                    (5.26)
Note that 𝐻 1 (𝑆, 𝐸) = 0 since 𝐸 is smooth over 𝑆 and 𝑆 is strictly henselian. Therefore
we have 𝐸(𝑆)/𝑝𝐸(𝑆) ≅ 𝐻 1 (𝑆, 𝐸[𝑝]). Given any point 𝑆 → 𝐸, the 𝐸[𝑝]-torsor over 𝑆 is
obtained by pulling back the short exact sequence (5.25) along 𝑆 → 𝐸.
            1
    For Ext𝑆 (𝐸[𝑝], 𝔾𝑚 ), consider the long exact sequence associated to the Kummer sequence
                                   0 → 𝜇𝑝 → 𝔾𝑚 → 𝔾𝑚 → 0.                                (5.27)
We get
                                                                  𝑝
             Hom(𝐸[𝑝], 𝔾𝑚 ) → Ext(𝐸[𝑝], 𝜇𝑝 ) → Ext(𝐸[𝑝], 𝔾𝑚 ) −  → Ext(𝐸[𝑝], 𝔾𝑚 )       (5.28)
Since Hom(𝐸[𝑝], 𝔾𝑚 ) is a sheaf, we get an exact sequence
                          0 → Hom(𝐸[𝑝], 𝔾𝑚 ) → Hom(𝐸[𝑝], 𝔾𝑚 )(𝑆).                       (5.29)
Note that Hom(𝐸[𝑝], 𝔾𝑚 )(𝑆) = 𝐸[𝑝](𝑆), which vanishes since 𝐸 is supersingular and 𝑆 is
integral. Therefore the first term Hom(𝐸[𝑝], 𝔾𝑚 ) in (5.28) vanishes. Note that since 𝐸[𝑝]
                                                 𝑝
is annihilated by 𝑝, the map Ext(𝐸[𝑝], 𝔾𝑚 ) −   → Ext(𝐸[𝑝], 𝔾𝑚 ) in (5.28) is the zero map.
Therefore we have Ext(𝐸[𝑝], 𝜇𝑝 ) ≅ Ext(𝐸[𝑝], 𝔾𝑚 ).
                                                  1
    Combining all the above results, we get Ext𝑆 (𝐸[𝑝], 𝜇𝑝 ) ≅ 𝐸(𝑆)/𝑝𝐸(𝑆).
    The last ingredient from Proposition 5.3.2 is the Ext group Ext(ℤ/𝑝ℤ, 𝜇𝑝 ). For this, we
have the following lemma:
                                         1                                                    𝑝
Proposition 5.3.6. The Ext group Ext𝑆 (ℤ/𝑝ℤ, 𝜇𝑝 ) is isomorphic to 𝐻 1 (𝑆, 𝜇𝑝 ) ≅ 𝒪∗𝑆 / (𝒪∗𝑆 ) .
                                               68


Proof. Consider the short exact sequence 0 → ℤ → ℤ → ℤ/𝑝ℤ → 0. This gives a long exact
sequence:
                                         1                     1
                      Hom(ℤ, 𝜇𝑝 ) → Ext (ℤ/𝑝ℤ, 𝜇𝑝 ) → Ext (ℤ, 𝜇𝑝 ) → 0.
Note that 𝑆 is reduced and of characteristic 𝑝. Therefore 𝒪𝑆 has no non-trivial 𝑝-th root of
unity and Hom(ℤ, 𝜇𝑝 ) = 0. On the other hand, it is easy to see that the extensions of ℤ by
𝜇𝑝 are completely freely determined by the 𝜇𝑝 -torsor over 1 ∈ ℤ. Therefore
                              1                  1
                         Ext𝑆 (ℤ/𝑝ℤ, 𝜇𝑝 ) ≅ Ext (ℤ, 𝜇𝑝 ) ≅ 𝐻 1 (𝑆, 𝜇𝑝 ).
                                   𝑝
    The isomorphism 𝒪∗𝑆 / (𝒪∗𝑆 ) ≅ 𝐻 1 (𝑆, 𝜇𝑝 ) is a classical result of Kummer theory. Con-
                                                      𝑝
sider the Kummer exact sequence 1 → 𝜇𝑝 → 𝔾𝑚 −        → 𝔾𝑚 → 1. Then we have the associated
long exact sequence
                                  𝑝
                   𝐻 0 (𝑆, 𝔾𝑚 ) −→ 𝐻 0 (𝑆, 𝔾𝑚 ) → 𝐻 1 (𝑆, 𝜇𝑝 ) → 𝐻 1 (𝑆, 𝔾𝑚 ).
Since 𝑆 is local, we have that 𝐻 1 (𝑆, 𝔾𝑚 ) = Pic(𝑆) = 0. Note that 𝐻 0 (𝑆, 𝔾𝑚 ) = 𝒪∗𝑆 , we
                                     𝑝
have that 𝐻 1 (𝑆, 𝜇𝑝 ) ≅ 𝒪𝑆∗ / (𝒪∗𝑆 ) .
    As a conclusion, we record all previous results as follows:
Theorem 5.3.7. The group scheme 𝐺 = 𝐺4 is a mixed extension of
                            (𝐹 ) ∶ 0 → 𝜇𝑝 → 𝐺3 → 𝐸[𝑝] → 0
                            (𝐸) ∶ 0 → 𝐸[𝑝] → 𝐺4 /𝐺1 → ℤ/𝑝ℤ → 0
as in (5.21). Here, the extension (𝐸) is explicitly given by Equation (5.19), (5.20) and (5.23).
The extension (𝐹 ) is the Cartier dual of (𝐸) and can alternatively obtained by Lemma 5.3.5.
                                     1
All mixed extensions form a Ext𝑆 (ℤ/𝑝ℤ, 𝜇𝑝 )-torsor
                            1
                   𝑤 ∶ Ext𝑆 (ℤ/𝑝ℤ, 𝜇𝑝 ) × Extpan(𝐸, 𝐹 ) → Extpan(𝐸, 𝐹 ),
                         1                         𝑝
where the ext group Ext𝑆 (ℤ/𝑝ℤ, 𝜇𝑝 ) ≅ 𝒪∗𝑆 / (𝒪∗𝑆 ) and the map 𝑤 is described in Proposition
5.3.2.
                                                69


5.4 Γ1 (𝑝) -cover over the Ordinary Locus
Now, let 𝑆𝜑 be the ordinary locus. The universal abelian surface 𝒳̄ 𝜑 is ordinary. In this
case, we have a powerful tool, namely the Serre–Tate theory, to help with analyzing the
Γ1 (𝑝)-cover. For details about Serre–Tate local moduli, see [17].
     Let 𝑥 ∈ 𝑆𝜑 be a geometric point which corresponds to a principally polarized abelian
variety 𝑋 over 𝔽̄𝑝 . Serre–Tate theory states that the deformation space of 𝑋 is canoni-
cally pro-represented by Spf 𝔽̄𝑝 J𝑡1 , 𝑡2 , 𝑡3 K. Alternatively, this means there is a compatible
system of universal principally polarized abelian schemes 𝑋𝑛 / Spec 𝔽̄𝑝 [𝑡1 , 𝑡2 , 𝑡3 ]/(𝑡1 , 𝑡2 , 𝑡3 )𝑛 .
By the definition of the Siegel threefold, this gives canonical isomorphisms 𝒪𝑥,𝑆                  ̂     /𝑚̂ 𝑛 →
                                                                                                      𝜑
                                              ̂
𝔽̄𝑝 [𝑡1 , 𝑡2 , 𝑡3 ]/(𝑡1 , 𝑡2 , 𝑡3 )𝑛 where 𝒪𝑥,𝑆      is the completion of the local coordinate ring at 𝑥 and
                                                  𝜑
𝑚̂ is the maximal ideal of 𝒪𝑥,𝑆        ̂    . This induces a canonical isomorphism 𝔽̄𝑝 J𝑡1 , 𝑡2 , 𝑡3 K ≅ 𝒪𝑥,𝑆̂     .
                                          𝜑                                                                      𝜑
     Consider 𝒳̂ ≔ 𝒳̄ 𝜑 ×𝑆𝜑 Spec 𝒪𝑥,𝑆           ̂
                                                   𝜑
                                                     . It is the universal abelian surface over the Serre–Tate
local moduli. The goal of this section is to give explicit description of 𝒳[𝑝]/            ̂           ̂
                                                                                               Spec 𝒪𝑥,𝑆       . To
                                                                                                           𝜑
do this, we will calculate the universal extension over the Serre–Tate local moduli.
     We first sketch the idea of Serre–Tate theory. Let 𝑋 be the principally polarized ordinary
abelian variety corresponding to 𝑥 ∈ 𝑆𝜑 . Let 𝑇𝑝 𝑋(𝑘) be the Tate module of 𝑋. By choosing
a basis, 𝑇𝑝 𝑋(𝑘) ≅ ℤ2𝑝 . Let 𝑋 𝑡 be the dual abelian variety of X. We also have 𝑇𝑝 𝑋(𝑘) ≅ ℤ2𝑝
after a choice of basis.
     The first result is that the deformation theory of 𝑋 is the same as the deformation theory
of the associated 𝑝-divisible group 𝑋[𝑝∞ ] (see [17, Theorem 1.2.1]). Therefore we only need
to work with 𝑋[𝑝∞ ]. Let 𝕏[𝑝∞ ] be a deformation of 𝑋[𝑝∞ ] over an Artin local ring 𝑅. Let
𝑋 𝑚𝑢𝑙 the maximal toroidal subgroup of 𝑋[𝑝∞ ] and let 𝕏𝑚𝑢𝑙 be the unique lift of 𝑋 𝑚𝑢𝑙 .
Then there is the canonical decomposition
                                    0 → 𝕏𝑚𝑢𝑙 → 𝕏[𝑝∞ ] → 𝑇𝑝 𝑋(𝑘) ⊗ ℚ𝑝 /ℤ𝑝 → 0.                              (5.30)
It turns out that the extension (5.30) can be obtained from the basic extension
                            0 → 𝑇𝑝 𝑋(𝑘) → 𝑇𝑝 𝑋(𝑘) ⊗ ℚ𝑝 → 𝑇𝑝 𝑋(𝑘) ⊗ ℚ𝑝 /ℤ𝑝 → 0.                             (5.31)
                                                              70


by pushing out along a unique homomorphism 𝜙𝕏/𝑅 ∶ 𝑇𝑝 𝑋(𝑘) → 𝕏𝑚𝑢𝑙 . On the other hand,
there is a pairing of group schemes over 𝑅
                                    𝐸𝕏 ∶ 𝕏𝑚𝑢𝑙 × 𝑇𝑝 𝑋 𝑡 (𝑘) → 𝔾̂ 𝑚 ,                        (5.32)
which is the unique lift of the paring of group schemes over 𝑘
                                   𝐸𝑋 ∶ 𝑋 𝑚𝑢𝑙 × 𝑇𝑝 𝑋 𝑡 (𝑘) → 𝔾̂ 𝑚                          (5.33)
that is induced from the Weil pairing 𝑋 𝑚𝑢𝑙 [𝑝] × 𝑇𝑝 𝑋 𝑡 [𝑝](𝑘) → 𝜇𝑝𝑛 . By composing the map
𝜙𝕏/𝑅 ∶ 𝑇𝑝 𝑋(𝑘) → 𝕏𝑚𝑢𝑙 with the pairing (5.32), we get a pairing
                            𝑞(𝕏/𝑅; −, −) ∶ 𝑇𝑝 𝑋(𝑘) ⊗ 𝑇𝑝 𝑋 𝑡 (𝑘) → 𝔾̂ 𝑚 (𝑅).                (5.34)
     It turns out that this 𝑞 contains all the information of the deformation 𝕏:
Proposition 5.4.1. ([17, Theorem 2.1]) The construction
                                        𝕏/𝑅 ↦ 𝑞(𝕏/𝑅; −, −)
gives a bijection of the set of isomorphism classes of deformations of 𝑋 and the group
Homℤ𝑝 (𝑇𝑝 𝑋(𝑘) ⊗ 𝑇𝑝 𝑋 𝑡 (𝑘), 𝔾̂ 𝑚 (𝑅)).
     Now we will calculate the universal extension. Take 𝑒1 , 𝑒2 as a basis for 𝑇𝑝 𝑋(𝑘) and
𝑓1 , 𝑓2 for 𝑇𝑝 𝑋 𝑡 (𝑘). Over the universal base 𝑆 = Spf 𝔽̄𝑝 J𝑡11 , 𝑡12 , 𝑡21 , 𝑡22 K, let
           𝜙 ∈ Hom𝑆 (𝑇𝑝 𝑋(𝑘) ⊗ 𝑇𝑝 𝑋 𝑡 (𝑘), 𝔾̂ 𝑚 ) = Hom𝑆 (𝑇𝑝 𝑋(𝑘), Hom(𝑇𝑝 𝑋 𝑡 (𝑘), 𝔾̂ 𝑚 ))
be the universal homomorphism given by 𝑒𝑖 ⊗ 𝑓𝑗 ↦ 𝑡𝑖𝑗 . By the Serre–Tate theory above, the
universal extension is the pushout of the basic extension (5.31) along 𝜙:
      0                 𝑇𝑝 𝑋(𝑘)              𝑇𝑝 𝑋(𝑘) ⊗ ℚ𝑝            𝑇𝑝 𝑋(𝑘) ⊗ ℚ𝑝 /ℤ𝑝      0
                            𝜙
      0         Hom(𝑇𝑝 𝑋 𝑡 (𝑘), 𝔾̂ 𝑚 )             𝑋̂                𝑇𝑝 𝑋(𝑘) ⊗ ℚ𝑝 /ℤ𝑝      0
                                                  71


     Note that Hom(𝑇𝑝 𝑋 𝑡 (𝑘), 𝔾̂ 𝑚 ) ≅ 𝔾̂ 𝑚 × 𝔾̂ 𝑚 by taking the images of 𝑓1 and 𝑓2 . We will
denote the elements in Hom(𝑇𝑝 𝑋 𝑡 (𝑘), 𝔾̂ 𝑚 ) by their images under this isomorphism. In this
way, the generators of Hom(𝑇𝑝 𝑋 𝑡 (𝑘), 𝔾̂ 𝑚 ) are denoted by (1+𝑡11 , 1+𝑡12 ) and (1+𝑡21 , 1+𝑡22 )
and a general element 𝑎𝑒1 +𝑏𝑒2 ∈ 𝑇𝑝 𝑋(𝑘) is mapped to ((1+𝑡11 )𝑎 (1+𝑡21 )𝑏 , (1+𝑡12 )𝑎 (1+𝑡22 )𝑏 )
by 𝜙.
     Now we consider the 𝑝-torsion 𝑋[𝑝]          ̂ of the fiber coproduct
                            𝑋̂ = (𝑇𝑝 𝑋(𝑘) ⊗ ℚ𝑝 ) ⊔𝑇𝑝 𝑋(𝑘) Hom(𝑇𝑝 𝑋 𝑡 (𝑘), 𝔾̂ 𝑚 ).
The points of 𝑋[𝑝]    ̂ are given by
  ̂
𝑋[𝑝](𝐴)      = {(𝑎, 𝑏, 𝑥, 𝑦)|𝑎, 𝑏 ∈ {0, … , 𝑝−1}, 𝑥𝑝 = (1+𝑡11 )𝑎 (1+𝑡21 )𝑏 , 𝑦𝑝 = (1+𝑡12 )𝑎 (1+𝑡22 )𝑏 }
                                                                                                     (5.35)
with group multiplication defined by
                       ̇ 2 , 𝑏2 , 𝑥2 , 𝑦2 ) =
  (𝑎1 , 𝑏1 , 𝑥1 , 𝑦1 )+(𝑎
    ⎧ (𝑎1 + 𝑎2 , 𝑏1 + 𝑏2 , 𝑥1 𝑥2 , 𝑦1 𝑦2 ) ,                              if 𝑎1 + 𝑎2 , 𝑏1 + 𝑏2 < 𝑝,
    {
    {                                    𝑥1 𝑥2          𝑦1 𝑦2
    { (𝑎1 + 𝑎2 − 𝑝, 𝑏1 + 𝑏2 ,                     ,             ),        if 𝑏1 + 𝑏2 < 𝑝 ≤ 𝑎1 + 𝑎2 ,
    {
    {                                 (1 + 𝑡11 ) (1 + 𝑡12 )𝑝
                                                𝑝
    {                                    𝑥1 𝑥2          𝑦1 𝑦2                                        (5.36)
    ⎨ (𝑎1 + 𝑎2 , 𝑏1 + 𝑏2 − 𝑝,                     ,             ),        if 𝑎1 + 𝑎2 < 𝑝 ≤ 𝑏1 + 𝑏2 ,
                                      (1 + 𝑡21 )𝑝 (1 + 𝑡22 )𝑝
    {
    {                                               𝑥1 𝑥2                    𝑦1 𝑦2
    { (𝑎1 + 𝑎2 − 𝑝, 𝑏1 + 𝑏2 − 𝑝,                                   ,                        ),
    {                                       (1 + 𝑡11 )𝑝 (1 + 𝑡21 )𝑝 (1 + 𝑡12 )𝑝 (1 + 𝑡22 )𝑝
    {
    {                                                                     if 𝑝 ≤ 𝑎1 + 𝑎2 , 𝑏1 + 𝑏2 .
    ⎩
     Note that we haven’t used the polarization structure on 𝐴 yet and hence the dimension
of the formal moduli is 4. Now we consider the polarization. From [17, Theorem 21. (4)],
the principal polarization 𝜆 ∶ 𝑋 → 𝑋 𝑡 lifts to 𝕏 → 𝕏𝑡 if and only if
                                      𝑞(𝕏/𝑅; 𝛼, 𝜆𝑡 (𝛽)) = 𝑞(𝕏𝑡 /𝑅; 𝜆(𝛼), 𝛽)                          (5.37)
for all 𝛼 ∈ 𝑇𝑝 𝑋(𝑘) and 𝛽 ∈ 𝑇𝑝 𝑋 𝑡𝑡 (𝑘) = 𝑇𝑝 𝑋(𝑘). From the symmetry formula in [17,
Theorem 21. (3)], we have
                                       𝑞(𝕏𝑡 /𝑅; 𝜆(𝛼), 𝛽) = 𝑞(𝕏/𝑅; 𝛽, 𝜆(𝛼)).                          (5.38)
                                                            72


By combining Equation (5.37), (5.38) and since 𝜆 = 𝜆𝑡 , we have
                                       𝑞(𝕏/𝑅; 𝛼, 𝜆(𝛽)) = 𝑞(𝕏/𝑅; 𝛽, 𝜆(𝛼)).                                  (5.39)
We choose a basis 𝑒1 , 𝑒2 for 𝑇𝑝 𝑋(𝑘) and let 𝑓𝑖 = 𝜆(𝑒𝑖 ), 𝑖 = 1, 2 be a basis of 𝑇𝑝 𝑋 𝑡 (𝑘).
Then Equation (5.39) is equivalent to 𝑞(𝕏/𝑅; 𝑒𝑖 , 𝑓𝑗 ) = 𝑞(𝕏/𝑅; 𝑒𝑗 , 𝑓𝑖 ). Recall that 𝑡𝑖𝑗 is
the image of 𝑒𝑖 ⊗ 𝑓𝑗 under the universal homomorphism 𝜙 ∈ Hom𝑆 (𝑇𝑝 𝑋(𝑘) ⊗ 𝑇𝑝 𝑋 𝑡 (𝑘), 𝔾̂ 𝑚 ).
Therefore the Serre–Tate local coordinates for principally polarized abelian surfaces are given
by 𝑡11 , 𝑡12 = 𝑡21 , 𝑡22 .
     Now we denote 𝑡1 ≔ 𝑡11 , 𝑡2 ≔ 𝑡12 = 𝑡21 and 𝑡3 ≔ 𝑡22 . After changing the variables, we
are now ready to state the final result:
Theorem 5.4.2. Let 𝑆𝜑 be the ordinary locus. Let 𝑆𝜑 be the ordinary locus and 𝑥 be a
closed point of 𝑆𝜑 . Let 𝒪𝑆̂ 𝜑 ,𝑥 be the completion of the local ring of 𝑆𝜑 at 𝑥. Then the base
change of 𝒳×                             ̂
              𝜑 [𝑝]/𝑆𝜑 to Spec 𝒪𝑆𝜑 ,𝑥 is isomorphic to
                                                                     𝑥𝑝                   𝑝
                                                                       1 −𝑃1 (𝑦1 ,𝑦2 ),𝑥2 −𝑃2 (𝑦1 ,𝑦2 ),
                  Spec 𝔽̄𝑝 J𝑡1 , 𝑡2 , 𝑡3 K[𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 ]/ (              𝑦1𝑝 −𝑦1 ,𝑦2𝑝 −𝑦2 ,        )
                                                                   (𝑦1𝑝−1 −1)(𝑦2𝑝−1 −1)Φ𝑝 (𝑥1 )Φ𝑝 (𝑥2 )
over Spec 𝔽̄𝑝 J𝑡1 , 𝑡2 , 𝑡3 K. Here, Φ𝑝 denotes the cyclotomic polynomial and the polynomials
𝑃1 , 𝑃2 ∈ 𝔽𝑝 J𝑡1 , 𝑡2 , 𝑡3 K[𝑦1 , 𝑦2 ] are interpolation polynomials characterized by
                                           𝑃1 (𝑖, 𝑗) = (1 + 𝑡1 )𝑖 (1 + 𝑡2 )𝑗 ,
                                           𝑃2 (𝑖, 𝑗) = (1 + 𝑡2 )𝑖 (1 + 𝑡3 )𝑗
for 0 ≤ 𝑖, 𝑗 ≤ 𝑝 − 1. The variables 𝑡1 , 𝑡2 , 𝑡3 are the Serre-Tate coordinates.
Proof. From the description of points of 𝑋[𝑝],             ̂    we can write
                                                ̂ =
                                              𝑋[𝑝]            ⨆         𝑋̂ 𝑎,𝑏
                                                          𝑎,𝑏=0,…,𝑝−1
where
             𝑋̂ 𝑎,𝑏 = Spec 𝑅[𝑥, 𝑦]/(𝑥𝑝 − (1 + 𝑡1 )𝑎 (1 + 𝑡2 )𝑏 , 𝑦𝑝 − (1 + 𝑡2 )𝑎 (1 + 𝑡3 )𝑏 ).
                                                             73


Note that in particular, 𝑋̂ 0,0 ≅ 𝜇𝑝 × 𝜇𝑝 .
    We want to modify the description of 𝑋[𝑝]                  ̂   to make it consistent with the forms in
the previous cases.           Note that the constant group scheme can be written as ℤ/𝑝ℤ ≅
Spec 𝑅[𝑥]/(𝑥𝑝 − 𝑥). Using this for the indices 𝑎, 𝑏 ∈ 0, … , 𝑝 − 1, we can write 𝑋[𝑝]                 ̂ as
   ̂ ≅ Spec 𝔽J𝑡
 𝑋[𝑝]             ̄ 1 , 𝑡2 , 𝑡3 K[𝑥1 , 𝑥2 , 𝑦1 , 𝑦2 ]/ (𝑥𝑝 − 𝑥1 , 𝑥𝑝 − 𝑥2 , 𝑦𝑝 − 𝑃1 (𝑥1 , 𝑥2 ), 𝑦𝑝 − 𝑃2 (𝑥1 , 𝑥2 )) .
                                                         1         2          1                  2
Here, the polynomials 𝑃1 , 𝑃2 ∈ 𝔽𝑝 J𝑡1 , 𝑡2 , 𝑡3 K[𝑥1 , 𝑥2 ] are defined by
                                                                                 (𝑥1 − 𝑘)(𝑥2 − 𝑙)
              𝑃1 (𝑥1 , 𝑥2 ) =       ∑         (1 + 𝑡11 )𝑖 (1 + 𝑡21 )𝑗 ( ∏                           ),
                                 0≤𝑖,𝑗≤𝑝−1                              𝑘≠𝑖,𝑙≠𝑗
                                                                                  (𝑖 − 𝑘)(𝑗 − 𝑙)
                                                                                 (𝑥1 − 𝑘)(𝑥2 − 𝑙)
              𝑃2 (𝑥1 , 𝑥2 ) =       ∑         (1 + 𝑡12 )𝑖 (1 + 𝑡22 )𝑗 ( ∏                           ).
                                 0≤𝑖,𝑗≤𝑝−1                              𝑘≠𝑖,𝑙≠𝑗
                                                                                  (𝑖 − 𝑘)(𝑗 − 𝑙)
They are interpolation polynomials so that we have 𝑃1 (𝑖, 𝑗) = (1+𝑡1 )𝑖 (1+𝑡2 )𝑗 and 𝑃2 (𝑖, 𝑗) =
(1 + 𝑡2 )𝑖 (1 + 𝑡3 )𝑗 for 0 ≤ 𝑖, 𝑗 ≤ 𝑝 − 1.
    For the primitive elements 𝑋̂ × [𝑝], we will use Lemma 2.3.3 again. Note that in this case,
the augmentation ideal is generated by 𝑥1 , 𝑥2 , 𝑦1 − 1, 𝑦2 − 1. Note that the constant terms
of 𝑃𝑖 (𝑥1 , 𝑥2 ) are equal to 0 since we have 𝑃 (0, 0) = 1 from interpolation conditions. Denote
             𝑥𝑝 − 1
Φ𝑝 (𝑥) ≔               for the cyclotomic polynomial. Therefore, using the notation in Lemma
             𝑥−1
2.3.3, the matrix 𝑀 with respect to generators 𝑥1 , 𝑥2 , 𝑦1 − 1, 𝑦2 − 1 is
                                         𝑥𝑝−1
                                            1    −1          0          ∗          ∗
                                       ⎛
                                       ⎜                  𝑝−1                           ⎞
                                               0         𝑥2 − 1         ∗          ∗ ⎟
                              𝑀 =⎜     ⎜                                                ⎟
                                       ⎜       0             0       Φ𝑝 (𝑦1 )      0 ⎟  ⎟
                                       ⎝       0             0          0       Φ𝑝 (𝑦2 )⎠
Therefore the primitive elements 𝑋̂ × [𝑝] ⊂ 𝑋[𝑝]            ̂ are defined by (det 𝑀 ) = (𝑥𝑝−1 − 1)(𝑥𝑝−1 −
                                                                                                  1           2
1)Φ𝑝 (𝑦1 )Φ𝑝 (𝑦2 ).
5.5     Applications
In the previous sections, we use the explicit descriptions of the Γ1 (𝑝)-cover to prove some
geometric properties of the Γ1 (𝑝)-cover over each stratum. In fact, these descriptions can
                                                             74


also be used to show some geometric properties of the whole integral model 𝒜 in mixed
characteristics. In this section, we will use the descriptions in Section 5.4 to prove that the
whole Γ1 (𝑝)-cover in mixed characteristics is not normal. More precisely, consider the Siegel
threefold 𝒜 = 𝒜2,1,𝑁 in mixed characteristics and let 𝒳 be the universal abelian surface
over 𝒜. We will prove:
Theorem 5.5.1. The universal Γ1 (𝑝)-cover (𝒳[𝑝])× over the Siegel threefold 𝒜 in mixed
characteristic is not normal.
Proof. By Serre’s criterion for normality, it is enough to prove that (𝒳[𝑝])× does not satisfy
the condition 𝑅1 , which says 𝒪𝒳̄ 𝑢𝑛 ,𝑥 is regular for any 𝑥 ∈ 𝒳̄ 𝑢𝑛 with codimension ≤ 1. In
fact, we will show that the local ring at the generic point of the special fiber, i.e. with respect
to the ideal (𝑝), is not regular.
    Recall our notation that 𝒜 is the Siegel threefold over Spec ℤ𝑝 and 𝒜 ̄ = 𝒜×Spec ℤ𝑝 Spec 𝔽𝑝
is the special fiber. Let 𝜉 be the generic point of the special fiber 𝒜 and let 𝒪𝒜,𝜉 to be the
local ring of 𝜉 in 𝒜. Let 𝐺𝜉 ≔ 𝒳[𝑝] ×𝒜 Spec 𝒪𝒜,𝜉 . Then 𝐺×         𝜉 is the universal Γ1 (𝑝)-cover
over Spec 𝒪𝒜,𝜉 . By Serre’s criterion, it suffices to prove that 𝒪𝐺×𝜉 is not regular.
    Let 𝑘(𝜉) ≔ 𝒪𝒜,𝜉 /𝑝𝒪𝒜,𝜉 be the residue field of 𝒪𝒜,𝜉 . Note that we also have 𝑘(𝜉) =
Frac(𝒪𝒜 ).̄  On the other hand, let 𝑥 ∈ 𝒜 ̄ be a geometric ordinary point. Let 𝒪𝒜,𝑥         ̄ be the
local ring of 𝒜 ̄ at 𝑥 and let 𝒪𝒜,𝑥
                                 ̂ ̄ be its completion. Then we have an inclusion
                                       𝒪𝒜 ̄ ⊂ 𝒪𝒜,𝑥     ̂ ̄ ,
                                                 ̄ ⊂ 𝒪𝒜,𝑥
which induces a field extension
                              𝑘(𝜉) = Frac(𝒪𝒜 )̄ ⊂ 𝐾̂ ≔ Frac(𝒪𝒜,𝑥̂ ̄ ).
Note that by Serre–Tate theory, we have a canonical isomorphism 𝒪𝒜,𝑥      ̂ ̄ ≅ 𝔽̄𝑝 J𝑡1 , 𝑡2 , 𝑡3 K.
                                                75


    Now consider the following Cartesian diagrams
                            ̂
                           𝐺×                   𝐺× ̄                    𝐺×                    𝒳× [𝑝]
                            𝜉                     𝜉                        𝜉
                              𝜉 ′′′                    𝜉 ′′         𝜉′
                                    □                        □                                             (5.40)
                                                                                       □
                                                                     𝜉
                      Spec 𝐾̂               Spec 𝑘(𝜉)               Spec 𝒪𝒜,𝜉                    𝒜
By construction, 𝐺×      ̂          ̂
                         𝜉 / Spec 𝐾 is the generic fiber of the group scheme in Theorem 5.4.2, given
by
                                                                         𝑥𝑝         𝑝
                                                                          1 −𝑥1 ,𝑥2 −𝑥2 ,
                    𝐺× ̂   ≅ Spec 𝐾[𝑥 ̂ 1 , 𝑥2 , 𝑦1 , 𝑦2 ]/ (   𝑦1𝑝 −𝑃1 (𝑥1 ,𝑥2 ),𝑦2𝑝 −𝑃2 (𝑥1 ,𝑥2 ), ) .   (5.41)
                      𝜉
                                                               (𝑥𝑝−1
                                                                 1    −1)(𝑥𝑝−1
                                                                            2   −1)Φ𝑝 (𝑦1 )Φ𝑝 (𝑦2 )
    Let 𝜉 ′′′ be the point of 𝐺×         ̂
                                         𝜉 in (5.41) corresponding to the maximal ideal 𝑚𝜉′′′ ≔
(𝑥1 , 𝑥2 , 𝑦1 − 1, 𝑦2 − 1). One can check directly from (5.41) that 𝑚𝜉′′′ /𝑚2𝜉′′′ is a 𝐾-vector          ̂
space generated by 𝑦1 − 1, 𝑦2 − 1. This proves
                                             dim𝐾̂ 𝑚𝜉′′′ /𝑚2𝜉′′′ = 2.                                      (5.42)
Let 𝜉 ′ , 𝜉 ′′ be the images of 𝜉 ′′′ as in (5.40). Then we have
                         dim𝑘(𝜉′ ) 𝑚𝜉′ /𝑚2𝜉′ ≥ dim𝑘(𝜉′′ ) ((𝑝) + 𝑚𝜉′ ) / ((𝑝) + 𝑚2𝜉′ )
                                               = dim𝑘(𝜉′′ ) 𝑚𝜉′′ /𝑚2𝜉′′
                                               = 𝑑𝑖𝑚𝐾̂ 𝑚𝜉′′′ /𝑚2𝜉′′′ = 2
Therefore 𝒪𝐺×𝜉 is not regular and we finish the proof by Serre’s criterion.
                                                            76


BIBLIOGRAPHY
      77


                                     BIBLIOGRAPHY
[1] P. Berthelot, Théorie de Dieudonné sur un anneau de valuation parfait, Ann. Sci. École
    Norm. Sup. (4), Volumn 13, 1980, No.2, 225–268.
[2] P. Berthelot, L. Breen and W. Messing, Théorie de Dieudonné cristalline. II, Lecture
    Notes in Mathematics, Vol. 930, Springer-Verlag, Berlin, 1982, x+261.
[3] C. Chai and P. Norman, Bad reduction of the Siegel moduli scheme of genus two with
    Γ0 (𝑝)-level structure, American Journal of Mathematics, Volume 112, 1990, No.6, 1003–
    1071.
[4] A. de Jong, Finite locally free group schemes in characteristic 𝑝 and Dieudonné modules,
    Invent. Math. Volume 114, 1993, 89–137.
[5] M. Demazure, Lectures on 𝑝-divisible groups, Lecture Notes in Mathematics, Vol. 302,
    Springer-Verlag, Berlin-New York, 1972, v+98.
[6] P. Deligne and D. Mumford, The irreducibility of the space of curves of given genus, Inst.
    Hautes Études Sci. Publ. Math. No. 36, 1969, 75–109.
[7] P. Deligne and G. Pappas, Les schémas de modules de courbes elliptiques, Compositio
    Math. Vol. 90, 1994, 59–79.
[8] P. Deligne and M. Rapoport, Singularités des espaces de modules de Hilbert, en les
    caractéristiques divisant le discriminant, Modular functions of one variable, II (Proc.
    Internat.Summer School, Univ. Antwerp, Antwerp, 1972), Lecture Notes in Math. Vol.
    349, Springer-Verlag, Berlin, 1973, 143–316.
[9] B. de Smit, K. Rubin and R. Schoof, Criteria for complete intersections, Modular forms
    and Fermat’s last theorem (Boston, MA, 1995), 343–356, Springer, New York, 1997.
[10] V. Drinfeld, Elliptic modules, Mat. Sb. (N.S.), Volume 94(136), 1974, 594–627, 656.
[11] A. Grothendieck, Groupes de monodromie en géométrie algébrique (SGA 7I), Lecture
    Notes in Mathematics, Vol. 288, Springer-Verlag, Berlin-New York, 1972, viii+523.
[12] A. Grothendieck and M. Demazure, Schémas en groupes I, II, III. (SGA 3), Lect. Notes
    Math. vols. 151, 152, 153, Berlin, Heidelberg, New York, Springer, 1971.
[13] C. Guan, Full level structure on some group schemes, Res. Number Theory, Vol. 7, 2021.
[14] H, Hida, Galois representations into GL2 (ℤ𝑝 [[𝑋]]) attached to ordinary cusp forms,
    Invent. Math., Vol. 85, no.3, 1986.
[15] H, Hida, Iwasawa modules attached to congruences of cusp forms, Ann. Sci. École Norm.
    Sup., Vol. 19, no.2, 1986.
                                              78


[16] T. Haines and M. Rapoport, Shimura varieties with Γ1 (𝑝)-level via Hecke algebra
    isomorphisms: the Drinfeld case, Ann. Sci. Éc. Norm. Supér. (4), Volume 45, 2012,
    No.5,719–785 (2013).
[17] N. Katz, Serre-Tate local moduli, Algebraic surfaces (Orsay, 1976–78), Lecture Notes
    in Math., Volumn 868, 138–202, 1981.
[18] T. Katsura and F. Oort, Families of supersingular abelian surfaces, Compositio Math.
    Vol. 62, 1987, 107–167.
[19] N. Katz and B. Mazur, Arithmetic moduli of elliptic curves, Annals of Mathematics
    Studies, Volume 108, Princeton University Press, Princeton, NJ, 1985, xiv+514.
[20] N. Koblitz, 𝑝-adic variation of the zeta-function over families of varieties defined over
    finite fields, Compositio Math., Vol. 31, 1975, 119–218.
[21] R. Kottwitz and P. Wake, Primitive elements for 𝑝-divisible groups, Research in Number
    Theory, Vol. 3, 2017, Art. 20.
[22] S. Kudla and M. Rapoport, Cycles on Siegel threefolds and derivatives of Eisenstein
    series, Ann. Sci. École Norm. Sup. (4), Vol. 33, 2000, 695–756.
[23] J. Lipman, Joseph and M. Hashimoto, Foundations of Grothendieck duality for diagrams
    of schemes, Lecture Notes in Mathematics, Vol. 1960, Springer-Verlag, Berlin, 2009,
    x+478.
[24] H. Matsumura, Commutative ring theory, Cambridge Studies in Advanced Mathemat-
    ics, Vol. 8, Cambridge University Press, Cambridge, 1986, xiv+320.
[25] L. Moret-Bailly, Familles de courbes et de variétés abéliennes sur ℙ1 , Seminar on Pencils
    of Curves of Genus at Least Two, Astérisque No. 86, 1981.
[26] V. B. Mehta and V. Srinivas, Varieties in positive characteristic with trivial tangent
    bundle, With an appendix by Srinivas and M. V. Nori, Compositio Math., Vol. 64, 1987,
    No. 2, 191–212.
[27] D. Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in
    Mathematics, No. 5, 1970, viii+242.
[28] D. Mumford and J. Fogarty and F. Kirwan, Geometric invariant theory, Ergebnisse der
    Mathematik und ihrer Grenzgebiete (2), Vol. 34, 3rd Edition, Springer-Verlag, Berlin,
    1994, xiv+292.
[29] T. Oda, The first de Rham cohomology group and Dieudonné modules, Ann. Sci. École
    Norm. Sup. (4), Vol. 2, 63–135, 1969.
[30] F. Oort, A stratification of a moduli space of abelian varieties, Moduli of abelian varieties
    (Texel Island, 1999), Progr. Math. Vol. 195, Birkhäuser, Basel, 2001.
                                               79


[31] F. Oort, A stratification of a moduli space of polarized abelian varieties in positive char-
    acteristic, Moduli of curves and abelian varieties, Aspects Math., E33, Friedr. Vieweg,
    Braunschweig, 2001.
[32] F. Oort, Which abelian surfaces are products of elliptic curves?, Math. Ann. Vol. 214,
    1975, 25–47.
[33] J. Tate and F. Oort, Group schemes of prime order, Ann. Sci. École Norm. Sup. (4),
    Volume 3, 1970, 1–21.
[34] F. Oort and T. Zink, Families of 𝑝-divisible groups with constant Newton polygon, Doc.
    Math., Vol. 7, 2002, 183–201.
[35] G. Pappas, Arithmetic models for Hilbert modular varieties, Compositio Math., Vol.
    98, 1995, 43–76.
[36] G. Pappas, Letter to Robert Kottwitz and Preston Wake, 2016.
[37] M. Rapoport and T. Zink, Period spaces for 𝑝-divisible groups, Annals of Mathematics
    Studies, Vol. 141, Princeton University Press, 1996, xxii+324.
[38] M. Raynaud, Schémas en groupes de type (𝑝, … , 𝑝), Bull. Soc. Math. France, Vol. 102,
    1974, 241–280.
[39] J.-P. Serre, Local fields, Graduate Texts in Mathematics, Vol. 67, Springer-Verlag, New
    York-Berlin, 1979, viii+241.
[40] T. Shioda, Supersingular 𝐾3 surfaces, Algebraic geometry (Proc. Summer Meeting,
    Univ. Copenhagen, Copenhagen, 1978), Lecture Notes in Math., Vol. 732, 564–591,
    Springer, Berlin, 1979.
[41] J. Voight, Introduction to finite group schemes, Online notes,
    https://math.dartmouth.edu/~jvoight/notes/274-Schoof.pdf.
[42] P. Wake, Full level structures revisited: pairs of roots of unity, Journal of Number
    Theory, Volume 168, 2016, 81–100.
[43] W. Waterhouse, Principal homogeneous spaces and group scheme extensions, Trans.
    Amer. Math. Soc., Vol. 153, 1971, 181–189.
                                              80