PRIMETORSIONINTHEBRAUERGROUPOFANELLIPTICCURVE
By
CharlotteUre
ADISSERTATION
Submittedto
MichiganStateUniversity
inpartialentoftherequirements
forthedegreeof
Mathematics|DoctorofPhilosophy
2019
ABSTRACT
PRIMETORSIONINTHEBRAUERGROUPOFANELLIPTICCURVE
By
CharlotteUre
TheBrauergroupisaninvariantinalgebraicgeometryandnumbertheory,thatcanbe
associatedtoaeld,variety,orscheme.Let
k
beaofcharacteristictfrom2or
3,andlet
E
beanellipticcurveover
k
.TheBrauergroupof
E
isatorsionabeliangroup
withelementsgivenbyMoritaequivalenceclassesofcentralsimplealgebrasoverthefunction

k
(
E
).TheMerkurjev-Suslintheoremimpliesthatanysuchelementcanbedescribed
byatensorproductofsymbolalgebras.Wegiveadescriptionofelementsinthe
d
-torsion
oftheBrauergroupof
E
intermsofthesetensorproducts,providedthatthe
d
-torsionof
E
is
k
-rationaland
k
containsaprimitive
d
-throotofunity.Furthermore,if
d
=
q
isa
prime,wegiveanalgorithmtocomputethe
q
-torsionoftheBrauergroupoveranyld
k
ofcharacteristictfrom2,3,and
q
containingaprimitive
q
-throotofunity.
Tomyfamily.
iii
ACKNOWLEDGMENTS
Thesepastsixyearsofgraduateschoolhavebeenanincredibleexperienceforme.Iam
stillamazedtoseehowmuchIhavegrownbothpersonallyandacademicallywiththe
encouragementandsupportIreceivedfromthepeopleIencountered.
Firstandforemost,IwouldliketothankmyadvisorRajeshKulkarni,withoutwhom
thisthesiswouldnotexistinthisform.Thankyouforalwayshavinganopendoorandfor
guidingmethroughoutthisprocess.ThankyouforlettingmestrugglewhenIneededto,
butespeciallyforencouragingmewhenIwantedtosurrender.Thankyouforbeingnotonly
myadvisorandmentor,butalsobecomingagoodfriend.
ThefacultyandatMichiganStateUniversityhavebeenaconstantsourceofsupport
overtheyears.Iwouldn'thavecometoMichiganStateUniversitywithoutCasimAbbas
whooriginallymademeawareoftheprogram.Mystudywaspartiallysupportedbythe
StudienstiftungdesDeutschenVolkes.AtMichiganStateUniversity,Iamespeciallygrateful
fortheworkofmycommitteemembers:IgorRapinchuk,MichaelShapiro,AaronLevin,and
GeorgePappas.
IhaveencounterednumerouswonderfulpeoplewithoutwhomthedaysinWellsHall
wouldhavebeenverydull.ThankyouReshmaandHiteshforcountlessconversations.
Thankyou

Akosforalwaysbrighteningmydayswithyourjokes.Thankyoutomy
AmericanroommatesandgoodfriendsSamanthaandAllisonformakingEastLansingfeel
likehome.ThankyouChristine,Christos,Dimitris,Michael,Mollee,Rami,Rani,Sami,
Sarah,Sebastian,Sugil,Tyler,andcountlessothersforagreattimetogether.Finally,thank
youtoallmyfellowparticipantsoftheStudentAlgebraSeminaratMSUwhoinspiredme
toworkharder.
iv
Iwouldalsoliketothankmyfamilyfortheircontinuedsupportandfortheiruncondi-
tionalbeliefinmyabilities.HereistomyparentsInaandEwaldwhoaremyrolemodels
forlifelonglearning.ThankyoutomysisterLenaforbeingtherewheneverIneededher.
ThankyouOmiInge,whoisalwaysinterestedinmystoriesandwhoseindependencecontin-
uestobeaninspirationtome.ThankyouOmaMariaandOpaSeppl,whoseunconditional
supportmeanstheworldtome.
MyspecialthanksgoestomyeSteviewhocanalwaysmakemesmile!
v
TABLEOFCONTENTS
KEYTOSYMBOLS
..................................
viii
Chapter1Introduction
...............................
1
Chapter2Background
................................
10
2.1EllipticCurves..................................10
2.1.1TorsionPoints...............................11
2.1.2Isogenies..................................12
2.1.3WeilPairing................................14
2.2Cohomology....................................15
2.2.1GroupCohomologyforabstractgroups.................15
2.2.2MapsonGroupCohomology.......................16
2.2.3GroupCohomologyforGroups................18
2.2.4Torsorsand
H
1
..............................19
2.2.5TheBrauerGroup,SymbolAlgebras,and
H
2
.............21
2.2.6TheBrauergroup......................24
2.3Anexactsequence................................25
Chapter3Torsorgivenbymultiplicationby
d
.................
28
3.1Overawithrationaltorsion........................28
3.2Overany...................................30
3.3GeneralArgumentthat

inducesthecorrectsplit...............33
3.3.1DigressiontoDerivedCategories....................35
3.3.2Applicationtoourcase..........................38
Chapter4Generatorsof
Br(
E
)...........................
40
4.1
M
is
k
-rational..................................42
4.2[
L
:
k
]iscoprimeto
q
..............................46
4.3[
L
:
k
]equals
q
..................................48
4.3.1TheImageoftheMap.....................49
4.3.2TheImageoftheRestrictionMap....................53
4.4
q
dividesthedegree[
L
:
k
]............................56
Chapter5Relations
.................................
58
5.1
M
is
k
-rational..................................58
5.2[
L
:
k
]iscoprimeto
q
...............................60
5.3[
L
:
k
]=
q
.....................................61
5.4
q
divides[
L
:
k
]..................................62
Chapter6Conclusions{TheAlgorithm
....................
64
vi
Chapter7Examples
.................................
67
7.1
M
is
k
-rationaloveranumber.......................67
7.2Degree
L=k
coprimeto
q
for
k
anumber.................68
7.3Degree
L=k
=
q
for
k
anumber......................74
7.4Positiverankoveranumber........................76
7.5Overalocal.................................78
7.6Overa.................................79
7.7Degree[
L
:
k
]divisibleby
q
...........................80
APPENDIX
........................................
82
BIBLIOGRAPHY
....................................
86
vii
KEYTOSYMBOLS
d;qd
integer

2,
q
isanoddprime
k
baseofcharacteristictfrom2or3,coprimeto
d
,
q
ˆ
primitive
d
-thor
q
-throotofunityin
k
F
aseparableclosureofthed
F
G
F
theabsoluteGaloisgroupofthe
F
E
anellipticcurveover
k

;

denotethepointadditionandsubtractionon
E
[
d
]
;
[
q
]multiplicationby
d
or
q
mapon
E
(
k
)
Md
-torsionor
q
-torsionof
E
(
k
)
e(
:;:
)theWeilpairingon
M
,section2.1.3
P;Q
generatorsof
M
withe(
P;Q
)=
ˆ
k
(
E
)Functionof
E
H
i
(
F;A
)the
i
-thgroupcohomologyof
G
F
withcotsinthe
G
F
-module
A
,
section2.2
restherestrictionmap,example2.2.3
corthecorestrictionmap,example2.2.6
infthemap,example2.2.4
[
:
]
ˆ
for
ˆ
i
2

d
,let

ˆ
i

ˆ
=
i
2
Z
=d
Z
:
[
:
]
Z
ˆ
for
ˆ
i
2

d
,let

ˆ
i

Z
ˆ
=
i
2f
0
;:::;d

1
gˆ
Z
:
viii
Chapter1
Introduction
TheBrauergroupisanimportantinvariantassociatedwitharing,variety,orascheme
ingeneral.OveraitselementsaregivenbyMoritaequivalenceclassesofcentralsim-
plealgebras.FirstintroducedbyBrauerinthe1920's,theBrauergroupsofglobal
werecompletelydescribedbyAlbert,Brauer,Hasse,andNoether[BNH32].TheBrauer
groupsofpurelytranscendentalextensionsofglobalareeasiertounderstand.They
werecalculatedbytheauthorsin[FSS79].Theydeducethatallrationalfunctionover
aglobalhaveisomorphicBrauergroups,independentlyoftheirtranscendence
degree.TheofaBrauergroupwasgeneralizedtocommutativeringsbyAuslan-
derandGoldman[AG60].Inthe1960's,GrothendieckdescribedaversionoftheBrauer
groupofaschemeusingetalecohomology[Gro68a].Itistobe
H
2
et
(
X;
G
m
),where
G
m
isthesheafofmultiplicativeunitsoftheringofregularfunctionson
X
.TheBrauer
groupalsohasanalgebraicincarnationthatwedenotebyBr(
X
).ItselementsareMorita
equivalenceclassesofsheavesofAzumayaalgebras.Thesearesheavesof
O
X
-algebrasthat
areetalelocallyisomorphictomatrixalgebras.ThisrealizationdescribestheBrauergroup
asatorsionabeliangroup.Ifascheme
X
admitsanampleinvertiblesheaf,thetorsionof
thecohomologicalBrauergroupcoincideswiththegroupofequivalenceclassesofAzumaya
algebras[dJ].ElementsintheBrauergroupmayalsobethoughtofastranscendentalcycles
{complementstoalgebraiccycles.WeareinterestedinBr(
E
)for
E
anellipticcurve.
1
TheBrauergrouphasprovenusefulinstudyinggeometricpropertiesofvarieties.Forexam-
ple,ArtinandMumfordutilizedittogiveanegativeanswertotheLurothproblemindimen-
sionthree.TheyconstructedunirationalvarietiesthatarenotrationalusingBrauerclasses
overthefunctionoftheprojectivespace
P
2
[AM72].TheBrauergroupcanalsodetect
arithmeticpropertiesoftheunderlyingvariety.Maninanobstructionlyinginthe
BrauergroupthatmeasuresthefailureoftheHasseprincipleforvarieties[Man71,Sko01].
UsinggeometricconstructionsofBrauerclasses,VirayandCreutzdescribedtheBrauer-
Maninobstructionexplicitlyinthecaseofhyperellipticcurves.Usingthis,theyconstructed
anfamilyofabeliansurfacesover
Q
withnontrivialTate-Shafarevichgroup[CV15].
Throughoutthisnote,let
k
beaofcharacteristictfrom2and3andlet
E
be
anellipticcurveover
k
.WewillexploretheBrauergroupof
E
.First,recallthattheBrauer
groupof
E
isatorsionabeliangroupandthereforeitwillbeenoughtodescribethetorsion
d
Br(
E
),where
d

2isaprimepower.TheBrauergroupof
E
isnaturallyisomorphicto
theBrauergroupofthefunction
k
(
E
)(section2.2.6and[CTS07,Theorem
5.11]).Furthermore,MerkurjevandSuslinrelatethesecondMilnor
K
2
withtheBrauer
groupinthefollowingtheoremfrom[MS82]and[Mer86].
Theorem1.0.1
(Merkurjev-Suslin)
.
Let
F
beaandlet
d

2
beaninteger.Assume
additionallythat
F
containsaprimitive
d
-throotofunity
ˆ
.Thereisanisomorphism
K
2
(
F
)
=dK
2
(
F
)
!
d
Br(
F
)
2
thattakesasymbol
f
a;b
g
tothesymbolalgebra
(
a;b
)
d;F
=(
a;b
)
d;F;ˆ
=
F
D
x;y
:
x
d
=
a;y
d
=
b;xy
=
ˆyx
E
:
Weapplythistheoremtothecase
F
=
k
(
E
)anddeducethateveryelementintheBrauer
groupof
k
(
E
)canbewrittenasatensorproductofsymbolalgebrasover
k
(
E
).Ourmain
goalisthefollowing.
Goal1.0.2.
Let
k
beaofcharacteristicentfrom
2
and
3
.Fixanellipticcurve
E
over
k
.Let
d

2
beanintegercoprimetothecharacteristicof
k
andassumeadditionally
that
k
containsaprimitive
d
-throotofunity.Describegeneratorsandrelationsof
d
Br(
E
)
astensorproductofsymbolalgebrasoverthefunction
k
(
E
)
.
Suchadescriptionisavailableforcertainintegers
d
and
k
.In[CG01],Chernousov
andGuletskidescribegeneratorsandrelationsof
2
Br(
E
)foranyellipticcurveoverany
ofcharacteristictfrom2.Inparticular,theyprovethefollowingtheorem[CG01,
Theorem3.6].
Theorem1.0.3.
Let
k
beaofcharacteristicentfrom
2
andlet
E
beanelliptic
curveover
k
dbytheequation
y
2
=(
x

a
)(
x

b
)(
x

c
)
with
a;b;c
2
k
.Then
2
Br(
E
)=
2
Br(
k
)

I
andeveryelementin
I
canbepresentedbya
tensorproduct
(
r;x

b
)
2
;k
(
E
)

(
s;x

c
)
2
;k
(
E
)
ofquaternionalgebraswith
r;s
2
k

.Any
suchalgebraistrivialin
I
ifandonlyifitissimilartooneofthefollowing
3

(
u

c;x

b
)
2
;k
(
E
)

(
u

b;x

c
)
2
;k
(
E
)
,where
u
isthe
x
-coordinateofapointin
E
(
k
)
suchthat
u

b
6
=0
and
u

c
6
=0
,

(
b

c;x

b
)
2
;k
(
E
)

((
b

c
)(
b

a
)
;x

c
)
2
;k
(
E
)
,or

((
c

a
)(
c

b
)
;x

b
)
2
;k
(
E
)

(
c

b;x

c
)
2
;k
(
E
)
.
ChernousovandGuletskifurtherdescribe
2
Br(
E
)iftheequation
y
2
=
f
(
x
)for
E
hasoneornorootsover
k
.Notethatthetwotorsionof
E
(
k
)is
k
-rationalifandonlyif
f
(
x
)
admitsthreerootsin
k
.Theauthorsin[CRR16,Section6]givegeneratorsofthe
d
-torsion
oftheBrauergroupoftheJacobianofacurveusingatmethod.Theyprovethe
followingtheorem.
Theorem1.0.4.
Let
C
beasmoothprojectivegeometricallyirreduciblecurveofgenus
g
overa
k
.Let
d

2
beanintegercoprimetothecharacteristicof
k
andsupposethat
k
containsaprimitive
d
-throotofunity.DenotetheJacobianof
C
by
J
.Supposethat
d
J
(
k
)
is
k
-rational.Fixabasis
P
1
;:::P
2
g
of
d
J
(
k
)
˘
=
(
Z
=d
Z
)
2
g
.Pickadivisor
^
P
i
representing
P
i
andlet
t
P
i
2
k
(
C
)
betheelementwithdivisor
d
^
P
i
.Then
d
Br(
k
(
C
))
ur
=
d
Br(
k
)

I
andeveryelementin
I
canbewrittenasatensorproduct

a
1
;t
m
1
P
1

d;k
(
C
)


a
2
g
;t
m
2
g
P
2
g

d;k
(
c
)
;
forsome
a
1
;:::;a
2
g
2
k

andsomeintegers
m
1
;:::;m
2
g
.
4
Inthisthesis,wewillgiveadescriptionof
d
Br(
E
)inthefollowingcases:
1.
Any
d
coprimetothecharacteristicof
k
,assumingthat
k
containsaprimitive
d
-th
rootofunityandthe
d
-torsion
M
of
E
(
k
)is
k
-rational(theorem1.0.5).
2.
d
=
q
anoddprime,thatiscoprimetothecharacteristicof
k
,assumingonlythat
k
containsaprimitive
q
-throotofunity(chapter6).
Notethatwerecovertheresultfrom[CG01]iftheellipticcurveissplitandalsotheresult
from[CRR16]inthecaseofanellipticcurve.Wenowproceedtogiveourdescriptioninthe
case.
Theorem1.0.5.
Let
k
beaofcharacteristicentfrom
2
and
3
.Fixaninteger
d

2
coprimetothecharacteristicof
k
andassumethat
k
containsaprimitive
d
-throotof
unity.Let
E
beanellipticcurveover
k
with
k
-rational
d
-torsion
M
.Fixtwogenerators
P
and
Q
of
M
,andlet
t
P
;t
Q
2
k
(
E
)
with
div(
t
P
)=
d
(
P
)

d
(0)
and
div(
t
Q
)=
d
(
Q
)

d
(0)
.
Additionally,assumethat
t
P

[
d
]
;t
Q

[
d
]
2

k
(
E
)


d
.Then
d
Br(
E
)=
d
Br(
k
)

I
andeveryelementin
I
canberepresentedbyatensorproduct
(
a;t
P
)
d;k
(
E
)

(
b;t
Q
)
d;k
(
E
)
forsome
a;b
2
k

.Furthermore,suchatensorproductistrivialifandonlyifitissimilar
tooneofthefollowing


t
Q
(
P
)
;t
P

k
(
E
)


t
P
(
P

Q
)
t
P
(
Q
)
;t
Q

k
(
E
)
,


t
Q
(
P

Q
)
t
Q
(
P
)
;t
P

k
(
E
)


t
P
(
Q
)
;t
Q

k
(
E
)
;
or


t
Q
(
R
)
;t
P

k
(
E
)


t
P
(
R
)
;t
Q

k
(
E
)
forsome
R
2
E
(
k
)
nf
0
;P;Q
g
.
5
Remark1.0.6.
Wemaydroptheassumptionthat
t
P

[
d
]
;t
Q

[
d
]
2

k
(
E
)


d
intheprevious
theorem.Inthiscaseourrelationsbecome

t
Q
(
R

S
)
t
Q
(
S
)
;t
P

k
(
E
)


t
P
(
R

S
)
t
P
(
S
)
;t
Q

k
(
E
)
forsome
R
2
E
(
k
)
and
S
2
E
(
k
)
isanypointsothatthefractionexistsandisnonzero.
Proof.
Let
t
P
;t
Q
2
k
(
E
)withdiv(
t
P
)=
d
(
P
)

d
(0)anddiv(
t
Q
)=
d
(
Q
)

d
(0).Fixsome
point
P
0
2
E
(
k
)with[
d
]
P
0
=
E
(
k
).Thereexists
g
P
2
k
(
E
)withdiv(
g
P
)=[
d
]

(
P
)

[
d
]

(0)=
P
R
2
M

P
0

R

:
NotethatthedivisorisinvariantundertheactionoftheGalois
group
G
k
andthereforewemaychoose
g
P
2
k
(
E
).Furthermore,sincethedivisorscoincide,
thereissome

2
k
sothat
g
d
P
=

P

[
d
]
2
k
(
E
)
d
.Forany
R
2
E
(
k
)

P
(
R
)=

P
(
R

S
)

P
(
S
)
=
t
P
(
R

S
)
t
P
(
S
)
;
where
S
isanyotherpointsothat
t
P
isnonzeroandwd.
Now,let
q
beanoddprimenotequaltothecharacteristicof
k
andsupposethat
k
containsaprimitive
q
-throotofunity.Wegiveanalgorithmtodescribe
q
Br(
E
)explicitly
inchapter6.ConsiderthestandardGaloisrepresentation
:
G
k
!
Aut(
M
)=
GL
2

F
q

anddenotetheofitskernelby
L
.Then
L
isthesmallestGaloisextensionof
k
so
that
M
is
L
-rational.Notethatthedegreeof
L
over
k
dividestheorderof
GL
2

F
q

,which
is(
q
+1)
2
q
(
q

1).Weconsiderthreecasesforthedegreeof
L
over
k
:
6
1.
q
-
[
L
:
k
]
Wedescribegeneratorsandrelationsin
q
Br(
E
)usingthatrestrictionfollowedbycore-
strictionisanisomorphism(section4.2andsection5.2).
2.
q
=[
L
:
k
]
Inthiscase,thecompositionofrestrictionandcorestrictionisthezeromap.Weuse
insteadtherestrictionexactsequencetodeterminegeneratorsandrelations
of
q
Br(
E
)(section4.3andsection5.3).
3.
q
j
[
L
:
k
]
Wecombinetheresultsfromtheprevioustwocasestogetadescriptionof
q
Br(
E
)
(section4.4andsection5.4).
WenowproceedtoreviewthemainideaswewillusetodeterminetheBrauergroup.
Foranyinteger
d

2,thereisasplitexactsequence
0
/
/
d
Br(
k
)
/
/
d
Br(
E
)
/
/
q
H
1
(
k;E
(
k
))
/
/
0
;
(1.1)
whichisinducedbytheHochschild-Serrespectralsequence(formoredetails,seesection2.3).
Weneedanexplicitsplittingtothissequenceontheright.ConsidertheKummersequence
0
/
/
M
/
/
E
(
k
)
[
d
]
/
/
E
(
k
)
/
/
0(1.2)
andtheinducedsequenceongroupcohomology
0
/
/
E
(
k
)
=
[
d
]
E
(
k
)
/
/
H
1
(
k;M
)

/
/
q
H
1
(
k;E
(
k
))
/
/
0
:
(1.3)
7
Wewillamap

:
H
1
(
k;M
)
!
d
Br(
E
)thatinducesthedesiredsplit.Thismapis
givenbythefollowingcomposition

:
H
1
(
k;M
)
˘
/
/
H
1
et
(Spec(
k
)
;M
)
p

/
/
H
1
et
(
E;M
)

[
T
]
/
/
H
2
et
(
E;M

M
)
e
/
/
H
2
et
(
E;
d
)
/
/
d
Br(
E
)
;
(1.4)
where
p

isthemorphisminducedbythestructuremap
p
:
E
!
Spec
k
,
T
isthetorsor
givenbymultiplicationby
d
ontheellipticcurve(seechapter3),andeisthemapinduced
bytheWeil-pairing(seesection2.1.3).Denoteby
I
theimageof

Wededucethat
d
Br(
E
)=
d
Br(
k
)

I:
Further,anelementin
H
1
(
k;M
)becomestrivialunder

ifandonlyifitisintheimageof

fromsequence1.1.
In[Sko01]and[Sko99],theauthordescribesthemap

abstractlyinthemoregeneral
settingofanabelianvariety
X
andanytorsor
T
on
X
.Hefurtherprovesthatsuchan

inducesthedesiredsplit.Wereviewhisabstractproofinsection3.3.Inthisthesis,wemake
thisconstructionexplicitandprovedirectlythatthemap

inducedbythecup-product
inducesasplit.
Thisthesisisorganizedasfollows.Inchapter2,wereviewbackgroundmaterialthat
willbeusedthroughoutthiswork.Forexample,wediscusssomefactsaboutellipticcurves,
cohomology,Brauergroups,andsymbolalgebrasasnecessaryfortheproofsinthefollowing
chapters.Wealsonotationthatwillbeusedthroughoutthiswork.Inchapter3,we
8
describethetorsorgivenbymultiplicationby
d
ontheellipticcurve.Wealsocalculate
thecocyclecorrespondingtothistorsoratthegenericpoint.Furthermore,wedescribethe
generaltheorybehindourmainresultandjustifyitusingderivedcategories.Inchapter4,
weexplorethealgorithmtocalculategeneratorsoftheBrauergroup.Wedescribethese
generatorsprovidedthat
M
is
k
-rational.Wethenproceedtodescribethegeneratorsofthe
primetorsioninvariouscases.Finally,inchapter5wegivetherelationsinthetorsionofthe
Brauergroup.Wesummarizeourresultsandgiveacompletealgorithmtodeterminethe
oddprimetorsionintheBrauergroupinchapter6.Lastly,wedeterminetheprimetorsion
oftheBrauergroupinvariousexamplesinchapter7.
9
Chapter2
Background
Thischaptercontainsbackgroundmaterialthatwillbeusedthroughoutthiswork.We
discusspropertiesofisogeniesofellipticcurvesandtheWeilpairing.Inasecondpart,we
reviewsomegeneralresultsongroupcohomologyandetalecohomology.Wefurtherconnect
thesetothenotionsoftorsorsandtheBrauergroup.Weproceedtotheunra
BrauergroupofaFinally,wedescribeanexactsequence,thatwillproveusefulto
determinetheBrauergroupofanellipticcurve.
2.1EllipticCurves
Let
E
beanellipticcurveovera
k
ofcharacteristictfrom2and3.Then
E
canbedescribedbyanequation
y
2
=
x
3
+
Ax
+
B
with
A;B
2
k
.Thediscriminant=

16

4
A
3
+27
B
2

of
E
isnonzero.Furthermore,
E
is
anabelianvarietywithidentitythepoint0=[0:1:0]aty.Wedenotethepointwise
additionon
E
(
k
)by

andthepointwisesubtractionby

.
10
2.1.1TorsionPoints
Let
d
beanaturalnumberandanalgebraicclosure
k
of
k
.Wedenoteby
[
d
]:
E
(
k
)
!
E
(
k
)
R
7!
R

R
|
{z
}
d
-times
themultiplicationby
d
mapontheellipticcurve.Thekernelof[
d
]arethe
d
-torsionpointsof
E
(
k
),whichisdenotedby
M
.Weextendourofmultiplicationto[

d
]=[

1]

[
d
].
Further,[0]istheconstantmapto0.
Wenowreviewdivisionpolynomials,whichareacomputationaltooltodetermine
M
.
Prop2.1.1.
Foranyinteger
d
,
d

2
and
P
=(
x
1
;y
1
)
2
E
(
k
)
,there
arepolynomials
 
n
2
k
[
x;y
]
for
n
2
Z
+
sothat
[
d
]
P
=
 
x
1

 
d

1
(
x
1
;y
1
)
 
d
+1
(
x
1
;y
1
)
 
2
d
(
x
1
;y
1
)
;
 
2
d
(
x
1
;y
1
)
2
 
4
d
(
x
1
;y
1
)
!
:
Thesepolynomialsareexplicitlygivenby
 
1
(
x;y
)=1
;
 
2
(
x;y
)=2
y;
 
3
(
x;y
)=3
x
4
+6
Ax
2
+12
Bx

A
2
;
 
4
(
x;y
)=4
y

x
6
+5
Ax
4
+20
Bx
3

5
A
2
x
2

4
ABx

8
B
2

A
3

;
 
2
m
+1
(
x;y
)=
 
m
+2
 
3
m

 
m

1
 
3
m
+1
;
for
m

2
and
 
2
m
(
x;y
)=

 
m
2
y


 
m
+2
 
2
m

1

 
m

2
 
2
m
+1

;
for
m

3
:
11
Proof.
Seeforinstance[Lan78,Ch.2
x
1].
Example2.1.2.
Let
P
=(
x
1
;y
1
)
2
E
(
k
)
.Then
P
isatwo-torsionpointifandonlyif
P
=

P
.If
E
isgivenbytheequation
y
2
=(
x

a
1
)(
x

a
2
)(
x

a
3
)
with
a
1
;a
2
;a
3
2
k
,
then
2
E
(
k
)=
f
0
;
(
a
1
;
0)
;
(
a
2
;
0)
;
(
a
3
;
0)
g
˘
=
Z
=
2
Z

Z
=
2
Z
:
Example2.1.3.
Let
P
=(
x
1
;y
1
)
2
E
(
k
)
.Then
P
isathree-torsionpointifandonlyif
P
=

[2]
P
.Thisisequivalentto
x
1
=
x
1

3
x
4
1
+6
Ax
2
1
+12
Bx
1

A
2
4
y
2
1
=
x
1

3
x
4
1
+6
Ax
2
1
+12
Bx
1

A
2
4

x
3
1
+
Ax
1
+
B

;
whichistrueifandonlyif
3
x
4
1
+6
Ax
2
1
+12
Bx
1

A
2
=0
:
Let
x
1
;:::;x
4
bethefourdistinctzerosofthispolynomial,andlet
y
i
=
q
x
3
i
+
Ax
i
+
B
,
then
3
E
(
k
)=
f
0
;
(
x
i
;y
i
)
;
(
x
i
;

y
i
):1

i

4
g
˘
=
Z
=
3
Z

Z
=
3
Z
:
2.1.2Isogenies
Let
E
1
and
E
2
betwoellipticcurvesovera
k
.Anisogenyisamorphism
˚
:
E
1
!
E
2
ofcurvessuchthat
˚
(0)=0.Itfollowsthatanisogenyisagrouphomomorphism
onthesetofpoints[Sil09,ChapterIII,Theorem4.8].Wedenoteby
˚

:
k
(
E
2
)
!
k
(
E
1
)
theinducedmaponfunctionIf
˚
isconstant,wesetthedegreeof
˚
tobezero.Ifit
12
isnotconstant,thedegreeof
˚
isthedegreeoftheextension
deg
˚
=[
k
(
E
1
):
˚

k
(
E
2
)]
:
Furthermore,wedenotebydeg
s
˚
theseparabledegreeoftheextensionandbydeg
i
˚
theinseparabledegree.Finally,let
P
2
E
1
(
k
).Theindexof
˚
at
P
,denoted
by
e
˚
(
P
)is
e
˚
(
P
)=ord
P

˚

t
˚
(
P
)

;
where
t
˚
(
P
)
isauniformizerat
˚
(
P
).Thefollowingclassicaltheoremcanforinstancebe
foundin[Sil09,Theorem4.10].
Proposition2.1.4.
Let
˚
:
E
1
!
E
2
beanonzeroisogenybetweenellipticcurves.
1.
Forevery
Q
2
E
2
(
k
)
,thenumberofpreimagesof
Q
under
˚
isequaltothedegree
deg
s
˚
.Furthermore,forevery
P
2
E
1
(
k
)
,
e
˚
(
P
)=deg
i
˚
.
2.
Thereisanisomorphism
ker(
˚
)
!
Aut

k
(
E
1
)
=˚

k
(
E
2
)

:
T
7!
˝

T
;
where
˝
T
isthetranslationby
T
map,
˝
T
:
E
(
k
)
!
E
(
k
):
R
7!
R

T
.
3.
Supposethat
˚
isseparable,then
˚
isunrd,thenumberofelementsinthekernel
isequaltothedegreeof
˚
,and
k
(
E
1
)
isaGaloisextensionof
˚

k
(
E
2
)
.
Example2.1.5.
Let
d

2
beanintegerandsupposethatthecharacteristicof
k
iscoprime
to
d
.Denotethe
d
-torsionof
E
(
k
)
by
M
˘
=
Z
=d
Z

Z
=d
Z
.Multiplicationby
[
d
]
isadegree
d
2
mapand
k
(
E
)
=
[
d
]

k
(
E
)
isaGaloisextensionofdegree
d
2
(formoredetails,seealso
13
[Sil09,ChapterIII]).Furthermore,
k
(
E
)
=
[
d
]

k
(
E
)
isGaloisofdegree
d
2
providedthat
M
is
k
-rational.
2.1.3WeilPairing
Let
E
beanellipticcurveover
k
andlet
D
=
P
n
R
R
2
Div(
E
)beadivisor.By[Sil09,
ChapterIII,Corollary3.5],
D
isaprincipaldivisorifandonlyif
P
R
2
E
(
k
)
n
R
=0and
P
R
2
E
(
k
)
[
n
R
]
R
=0,wherethesumisregularadditionintheintegersandthesecond
sumisgivenbyadditionontheellipticcurve.
AkeytoolinthestudyofellipticcurvesistheWeilpairingwhoseconstructionwewillnow
review.Formoredetails,seealso[Sil09,SectionIII.8].Let
d

2beanintegerandlet
P
be
a
d
-torsionpointon
E
.Asbefore,denoteby
M
the
d
-torsionof
E
(
k
).Asdiscussedbefore,
thereexistssome
f
P
2
k
(
E
)sothat
div(
f
P
)=
d
(
P
)

d
(0)
:
Nowlet
P
0
2
E
(
k
)suchthat[
d
]
P
0
=
P
.Since
P
R
2
M
(
P
0

R
)=[
d
2
]
P
0
=[
d
]
P
=0,there
existssome
g
P
2
k
(
E
)suchthat
div(
g
P
)=[
d
]

(
P
)

[
d
]

(0)=
X
R
2
M

(
P
0

R
)

(
R
)

:
Sincethedivisorscoincide,wemayassumethat
f
P

[
d
]=
g
d
P
.For
Q
2
M
the
Weil-pairingof
P
and
Q
as
e(
Q;P
)=
g
P
(
X

Q
)
g
P
(
X
)
;
14
where
X
2
E
(
k
)isanypointsothat
g
P
(
X
)and
g
P
(
X

Q
)isandnonzero.The
Weil-pairingtakesvaluesinthesetof
d
-throotsofunity.
Example2.1.6
(example2.1.5,contd.)
.
Let
P;Q
2
M
andlet
g
P
and
g
Q
bedas
before,then
˝

Q
(
g
P
)=e(
P;Q
)
g
P
;
(2.1)
where
˝
Q
isthetranslationby
Q
map,
˝
Q
:
E
(
k
)
!
E
(
k
):
R
7!
R

Q
.
2.2Cohomology
Inthissection,wesetupourcohomologicalnotation.Formoredetails,see[Ser79,Ch.VII
x
5],[NSW08],or[GS17].
2.2.1GroupCohomologyforabstractgroups
Throughoutthissectionlet
G
beagroupandlet
A
beanabeliangroup.Assumethat
G
actson
A
ontheleftvia
g:a
.
Let
C
i
bethesetofofmaps

n
i
=1
G
!
A
.Thecoboundarymap
d
:
C
i
!
C
i
+1
isgiven
by
df
(
g
1
;:::;g
i
+1
)=
g
1
:f
(
g
2
;:::;g
i
+1
)
+
j
=
i
X
j
=1
(

1)
j
f

g
1
;:::;g
j
g
j
+1
;:::;g
i
+1

+(

1)
i
+1
f
(
g
1
;:::;g
i
)
:
(2.2)
Adirectcomputationshowsthat
d
2
=0andthus(
C
i
;d
)isacomplex.Elementsinthe
imageof
d
arecalledcocyclesandelementsinthekernelof
d
arecochains.The
i
-thgroup
cohomology,denoted
H
i
(
G;A
),isthequotientofcocyclesbycochainsin
C
i
.
15
Example2.2.1.
A
1
-cocycleisamap
f
:
G
!
A
satisfying
f
(
gh
)=
g:f
(
h
)+
f
(
g
)
forall
g;h
2
G
.Wealsocalltheseelementscrossedhomomorphism.Acrossedhomomorphismis
trivialifthereexistssome
a
2
A
suchthat
f
(
g
)=
g:a

a
forall
g
2
G
.
Example2.2.2.
A
2
-cocycleisamap
f
:
G

G
!
A
suchthat
g:f
(
g
0
;g
00
)

f
(
gg
0
;g
00
)+
g
(
g;g
0
g
00
)

f
(
g;g
0
)=0
forall
g;g
0
;g
00
2
G:
Suchacocycleistrivialifandonlyifthereis
amap
~
f
:
G
!
A
suchthat
f
(
g;h
)=
g:
~
f
(
h
)

~
f
(
gh
)+
f
(
g
)
forall
g;h
2
G
.
2.2.2MapsonGroupCohomology
Let
G
and
G
0
begroups.Fixa
G
-module
A
anda
G
0
-module
A
0
.Let
˚
:
G
!
G
0
and
 
:
A
!
A
0
begrouphomomorphisms.Wesaythat
˚
and
 
arecompatibleif
 
(
˚
(
g
)
:a
)=
g: 
(
g
)
forall
g
2
G
and
a
2
A
.Apair(
˚; 
)ofcompatiblemorphismsinducesamaponcohomology
givenbypost-andprecomposition
(
˚; 
)

i
:
H
i
(
G;A
)
f
7!
f

˚
/
/
H
i
(
G
0
;˚

A
)
f
7!
 

f
/
/
H
i
(
G
0
;A
0
)
:
Example2.2.3
(Restriction)
.
Supposethat
H
isasubgroupof
G
.Theinclusionof
H
into
G
iscompatiblewiththeidentityon
A
.Theinducedhomomorphism
res:
H
i
(
G;A
)
!
H
i
(
H;A
)
iscalledtherestrictionhomomorphism.
Example2.2.4

.
If
H
isanormalsubgroupof
G
,wedenoteby
A
H
thesubgroup
of
A
ofelementsdby
H
.Theidentityon
G
iscompatiblewiththeinclusionof
A
H
16
into
A
.Theinducedhomorphism
inf:
H
i

G=H;A
H

!
H
i
(
G;A
)
iscalledthe
homomorphism.
Example2.2.5
(Conjugation,actionon
H
n
)
.
Let
H
ˆ
G
beasubgroup,
A
a
G
-module,and
B
an
H
-submoduleof
A
.Foranyd
˙
2
G
themorphisms
˙

1
H˙
!
H
:
h
7!
˙h˙

1
and
B
!
˙

1
B
:
b
7!
˙

1
b
arecompatibleandinduceisomorphisms
˙

:
H
n
(
H;B
)
!
H
n
(
˙

1
H˙;˙

1
B
)
called
conjugation
.Thisanactionof
G
(orif
H
isnormalin
G
of
G=H
)on
H
n
(
H;A
)
.
Example2.2.6
(Corestriction)
.
Supposethat
H
isasubgroupof
G
.Foreveryrightcoset
c
2
H
n
G
acosetrepresentative
c
2
c
ˆ
G
.Onthelevelofcocyclesthecorestriction
cor:
C
i
(
H;A
)
!
C
i
(
G;A
)
isgivenby
cor(
f
)(
g
1
;:::;g
i
)=
X
c
2
H
n
G
c

1
f

cg
1
cg
1

1
;:::;
cg
i
cg
i

1

:
Thefollowingpropositionsrelatingirestriction,andcorestrictionwillbeuseful
forourcalculationsinchapter4.
Proposition2.2.7.
Let
H
beanormalsubgroupof
G
andlet
A
bea
G
-module.Thenthe
followingsequenceisexact
0
/
/
H
1
(
G=H;A
H
)
inf
/
/
H
1
(
G;H
)
res
/
/
H
1
(
H;A
)
:
Proof.
See[Ser79,Ch.VII
x
6,Proposition4].
17
Proposition2.2.8.
Let
H
beasubgroupof
G
ofindexandlet
A
bea
G
-module.Then
thecomposition
H
i
(
G;A
)
res
/
/
H
i
(
H;A
)
cor
/
/
H
i
(
G;A
)
coincideswithmultiplicationby
[
G
:
H
]
.
Proof.
See[Ser79,Ch.VII
x
7,Proposition6].
2.2.9
(cup-product)
.
Let
G
beagroupand
A
and
B
be
G
-modules.The
cup-
product
onthelevelofcocyclesisgivenby
H
n
(
G;A
)

H
m
(
G;B
)
!
H
n
+
m
(
G;A

B
)
f
[
g
(
˙
1
;:::˙
n
+
m
)=
f
(
˙
1
;:::;˙
n
)
˙
n
g
(
˙
n
+1
;:::˙
m
+
n
)
:
2.2.3GroupCohomologyforGroups
Inthissection,weextendthepreviousofgroupcohomologytonitegroups.
Recallthatagroup
G
istheinverselimitlim
 
G

ofaninversesystemof
groups
f
G

g
.Withoutlossofgeneralitywemaytake
G

tobeaquotient
G=U

of
G
byan
opennormalsubgroup
U

.
Example2.2.10.
Let
K
over
k
beaGaloisextension.TheGaloisgroup
Gal(
K=k
)
isthe
inverselimitoftheinversesystemoftheGaloisgroupsofsubextensionsandassuch
isaprgroup.
Aitegroup
G
=lim
 
G

admitsanaturaltopologyasfollows.Considerthediscrete
topologyon
G

andtheinducedproducttopologyon
Q
G

.Weendow
G
withthesubspace
topolgyoftheproduct.Acontinuous
G
-moduleisa
G
-module
A
sothatthestabilizerof
each
a
2
A
isopenin
G
.
18
2.2.11.
Let
G
=lim
 
G

beaprgroupwith
G

=
G=U

andlet
A
bea
continuous
G
-module.Then
A
U

isa
G

-module.Considerthemaps
inf


:
H
i

G

;A
U


!
H
i

G

;A
U


asinexample2.2.4.the
i
-thgroupcohomology
H
i
cont
(
G;A
)
asthedirectlimitofthe
system

H
i

G

;A
U


;
inf



.
FortheabsoluteGaloisgroup
G
k
ofa
k
,wedenote
H
i
(
k;A
)=
H
i
cont
(
G
k
;A
)
.
Remarkthatbyconstructionofdirectlimitsasaquotientofthedirectsum,forevery
f
2
H
i
cont
(
G;A
)thereexistssome

sothat
f
2
H
i

G

;A
U


.Weusethisidenation
throughoutthefollowingchapters,particularlyfor
H
i
(
k;A
).
Let
H
beaclosedsubgroupofaitegroup
G
and
A
acontinuous
G
-module.The
restrictionmapres:
H
i
cont
(
G;A
)
!
H
i
cont
(
H;A
)isasthedirectlimitoftheusual
restrictionongroupcohomology(example2.2.3).If
H
isopenin
G
,thecontinuouscorestric-
tionissimilarlyandif
H
isaclosednormalsubgroup,wecanconstructcontinuous
maps.Furthermore,thereisacontinuouscupproductinducedbythecupprod-
uctin2.2.9.Finally,propositions2.2.7and2.2.8canberecoveredfor
cohomology[GS17,Chapter4.2and4.3].
2.2.4Torsorsand
H
1
Wenowproceedtodescribethecorrespondencebetweentorsorsandelementsinthe
cohomologygroup.Formoredetails,see[Sko01].Let
A
beanalgebraicgroupover
a
k
.A
k
-torsorunder
A
isanon-empty
k
-variety
T
equippedwitharight-actionof
A
19
sothat
T
(
k
)isaprincipalhomogeneousspaceunder
A
(
k
).Thismeans,thatthemap
T
(
k
)

A
(
k
)
!
T
(
k
)

T
(
k
)
(
t;a
)
7!
(
t;t:a
)
isanisomorphism.
Thereisabijection
H
1
(
k;A
)
$
8
>
>
<
>
>
:
k
-torsorsunder
A
uptoisomorphism
9
>
>
=
>
>
;
thatisexplicitlygivenasfollows.Let
T
bea
k
-torsorunder
A
.Choosea
k
-point
x
0
of
T
.Bytheof
k
-torsor,forany
˙
2
G
k
,thereexistsaunique
a
˙
2
A
(
k
)sothat
˙
(
x
0
)=
x
0
:a
˙
.Themap
˙
7!
a
˙
thecocyclein
H
1
(
k;A
)correspondingto
T
.
Let
X
beavarietyover
k
.An
X
-torsorunderan
X
-groupscheme
A
isascheme
T
over
X
togetherwithan
A
-actioncompatiblewiththeprojectionto
X
thatisetale-locallytrivial.
Asbefore,thereisaone-to-onecorrespondencebetween
X
-torsorsunder
A
andelementsof
theetalecohomology
H
1
et
(
X;
A
).
Theorem2.2.12
(Colliot-Thelene{Sansuc)
.
Assumethat
k
[
X
]

=
k

.Let
S
bea
G
k
-
moduleanddenoteby
M
=
Hom
k

groups
(
S
;
G
m
)
thedualof
S
.Thereisanexactsequence
0
/
/
H
1
et
(Spec
k;
M
)
p

/
/
H
1
et
(
A
;
M
)
type
/
/
Hom
k

M
;
Pic(
A
)

/
/
H
2
et
(Spec
k;
M
)
;
where
p

isthemapinducedbythestructuremorphism
p
:
A!
Spec
k
.
If
type
(
T
)=

foratorsor
T
,wesaythat
T
has
type

.
20
Proof.
Thisisthesequenceoflowdegreetermsforthespectralsequenceoflocaltoglobal
Ext.Formoredetails,seeforinstance[CTS87,Theorem1.5.1andEquation2.0.2]or[Sko01,
Theorem2.3.6andCorollary2.3.9].
Example2.2.13.
A
d
-coveringofanabelianvariety
X
isapair
(
T
; 
)
,where
T
isa
k
-torsorunder
X
and
 
:
T!
X
isamorphismsuchthat
 
(
x:t
)=
dx
+
 
(
t
)
forany
t
2T
(
k
)
;x
2
X
(
k
)
:
Let

bethecomposition

:
d
X
_
(
k
)
/
/
X
_
(
k
)=Pic
0
(
X
)
/
/
Pic(
X
)
ofthenaturalinjectionfollowedbytheinclusion,where
X
_
denotesthedualabelianvariety
of
X
.By[Sko01,Proposition3.3.4(a)],any
d
-coveringisan
X
-torsorunder
d
X
of
type

,
andviceversa.
Example2.2.14.
Multiplicationby
d
on
X
determinesa
d
-covering
(
X;
[
d
])
,andtherefore
an
X
-torsorof
type

bythepreviousproposition.
2.2.5TheBrauerGroup,SymbolAlgebras,and
H
2
Inthissection,wedescribethecorrespondencebetweentheBrauergroupandthesecond
cohomologygroup.Let
F
bealdandlet
d

2.Wesaythattwocentralsimplealgebras
A
and
B
areMoritaequivalentifthereexistsome
n;m
2
N
0
sothat
A

M
n
(
F
)and
B

M
m
(
F
)
areisomorphicas
F
-algebras.ElementsintheBrauergrouparegivenbyequivalenceclasses
ofcentralsimplealgebrasmoduloMoritaequivalenceandthegroupstructureisgivenby
thetensorproduct.ThereisagroupisomorphismbetweenBr(
F
)and
H
2
(
F;
F

).Wewill
describethecorrespondenceforaGaloisextension
K=F
.Forfurtherdetailsseefor
instance[GS17].
21
2.2.15
(CrossedProductAlgebra)
.
Let
K=F
beaGaloisextensionwith
Galoisgroup
G
andlet
f
beacocyclerepresentinganelementin
H
2
(
G;K

)
.Considerthe
F
-vectorspace
A
=
F

x
g
:
g
2
G

withmultiplication

g
=
x
g
g
(
a
)
and
x
g
x
h
=
f
(
g;h
)
x
gh
.
Thisturns
A
intoadimensionalcentralsimplealgebraover
F
.
Fromnowonsupposethatthe
F
containsaprimitive
d
-throotofunity
ˆ
.Fix
anisomorphism[
:
]
ˆ
:

d
!
Z
=d
Z
with

ˆ
i

ˆ
=
i
.Furthermore,identify
Z
=d
Z
withthe
subset
f
0
;:::;d

1
g
oftheintegersanddenotetheimageof
ˆ
i
underthecompositionby

ˆ
i

Z
ˆ
=
i
2
Z
.
2.2.16.
Let
a;b
2
F

.Thesymbolalgebraisthe
F
-algebra
(
a;b
)
d;F
=(
a;b
)
d;F;ˆ
:=
F
D
x;y
:
x
d
=
a;y
d
=
b;xy
=
ˆyx
E
:
Itiseasytoshowthat
(
a;b
)
d;F
isacentralsimplealgebraover
F
.
Example2.2.17.
Let
a;b
2
F

.Theelementin
H
2
(
F;
F

)
correspondingtothesymbol
algebra
(
a;b
)
d;F;ˆ
canberepresentedbythecocycle
(
;˝
)
7!
8
>
>
>
>
<
>
>
>
>
:
a
if
"


d
p
a

d
p
a
#
Z
ˆ
+
"
˝

d
p
b

d
p
b
#
Z
ˆ

d
1
else
:
Formoredetails,see[Rei03,Chapter7
x
29].
Thefollowingcocyclerepresentingthesymbolalgebra(
a;b
)
d;F
willprovemoreusefulfor
ourpurposes.
22
Proposition2.2.18.
Let
M
bethe
d
-torsionofanellipticcurve
E
withgenerators
P
and
Q
.AssumethattheWeil-pairing
e(
P;Q
)=
ˆ
.Let
a;b
2
F

,thenthesymbol
algebra
(
a;b
)
d;F
canberepresentedbythecocycle
(
;˝
)
7!
e
0
@


d
p
a


d
p
a

P;


d
p
b


d
p
b

Q
1
A

1
:
Proof.
Considerthemap
g
:

!
d
p
a
2
4


d
p
b


d
p
b

3
5
Z
ˆ
:
Thetialof
g
is
dg
(
;˝
)=

0
B
B
B
B
B
@
d
p
a
2
4
˝

d
p
b


d
p
b

3
5
Z
ˆ
1
C
C
C
C
C
A
d
p
a
2
4


d
p
b


d
p
b

3
5
Z
ˆ
d
p
a

2
4
˝

d
p
b


d
p
b

3
5
Z
ˆ
=
8
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
:
a
 


d
p
a

d
p
a
!
2
4
˝

d
p
b


d
p
b

3
5
ˆ
if
"


d
p
a

d
p
a
#
Z
ˆ
+
"
˝

d
p
b

d
p
b
#
Z
ˆ

d
 


d
p
a

d
p
a
!
2
4
˝

d
p
b


d
p
b

3
5
ˆ
else
=
8
>
>
>
>
>
<
>
>
>
>
>
:
a
e
 


d
p
a


d
p
a

P;


d
p
b


d
p
b

Q
!
if
"


d
p
a

d
p
a
#
Z
ˆ
+
"
˝

d
p
b

d
p
b
#
Z
ˆ

d
e
 


d
p
a


d
p
a

P;


d
p
b


d
p
b

Q
!
else
Subtractingthistrivialcocyclefromthecocycleinexample2.2.17givesthedesiredresult.
Asmentionedbefore,thesecanbegeneralizedtotheBrauergroupofavariety
23
orascheme,see[Mil80,ChapterIV]fordetails.AfamousresultofGabber[dJ]statesthat
theBrauergroupasequivalenceclassesofAzumayaalgebrascoincideswiththe
torsionof
H
2
et
(
X;
G
m
)if
X
canbeendowedwithanampleinvertiblesheaf.
2.2.6TheBrauergroup
AnimportantsubgroupoftheBrauergroupofaistheBrauergroup.In
thissection,wereviewitstionandsomebasicfactsaboutit.Formoredetails,see
[Sal99,Chapter10].First,let
R
beadiscretevaluationdomaindomainwithoffractions
K
.Denoteby
v
:
K

!
Z
thevaluationby
R
.Let
ˇ
beaprimewith
v
(
ˇ
)=1.
Denoteby
^
K
thecompletionof
K
withrespectto
v
.Let
R
=
R=ˇR
anddenoteby
p
the
characteristicof
R
.Let
K
ur
bethemaximalextensionofthecompletion
^
K
.The
followingresultgivenin[Sal99,Theorem10.1]willhelpusthemap.
Theorem2.2.19.
Anyelementin
Br(
^
K
)
oforderprimeto
p
issplitby
K
ur
.
Foraprime
p
andanabeliangroup
A
,denoteby
A
0
theprime-to-
p
part.Extendthe
valuation
v
to
v
:
K
ur
!
Z
.By[Ser79,p.28]thevaluationmapiscompatiblewiththe
actionofGal

K
ur
=
^
K

,wheretheactionon
Z
istrivial.nethemapasthe
composition
ram
R
:
Br(
K
)
0
/
/
Br(
^
K
)
0
˘
theorem2.2.19
/
/
H
2

Gal

K
ur
=
^
K

;K

ur

0
v
/
/
H
2

Gal

K
ur
=
^
K

;
Z

0
˘
/
/
H
1

Gal

K
ur
=
^
K

;
Q
=
Z

0
˘
/
/
Hom

Gal

K
ur
=
^
K

;
Q
=
Z

0
;
wherethesecondtolastmapistheinverseofthecoboundaryinducedbytheexactsequence
0
!
Z
!
Z
!
Q
=
Z
!
0.Itcanbeshown[Sal99,Theorem10.3]thatthe
24
intheexactsequence
0
/
/
Br(
R
)
0
/
/
Br(
K
)
0
ram
R
/
/
Hom

Gal

K
ur
=
^
K

;
Q
=
Z

0
/
/
0
:
Wewillnowreturntothegeneralsetting.Let
k
beaground,andlet
F
be
aextensionof
k
.Let
R
F
bethesetofdiscretevaluationringscontaining
k
thathave
offraction
F
.
2.2.20.
TheunrdBrauergroupof
F
(withrespectto
k
)is
Br
ur
(
F
)=
\
R
2R
F
(
Imageof
Br(
R
)
!
Br(
F
))
:
Wewillusethefollowingidenthroughoutthiswork.
Theorem2.2.21.
Let
X
beaprojectiveregularvarietyover
k
withfunction
k
(
X
)
.
Then
Br(
X
)
equals
Br
ur
k
(
X
)
.
Proof.
Seeforinstance[CTS07,Theorem5.11]or[Sal99,Proposition10.5(c)].
2.3Anexactsequence
Inthissection,wedescribethemapsintheexactsequence1.1explicitly.Considerthe
Hochschild-Serrespectralsequence[Mil80,III.2.20]
H
i

k;H
j

E;
G
m


)
H
i
+
j
(
E;
G
m
)
:
25
Itssequenceoflowdegreetermsis
0
/
/
Br(
k
)
i
/
/
Br(
E
)
r
/
/
H
1

k;E

k

/
/
0
;
(2.3)
wherethemapsendstheclassofacentralsimplealgebra
A
totheclassof
A

k
(
E
).For
moredetailsonthissequence,seealso[Fad56]and[Lic69].Thesecondmapismorecompli-
cated.Let

2
Br(
E
).UsingTsen'stheoremweview

asanelementin
H
2

G
k
;
k
(
E
)


˘
=
Br
k
(
E
).Considertheexactsequence
0
/
/
k
(
E
)

/
/
Prin(
E
)
/
/
Div(
E
)
/
/
0
;
wherePrin(
E
)denotesthesetofpricipaldivisorson
E
andDiv(
E
)isthesetofdivisorson
E
.Thesequenceinducedongroupcohomologyis
H
2
(
G
k
;
k
(
E
)

)
/
/
H
2
(
G
k
;
Prin(
E
))
/
/
H
2
(
G
k
;
Div(
E
))
;
wherethemaptakes

tosome

0
inthekernelofthesecondmap.Nowconsiderthe
degreesequence
0
/
/
Div
0
(
E
)
/
/
Div(
E
)
/
/
Z
/
/
0
;
whereDiv
0
(
E
)isthegroupofdegreezerodivisors.Notethat
H
1
(
G
k
;
Z
)=0andtherefore
themap
H
2
(
G
k
;
Div
0
(
E
))
/
/
H
2
(
G
k
;
Div(
E
))
26
isinjective.Finally,theexactsequence
0
/
/
Prin(
E
)
/
/
Div
0
(
E
)
/
/
E
(
k
)
/
/
0
inducesanexactsequence
1
/
/
H
1
(
G
k
;E
(
k
))
/
/
H
2
(
G
k
;
Prin(
E
))
/
/
H
2
(
G
k
;
Div
0
(
E
))
:
Theelement

0
isinthekernelofthesecondmap,andthereforethereexistsaunique

00
2
H
1
(
G
k
;E
(
k
))withimage

0
.Set
r
(

)=

00
.
Thiscompletesthedescriptionoftheexactsequence(2.3).Wewillusethisexplicit
descriptiontoprovethatthemap

inducedbythecupproduct(eq.(4.1))inducesasplit.
27
Chapter3
Torsorgivenbymultiplicationby
d
Let
k
beaofcharacteristictfrom2or3.Let
d

2beanintegercoprimetothe
characteristicof
k
.Assumeadditionallythat
k
containsaprimitive
d
-throotofunity
ˆ
.Fix
anisomorphism[
:
]
ˆ
:

d
!
Z
=d
Z
with

ˆ
i

=
i
.Furthermore,for
ˆ
i
2

d
,let

ˆ
i

Z
ˆ
=
i
2
f
0
;:::;d

1
gˆ
Z
.Let
E
beanellipticcurveover
k
anddenoteits
d
-torsionby
M
.This
chaptercontainsacocycledescriptionofthetorsorgivenbymultiplicationby
d
on
E
.Fix
twogenerators
P
and
Q
of
M
.Denotebye(
:;:
)theWeilpairing(section2.1.3)andassume
thate(
P;Q
)=
ˆ
.Let
t
P
;t
Q
2
k
(
E
)withdiv(
t
P
)=
d
(
P
)

d
(0)anddiv(
t
Q
)=
d
(
Q
)

d
(0).
3.1Overawithrationaltorsion
Assumethroughoutthissectionthat
M
is
k
-rational.Wemayassumethat
t
P
;t
Q
2
k
(
E
)
sinceitsdivisorisinvariantundertheGaloisactionof
G
k
.Let
T
bethetorsorgivenby
multiplicationby
d
on
E
asinsection2.2.4.
Proposition3.1.1.
Thepull-back


(
T
)
alongthegenericpoint

:Spec
k
(
E
)
!
E
corre-
spondstotheelementin
H
1
(
k
(
E
)
;M
)
givenbythecocycle

7!
"



Q


Q
#
ˆ
P



(

P
)

P

ˆ
Q;
where

P
;
Q
2
k
(
E
)
with

d
P
=
t
P
and

d
Q
=
t
Q
.
28
Proof.
Forthecorrespondencebetweentorsorsandelementsin
H
1
seesection2.2.4.Let
P
0
2
E
(
k
)sothat[
d
]
P
0
=
P
.Thenthereissome
g
P
2
k
(
E
)with
div(
g
P
)=[
d
]

(
P
)

[
d
]

(0)=
X
R
2
M

(
P
0

R
)

(
R
)

:
Notethatwemaychoose
g
P
2
k
(
E
)sincethedivisorisinvariantundertheactionofthe
absoluteGaloisgroupof
k
.Nowdiv

g
d
P

=div([
d
]

t
P
)andthuswemayassumethat
g
d
P
=[
d
]

t
P
.Similarlywe
g
Q
2
k
(
E
)with
g
d
Q
=[
d
]

t
Q
.Nowconsiderthepullbackof
T
alongthegenericpoint

:Spec
k
(
E
)
!
Spec
k
.Fixa
k
(
E
)-point
x
0
ofthispullback,i.e.
amapofalgebrassothat
x
0

[
d
]

=

,where

:
k
(
E
)
!
k
(
E
)istheinclusion.
Spec
k
(
E
)

/
/
[
d
]


E
[
d
]


Spec
k
(
E
)
8
8
/
/
Spec
k
(
E
)

/
/
E
k
(
E
)
x
0
|
|
k
(
E
)
k
(
E
)
[
d
]

O
O

o
o
Afterpossiblyrenaming

P
and

Q
,wemayassumethat
x
0
(
g
P
)=

P
and
x
0

g
Q

=

Q
.
Byproposition2.1.4thereisagroupisomorphism
M
!
Gal(
k
(
E
)
=
[
d
]

k
(
E
)):
R
7!
˝

R
;
where
˝
R
:
E
!
E
isthetranslationby
R
-map;
˝
R
:
E
!
E
:
S
7!
S

R
.Bythe
oftheWeil-pairinge(
R;P
)=
g
P
(
X

S
)
g
P
(
X
)
=
˝

S
(
X
)
g
P
(
X
)
;
forany
R
2
M
,
X
2
E
(
k
)anypointso
that
g
P
(
X
)and
g
P
(
X

S
)areTheanalogousresultholdsfor
g
Q
aswell.Finally,
29
wecalculate
x
0

˝

"

(

Q
)

Q
#
ˆ
P



(

P
)

P

ˆ
Q
(
g
P
)=
x
0
 
e
 


(

Q
)

Q

ˆ
P



(

P
)

P

ˆ
Q;P
!
g
P
!
=
x
0
 
e
 



(

P
)

P

ˆ
Q;P
!
g
P
!
=
x
0


(

P
)

P
g
P

=

(

P
)

P

P
=

(

P
)
and
x
0

˝

"

(

Q
)

Q
#
ˆ
P



(

P
)

P

ˆ
Q

g
Q

=
x
0
 
e
 


(

Q
)

Q

ˆ
P



(

P
)

P

ˆ
Q;Q
!
g
Q
!
=
x
0
 
e
 


(

Q
)

Q

ˆ
P;Q
!
g
Q
!
=
x
0


(

Q
)

Q
g
Q

=

(

Q
)

Q

Q
=

(

Q
)
:
Thestatementfollowssince
k
(
E
)
=
[
d
]

k
(
E
)isgeneratedby
g
P
and
g
Q
.
3.2Overany
Let
k
beanyConsidertheGaloisrepresentation
:
G
k
!
Aut(
M
)=
GL
2
(
F
d
)
30
givenbytheactionon
M
anddenotetheofitskernelby
L
.Considerthetower
ofextensions
k
(
E
)
L
(
E
)
M
Gal(
L=k
)
[
d
]

L
(
E
)
Gal(
L=k
)
k
(
E
)
[
d
]

k
(
E
)
Fixaset
~
G
L=k
ˆ
G
k
(
E
)
ofcosetrepresentativesof
G
k
(
E
)
=G
L
(
E
)
˘
=
Gal(
L=k
).Notethat
~
G
L=k
isalsoasetofcosetrepresentativesof
G
[
d
]

k
(
E
)
=G
[
d
]

L
(
E
)
˘
=
Gal(
L=k
)andev-
ery~
˙
2
~
G
L=k

k
(
E
).Let

2
G
[
d
]

k
(
E
)
,then

decomposesas

=

0
~
˙
forsome

0
2
G
[
d
]

L
(
E
)
andsome~
˙
2
~
G
L=k
.
Let
T
bethetorsorgivenbymultiplicationby
d
on
E
.Considerthepullback


k
(
E
)
of
T
tothegenericpoint.Notethatxinga
k
(
E
)-pointof


T
isthesameasan
isomorphism
˚
0
:
k
(
E
)
!
k
(
E
)makingthefollowingdiagramcommute{orequivalentlyan
elementinGal(
k
(
E
)
=
[
d
]

k
(
E
)).
k
(
E
)

1
/
/
k
(
E
)
[
d
]

k
(
E
)
O
O

1
/
/
k
(
E
)
k
(
E
)
˘
[
d
]

O
O

1
/
/
k
(
E
)
˘
˚
d
O
O
:
31
Wewillidentify
G
[
d
]

k
(
E
)
with
G
k
(
E
)
and
G
[
d
]

L
(
E
)
with
G
L
(
E
)
.Fixasetofcosetrep-
resentatives
~
G
L=k
asbefore.Thenevery

2
G
k
(
E
)
canbedecomposedas

0
~
˙
forsome

0
2
G
L
(
E
)
andsome~
˙
2
~
G
L=k
.Furthermore,~
˙
theimageof
k
(
E
)
(topleftcornerof
thediagram)byconstruction.
Wewanttodescribethecocyclecorrespondingtothepullbackof
T
alongthegeneric
pointusingthecorrespondenceinsection2.2.4.Let
x
1
bea
k
(
E
)point.Wemayassume,
that
x
0
=

1

x
0
forsome
x
0
asinthefollowingcommutativediagram.
k
(
E
)

1
/
/
x
1


L
(
E
)
x
0
!
!
k
(
E
)
[
d
]

O
O

1
/
/
L
(
E
)
[
d
]

O
O

/
/
k
(
E
)
:
Any

=

0
~
˙
2
G
k
(
E
)
with

0
2
G
L
(
E
)
;
~
˙
2
~
G
L=k
actson
x
1
by
:x
1
=

0

~
˙x
0


1
=

0
:x
1
:
Finally,

0
:x
1
canbecomputedasinproposition3.1.1.Summarizingthisweconcludethe
followingproposition.
Proposition3.2.1.
Thepull-backof
T
tothegenericpointcorrespondstotheelementin
H
1
(
k
(
E
)
;M
)
givenbythecocycle
G
k
(
E
)
!
M
:

7!
"

0
(

Q
)

Q
#
ˆ
P



0
(

P
)

P

ˆ
Q;
32
where

=

0
~
˙
asabove,forsome
~
˙
2
~
G
L=k
and

0
2
G
L
(
E
)
,

P
;
Q
2
k
(
E
)
sothat

d
P
=
t
P
and

d
Q
=
t
Q
.
Wenowproceedtogiveanexplicitdiscriptionoftheelements
t
P
and
t
Q
.Remarkthat
thesecanbechosenin
k
(
E
)

if
M
is
k
-rational.Formoredetailsontheconstruction,see
also[Mil04,Section4.1].Foranytwopoints
R;S
2
E
(
k
),denoteby
L
R;S
thenormalized
functionsuchthat
L
R;S
=0givestheequationofthelinethrough
R
and
S
.Itsdivisoris
div(
L
R;S
)=(
R
)+(
S
)+(

(
R

S
))

3(0).anelement
h
R;S
=
L
R;S
L
R

S;

(
R

S
)
2
k
(
E
)

withdiv
h
R;S
=(
R
)+(
S
)

(
R

S
)

(0).Nowthefunction
t
P
=
Q
d
i
=1
h
P;
[
i
]
P
2
k
(
E
)has
divisor
div(
t
P
)=
d

1
X
i
=1
(
P
)+([
i
]
P
)

([
i
+1]
P
)

(0)
=(
d

1)(
P
)+
P

([
d
]
P
)

(
d

1)(0)
=
d
(
P
)

d
(0)
:
Similarly,weconstruct
t
Q
.Notethatinthecase
q
=3,wecanchoosethenormalized
function
t
P
;t
Q
2
k
(
E
)

suchthat
t
P
=0and
t
Q
=0givethetangentlinesat
P
,and
Q
respectively.
3.3GeneralArgumentthat

inducesthecorrectsplit
Inthissectionwereviewanabstractargumentgivenin[Sko01,Chapter4]toprovethat
themap

beforeinducesthedesiredsplit.Wewillreprovethisinchapter4inour
spcaseusingexplitmethods.
33
Let
k
beawithalgebraicclosure
k
andlet
X
bea
k
-variety.Denotethestructure
mapby
p
:
X
!
Spec
k
.Further,let
X
=
X

Spec
k
Spec
k
.Assumethat
M
isa
G
k
-module
thatisgeneratedasanabeliangroup.Assumethattheorderofthetorsionof
M
is
coprimetothecharacteristicof
k
.Denoteby
S
=Hom(
S;
G
m
)thedual
k
-groupof
M
.Let
T
bean
X
-torsorof
type

forsome

inHom
k

M;
Pic(
X
)

(theorem2.2.12)andassume
that
k
[
X
]

=
k

.ConsiderthelongexactsequencefromthespectralsequenceofExt's
0
/
/
Pic(
X
)
/
/
Pic(
X
)
G
k
/
/
Br(
k
)
/
/
Br
1
(
X
)
r
/
/
H
1
(
k;
Pic(
X
))
/
/
H
3
(
k;
G
m
)
;
whereBr
1
(
X
)denotesthekernelofthenaturalmapBr(
X
)
!
Br(
X
)
G
k
:

Br

(
X
):=
r

1



H
1
(
k;M
)


Br
1
(
X
)
:
Theorem3.3.1.
Thecup-product
p

(

)
[
[
T
]
isanelementof
Br

(
X
)
forany

2
H
1
(
k;M
)
and
r
(
p

(

)
[
[
T
])=


(

)
2
H
1

k;
Pic(
X
)

:
Thatis,thefollowingdiagramcommutes
H
1
(
k;M
)




p

(
:
)
[
[
T
]
w
w
/
/
Br(
k
)
/
/
Br
1
(
X
)
r
/
/
H
1
(
k;
Pic(
X
))
/
/
H
3
(
k;
G
m
)
:
(3.1)
Furthermore,any
A
2
Br

(
X
)
canbewrittenas
A
=
p

(

)
[
[
T
]+
p

(
A
0
)
34
forsome

2
H
1
(
k;M
)
andsome
A
0
2
Br(
k
)
.
Thisis[Sko01,Theorem4.11].Wewillreviewtheproofhereforcompletion.Notethat
itisenoughtoshowthatthefollowingdiagramcommutes
H
1
(
X;S
)
Ext
1
X
(
p

M;
G
m
)
type
/
/
d


Hom
k
(
M;
Pic(
X
))
d


Br
1
(
X
)
r
/
/
H
1
(
k;
Pic(
X
))
;
(3.2)
where
d
istheconnectinghomomorphismofthelongexactsequenceofExt.Theproofwill
requiresomefactsonthederivedcategoryof
X
from[Wei94],thatwewillreview
3.3.1DigressiontoDerivedCategories
Let
X
bea
k
-variety.Considerthecategories
Sh
(
X
)ofetalesheaveson
X
,
G
k
-mod
of
G
k
modules,and
Ab
ofabeliangroups.Denoteby
D
+
(
X
)
;
D
+
(
k
),and
D
+
(
Ab
)the
correspondingderivedcategoriesofboundedbelowcomplexes.Notethatallthesecategories
haveenoughinjectives.For
M
inoneofthesecategories,let
M

betheelementinthe
derivedcategorywith
M
i
=
8
>
>
>
<
>
>
>
:
Mi
=0
0else
:
Forany
n
2
Z
,wedenoteby
˝

n
and
˝

n
thetruncationfunctors.Thatis,foracomplex
F
we
˝

n
(
F
)asthecomplex
!
0
!
F
n
=
Im(
d
)
!
F
n
+1
!
35
Furthermore,denote
˝
[
m;n
]
(
F
)=
˝

m
˝

n
(
F
)and
˝
[
n
]
(
F
)=
˝
[
n;n
]
(
F
).
Let0
/
/
A
/
/
B
/
/
C
/
/
0beanexactsequenceof
G
k
-modulesandlet
F
be
anobjectin
D
+
(
k
).Thenthefollowingdiagramin
D
+
(
Ab
)commutes
R
Hom
k

A;˝

1
(
F
)

/
/
d


R
Hom
k

A;˝
[1]
(
F
)

d


R
Hom
k

C;˝

1
(
F
)

[1]
/
/
R
Hom
k

C;˝
[1]
(
F
)

[1]
;
(3.3)
wheretheverticalmapsaretheconnectinghomomorphismsinducedbytheexactsequence.
Considerthestructuremorphism
p
:
X
!
Spec
k
.Thedirectimagefunctor
p

assigns
toanetalesheafon
X
anetalesheafonSpec
k
.Fixthegeometricpoint
s
:Spec
k
!
Spec
k
inducedbytheinclusion
k

k
.By[Sta19,Lemma54.58.1,Tag04JQ],thestalkfunctor
F7!F
s
inducesanequivalenceofcategoriesbetweenthecategoryofetalesheavesover
X
andthecategoryof
G
k
-modules.Considerthecompositionof
p

withthisequivalenceof
categories.Abusingnotation,wewilldenotethiscompositionby
p

aswell.Furthermore,
thereisafunctorHom
k
(
M;

):
G
k
-mod
!
Ab
.For
M
=
Z
,thisfunctorisgivenby
sendinga
G
k
-module
N
toits
G
k
-invariants
N
G
k
.Denotetheassociatedderivedfunctors
by
R
p

;
R
Hom
k
(
M;

),and
H
(
k;

)=
R
Hom
k
(
Z
;

).Recall,thatunderourequivalence
ofcategoriesgivenbythestalkfunctor
F7!F
s
,thesheaf
R
i
p

G
m
correspondstothe
G
k
-module
^
H
i
(
X;
G
m
)
s
=
H
i
(
X;
G
m
)
;
(3.4)
by[Har77,IIIProposition8.5]andthesecondequalityholdstrue,becausetakingstalks
36
commuteswithcohomology.
Thefunctor
p

admitsaleft-adjoint
p

sothatHom
k
(
M;p

F
)=Hom
X
(
p

M;F
)forany
G
k
-module
M
andanyetalesheaf
F
on
X
.Wewillusethefollowingidentitiesofadjoint
functorsthroughout
R
Hom
k
(
M;

)

R
p

=
R
Hom
X
(
p

M;

)(3.5)
H
(
k;

)

R
p

=
H
(
X;

)
:
(3.6)
Wenowreviewthedeofthehypercohomologyfunctorasgivenin[Mil80,Ap-
pendixC].Let
f
:
A!B
bealeft-exactfunctorbetweenabeliancategoriesandassume
that
A
hasenoughinjectives.Denoteby
C
+
(
A
)thecategoryofcomplexesboundedbelow.
Let
A

2
C
+
(
A
).Thenthereexistsacomplex
I

2
C
+
(
A
)whoseobjectsareinjectives,
thatisquasi-isomorphicto
A
.Theright-hyperderivedfunctor
R
i
f
of
f
assignsto
A

the
object
H
i
(
fI

)in
B
.Denoteby
H
i
(
k;

)thehyperderivedfunctorof
H
(
k;

).
Fora
G
k
-module
M
thatisgeneratedasanabeliangroupwithtorsioncoprime
tothecharacteristicof
k
,asequence
0
/
/
M
/
/
N
/
/
Z
/
/
0
of
G
k
-modules,where
G
k
actstriviallyon
Z
.Thecommutativediagram3.3givesthe
37
followingaftersetting
F
=
R
p

G
m
andtakingcohomology.
R
1
Hom
k

M;˝

1
(
R
p

G
m
)

/
/
d


R
1
Hom
k

M;˝
[1]
(
R
p

G
m
)

d


H
1

k;˝

1
(
R
p

G
m
)[1]

/
/
H
1

k;˝
[1]
(
R
p

G
m
)[1]

:
(3.7)
Itremainstoshowthatthediagrams3.2and3.7areisomorphic,i.e.thereexistisomor-
phismsbetweentheirobjectssothatthefollowingdiagramcommutes.
R
1
Hom
k

M;˝

1
(
R
p

G
m
)

/
/


˘
'
'
R
1
Hom
k

M;˝
[1]
(
R
p

G
m
)

˘
(
(


Ext
1
X
(
p

M;
G
m
)
/
/


Hom
k
(
M;
Pic(
X
))
d


H
1

k;˝

1
(
R
p

G
m
)[1]

/
/
˘
'
'
H
1

k;˝
[1]
(
R
p

G
m
)[1]

˘
(
(
Br
1
(
X
)
r
/
/
H
1
(
k;
Pic(
X
))
(3.8)
Fortheproofswereferthereaderto[Mil80,AppendixC],[Sko01,p.67or[Wei94,
Corollary10.8.3].
3.3.2Applicationtoourcase
Let
E
beanellipticcurveovera
k
ofcharacteristicprimeto
d
.Denoteby
M
the
d
-torsionof
E
(
k
).Considerthetorsor
T
givenbymultiplicationby
d
asinexample2.2.14.
It's
type
isthecomposition

:
M
!
E
(
k
)=Pic
0
(
E
)
!
Pic(
E
)
:
38
Furthermore,recallthatPic(
E
)=Pic
0

E


Z
and
H
1
(
k;
Z
)=0.Thenthecohomology
becomes
H
1

k;
Pic(
E
)

=
H
1

k;
Pic
0
(
E
)

=
H
1
(
k;E
(
k
))
:
Thediagram3.1becomes
H
1
(
k;M
)




x
x
0
/
/
Br(
k
)
/
/
Br(
E
)
r
/
/
H
1
(
k;E
(
k
))
/
/
0
:
Notethatboth

and
r
aresurjectiveandthereforeby
Br

(
E
)=
d
Br(
E
)
:
Bytheorem3.3.1,everyelementin
d
Br(
E
)canberepresentedasacupproduct
p

(

)
[
[
T
]+
p

(
A
0
)
forsome

2
H
1
(
k;M
)and
A
0
2
Br(
k
).Furthermore,thealgebra
p

(

)
[
[
T
]istrivialif
andonlyif

isintheimageoftheKummermap

:
E
(
k
)
=
[
d
]
E
(
k
)
!
H
1
(
k;M
).
39
Chapter4
Generatorsof
Br(
E
)
Let
k
beadofcharacteristictfrom2or3.Let
d

2beanintegercoprimeto
thecharacteristicof
k
andassumeadditionallythat
k
containsaprimitive
d
-throotofunity
ˆ
.Let
E
beanellipticcurveover
k
.Recallthatourmethodofcomputinggeneratorsof
d
Br(
E
)isviaanexplicitsplittotheexactsequence
0
/
/
d
Br(
k
)
i
/
/
d
Br(
E
)
r
/
/
d
H
1

k;E

k

/
/
0
describedinsection2.3.ConsidertheKummersequence
0
/
/
M
/
/
E
(
k
)
[
d
]
/
/
E
(
k
)
/
/
0
:
Thesequenceinducedoncohomologyis
0
/
/
E
(
k
)
=
[
d
]
E
(
k
)

/
/
H
1
(
k;M
)

/
/
d
H
1

k;E
(
k
)

/
/
0
:
Wewilldescribeamap

:
H
1
(
k;M
)
!
d
Br(
E
),thatinducesasplitofthesequencein
section2.3.Denoteby
T
thetorsorgivenbymultiplicationby
d
on
E
asinchapter3.
40


:
H
1
(
k;M
)
!
d
Br(
E
)asthefollowingcomposition

:
H
1
(
k;M
)
˘
/
/
H
1
et
(Spec(
k
)
;M
)
p

/
/
H
1
et
(
E;M
)

[
T
]
/
/
H
2
et
(
E;M

M
)
e
/
/
H
2
et
(
E;
d
)
/
/
d
Br(
E
)
:
(4.1)
In[Sko01,Theorem4.1.1],theauthorprovesabstractlythat

inducessuchasplitusing
generalpropertiesoftorsorsandthecup-product.Inthischapter,wewilldetermine

explicitlyandprovedirectlythatthemapinducesthedesiredsplit.
Proposition4.0.1.
Onthelevelofcocycles

coincideswiththemapthatassignstoa
1-cocycle
f
:
G
k
!
M
the2-cocycle

(
f
):
G
k
(
E
)

G
k
(
E
)
!
k
(
E
)

(
;˝
)
7!
e
0
@
f
(

)
;
0
@
"
˝
0
(

Q
)

Q
#
ˆ
P


˝
0
(

P
)

P

ˆ
Q
1
A
1
A
;
(4.2)
for
;˝
2
G
k
(
E
)
;˝
=
˝
0
~
˙
forsome
~
˙
2
~
G
L=k
and
˝
0
2
G
k
(
E
)
.(Foradescriptionof
~
G
L=k
seesection3.2).
Proof.
Considerthefollowingdiagram
H
1
(
k;M
)
p

/
/
H
1
et
(
E;M
)





[
T
]
/
/
H
2
et
(
E;M

M
)
e
˘
/
/




H
2
et
(
E;
d
)


H
1
et
(
k
(
E
)
;M
)

[


T
]
/
/
H
2
et
(
k
(
E
)
;M

M
)
e
˘
/
/
d
Br
k
(
E
)
Ithascommutativesquaresasthecup-productcommuteswith


[Bre97,Chapter2,8.2].
Foreverycocycle
f
:
G
k
!
M
,theBrauerclasse

(


p

([
f
])
[
[


T
])canbedescribed
bythecocycleineq.(4.2)byproposition3.2.1andbytheofthecup-productin
41
groupcohomology(see2.2.9).Recallthatby[CTS07,Theorem5.11]themapon
therightisgivenbytheinjectionthatidenBr(
E
)withtheunBrauergroupof
k
(
E
)(fortheBrauergroupseealsosection2.2.6).
4.1
M
is
k
-rational
Assumethroughoutthissectionthat
M
ˆ
E
(
k
)withgenerators
P
and
Q
sothate(
P;Q
)=
ˆ
.
Wearenowreadytocalculateasetofgeneratorsof
d
Br(
E
).ByKummer-theorythereisan
isomorphism
˚
:

k

=
(
k

)
d



k

=
(
k

)
d

!
H
1
(
k;M
)
(
a;b
)
7!
c
a;b
;
(4.3)
where
c
a;b
canberepresentedbythecocycle
G
k
!
M

7!
"


d
p
a

d
p
a
#
ˆ
P

2
4


d
p
b

d
p
b
3
5
ˆ
Q:
Proposition4.1.1.
Thecomposition


˚
(
a;b
)
correspondstotheBrauerclassofthetensor
productofsymbolalgebras
(
a;t
P
)
d;k
(
E
)


b;t
Q

d;k
(
E
)
forany
(
a;b
)
2
k

=
(
k

)
d

k

=
(
k

)
d
.
Proof.
Observethattheclassof

k

˚
(
a;b
)canberepresentedbythecocyclethattakesthe
42
pair(
;˝
)
2
G
k
(
E
)

G
k
(
E
)
to
e
0
B
@
"


d
p
a

d
p
a
#
ˆ
P

2
4


d
p
b

d
p
b
3
5
ˆ
Q;

˝
(

Q
)

Q

ˆ
P


˝
(

P
)

P

ˆ
Q
1
C
A
=e
0
@
"


d
p
a

d
p
a
#
ˆ
P;

˝
(

P
)

P

ˆ
Q
1
A

1
e
0
B
@
2
4


d
p
b

d
p
b
3
5
ˆ
Q;

˝
(

Q
)

Q

ˆ
P
1
C
A
:
Thestatementfollowsfromproposition2.2.18.
Recallthatweneedtoprovethat

inducesasplittothesequenceinsection2.3onthe
right,i.e.weneedtoshowthat
r


=

and

(ker(

))=0.Fortheof
r
and

seesection2.3.Westprovethat
r


=

.
Proposition4.1.2.
r


=

.
Proof.
Wewillonlyprovethat
r



˚
(
a;
1)=


˚
(
a;
1)
:
Theothercasesaresimilar.We
showedpreviouslythat


˚
(
a;
1)=(
a;t
P
)
d;k
(
E
)
,whichcorrespondstothecocycle
(
;˝
)
7!
8
>
>
>
>
<
>
>
>
>
:
1


(
d
p
a
)
d
p
a

Z
ˆ
+

˝
(
d
p
a
)
d
p
a

Z
ˆ
<d
t
P
else
(

)
in
H
2
(
G
k
;
k
(
E
)

).Thisgivesanelementin
H
2
(
G
k
;
Prin(
E
))via
(
;˝
)
7!
8
>
>
>
>
<
>
>
>
>
:
1


(
d
p
a
)
d
p
a

Z
ˆ
+

˝
(
d
p
a
)
d
p
a

Z
ˆ
<d
d
(
P
)

d
(0)else
:
43
Ontheotherhandforany

2
G
k
,

(
˚
(
a;
1))(

)=
˚
(
a;
1)(

)=
"

(
d
p
a
)
d
p
a
#
Z
ˆ
P:
Nowwefollowtheproofofthesnakelemmatocalculatetheimageof

(
f
)underthe
connectinghomomorphism
H
1
(
k;E
(
k
))
!
H
2
(
k;
Prin(
E
))
inducedbythesequence
0
!
k
(
E
)

!
Prin(
E
)
!
Div(
E
)
!
0
:
Firstliftitto

7!
"

(
d
p
a
)
d
p
a
#
Z
ˆ
((
P
)

(0))
2
Div
0
(
E
)
:
Nowusetheboundarymaptoget
(
;˝
)
7!

0
@
"
˝
(
d
p
a
)
d
p
a
#
Z
ˆ
((
P
)

(0))
1
A

"
˝
(
d
p
a
)
d
p
a
#
Z
ˆ
((
P
)

(0))+
"
˝
(
d
p
a
)
d
p
a
#
Z
ˆ
((
P
)

(0))
=
"
˝
(
d
p
a
)
d
p
a
#
Z
ˆ
((
P
)

(0))

"
˝
(
d
p
a
)
d
p
a
#
Z
ˆ
((
P
)

(0))+
"
˝
(
d
p
a
)
d
p
a
#
Z
ˆ
((
P
)

(0))
=
8
>
>
>
>
<
>
>
>
>
:
d
(
P
)

d
(0)if


(
d
p
a
)
d
p
a

Z
ˆ
+

˝
(
d
p
a
)
d
p
a

Z
ˆ

d
1else
;
whichcoincideswithwhatwecalculatedin(

).Thestatementfollows.
44
Proposition4.1.3.

(ker(

))=0
:
Proof.
Recallthatker(

)=Im(

)andlet
R
2
E
(
k
).Bythepreviousproposition
r




(
R
)=



(
R
)istrivial.Thusthealgebra



(
R
)isintheimageoftheBr(
k
)
!
Br(
E
).Itremains
toshowthatthespecializationof



(
R
)at0istrivial.Thecup-productcommuteswith
specializationataclosedpoint[Bre97,Chapter2,8.2],i.e.

[
T
0
]
[
[
T
]

S
=

[
T
0
]

S
[
([
T
])
S
forevery
S
2
E
(
k
)andevery[
T
0
]
2
H
1
et
(
E;M
).Bydeof

,
(



(
R
))
S
=

(
R
)
[T
S
2
Br(
k
)
forany
S
2
E
(
k
).Inparticular,(



(
R
))
0
=

(
R
)
[T
0
.Thespecializationof
T
at0admits
apoint(thepoint0)andisthereforethetrivialtorsor.Wededucethat(



(
R
))
0
istrivial
andthussois



(
R
).
Theorem4.1.4.
Supposethatthe
d
-torsion
M
of
E
is
k
-rational.Fixtwogenerators
P
and
Q
of
M
.Let
t
P
;t
Q
2
k
(
E
)
withdivisors
div(
t
P
)=
d
(
Q
)

(0)
and
div(
t
Q
)=
d
(
Q
)

q
(0)
.
Thenthe
d
-torsionof
Br(
E
)
decomposesas
d
Br(
E
)=
d
Br(
k
)

I
andeveryelementin
I
canberepresentedasatensorproduct
(
a;t
P
)
d;k
(
E
)


b;t
Q

d;k
(
E
)
with
a;b
2
k

.
Thisresultwaspreviouslyknownandwasprovedusingtmethodsforinstancein
45
[CRR16,Remark6.3].
Proof.
Proposition4.1.2andproposition4.1.3implythat

inducesthedesiredsplit.There-
fore
d
Br(
E
)=
d
Br(
k
)

Im(

)
:
Thetheoremfollowsfromproposition4.1.1.
4.2
[
L
:
k
]
iscoprimeto
q
Fromnowonlet
q
=
d
beanoddprimeanddroptheassumptionthat
M
is
k
-rational.
ConsiderthenaturalGaloisrepresentation
:
G
k
!
Aut(
M
)=
GL
2
(
F
q
)
:
Denoteby
L
theofthekernelofThedegreeoftheGaloisextension
L
over
k
dividestheorderof
GL
2
(
F
q
),whichis(
q
+1)
q
(
q

1)
2
.
Let
T
bethetorsorgivenbymultiplicationby
d
on
E
.Denotethegenericpointsof
E
and
E

Spec
L
by

:Spec
k
(
E
)
!
E
and

L
:Spec
L
(
E
)
!
E

Spec
L
,respectively.
Considerthepull-back


L
(
T
)of
T
to
L
(
E
)andthepull-back


k
(
T
)of
T
to
k
(
E
).By
proposition3.1.1andproposition3.2.1weseeimmediatelythatres



k
(
T
)

=


L
(
T
).By
[NSW08,Ch.1,Proposition1.5.3(iii)and(iv)]andtheconstructionof

,thefollowing
diagramcommutes
H
1
(
k;M
)
res
/
/

k


H
1
(
L;M
)
cor
/
/

L


H
1
(
k;M
)

k


q
Br(
E
)
res
/
/
q
Br(
E

L
)
cor
/
/
q
Br(
E
)
:
Throughoutthissection,weassumethat
q
doesnotdividetheorder[
L
:
k
].The
46
corestrictionmap
cor:
q
Br(
E

L
)
!
q
Br(
E
)
issurjectiveandeveryelementin
I
canbewrittenascor(
A
)with
A
2
q
Br(
E
).Wesummarize
thisobservationinthefollowingtheorem.
Theorem4.2.1.
Let
t
P
;t
Q
2
L
(
E
)
withdivisors
div(
t
P
)=
q
(
Q
)

q
(0)
and
div(
t
Q
)=
q
(
Q
)

q
(0)
.Thenthe
q
-torsionof
Br(
E
)
decomposesas
q
Br(
E
)=
q
Br(
k
)

I
andeveryelementin
I
canberepresentedasatensorproduct
cor(
a;t
P
)
q;L
(
E
)

cor

b;t
Q

q;L
(
E
)
with
a;b
2
L

.
Remark4.2.2.
Notethatcorestrictionisingeneralnotinjective.Togetasmallersetof
generators,observethatby[NSW08,Ch,1,Corollary1.5.7]theimageoftherestrictionmap
H
1
(
k;M
)
!
H
1
(
L;M
)
coincideswiththeimageoftheNormmap
N
L=k
:
H
1
(
L;M
)
!
H
1
(
L;M
)
:
NowbyKummertheory
H
1
(
L;M
)
˘
=
L

=
(
L

)
q

L

=
(
L

)
q
viatheisomorphism
˚
.Let
g
2
G
k
and
(
a;b
)
2
L

=
(
L

)
q

L

=
(
L

)
q
.Supposethat
g

1
(
P
)=
c
1
P

c
2
Q
and
47
g

1
(
Q
)=
c
3
P

c
4
Q
.Theactionof
g
compatiblewith
˚
is
g:
(
a;b
)=
˚

1
g:˚
(
a;b
)
=
˚

1
0
B
@
g:
0
B
@

7!


(
q
p
a
)
q
p
a

ˆ
P

2
4


q
p
b

q
p
b
3
5
ˆ
Q
1
C
A
1
C
A
=

g

1
(
a
)

c
1

g

1
(
b
)

c
3
;

g

1
(
a
)

c
2

g

1
(
b
)

c
4

:
Nowtheimageoftherestrictionfollowedby
˚
coincideswiththeimageofthenormon
L

=
(
L

)
q

L

=
(
L

)
q
undertheaboveaction.
4.3
[
L
:
k
]
equals
q
Inthissection,weassumethat
L
isofdegree
q
over
k
.Afterrenaming
P
and
Q
wemay
assumewithoutlossofgeneralitythatthereissome
˙
2
G
k
suchthat
˙
(
Q
)=
P

Q
and
˙
generates
G
k
=G
L
.Fixacosetrepresentative~
˙
of
˙
in
G
k
(
E
)
(comparesection3.2).To
avoidconfusion,wewilladdsubscriptstothemaps
;r
,and

todenotetheirof
Additionallydenoteaprimitiveelementfortheextension
L=k
by
l
.
Considerthediagram
0
/
/
H
1
(
G
k
=G
L
;M
)
inf
/
/
H
1
(
G
k
;M
)
res
/
/

k


H
1
(
G
L
;M
)
G
k
=G
L

L


q
Br(
E
)
res
/
/
q
Br(
E

k
Spec
L
)
;
wheretherowistherestrictionexactsequence.Thediagramcommutesby
48
constructionof

andsincetherestrictionmapandthecup-productcommute[NSW08,Ch.
1Proposition1.5.3(iii)].Wewilldescribetheimageofthemap,andthen
exploretherestrictionafterwards.Wewillapplythefollowingtechnicallemmathroughout.
Lemma4.3.1.
P
q

1
i
=0
˙
i
(
R
)=0
forevery
R
2
M
.
Proof.
Let
R
=
mP

nQ
2
M
.Wecalculatedirectlythat
q

1
X
i
=0
˙
i
(
mP

nQ
)=
q

1
X
i
=0
(
mP

inP

nQ
)=
mqP

q
(
q

1)
2
nP

nqQ
=0
:
4.3.1TheImageoftheMap
Lemma4.3.2.
Thegroup
H
1
(
G
k
=G
L
;M
)
iscyclicofrank
q
withgenerator
f
L
dby
f
L
(
˙
)=
Q
.
Proof.
Lemma4.3.1impliesthat
f
L
(
˙
q
)=
P
q

1
i
=0
˙
i
f
L
(
˙
)=0andthus
f
L
acocycle.
Since
G
k
=G
L
iscyclicwithgenerator
˙
,everyelement
f
in
H
1
(
G
k
;G
L
;M
)isdetermined
by
f
(
˙
).Furthermore,if
f
(
˙
)=
mP

nQ
,then
f
(
˙
)

˙
(
mP
)=
mP

nQ

mP
=
nQ
=
f
b
L
(
˙
)
:
Thestatementfollows.
Let

Q
2
k
(
E
)with

q
Q
=
t
Q
.Consider
n
Q
=
Q
q

1
i
=0
~
˙
i


Q

.Notethat
n
Q
is
tobetheelementin
k
(
E
)with
n
q
Q
=
N
L
(
E
)
=k
(
E
)
(
t
Q
).Furthermore,notethatdiv
n
Q
=
P
q

1
i
=0

˙
i
(
Q
)

(0)

.
49
Proposition4.3.3.

k
(inf(
f
L
))
istheinverseoftheBrauerclassofthesymbolalgebra

l
q
;n
Q

q;k
(
E
)
;
where

Q
2
k
(
E
)
with

q
Q
=
t
Q
.
Proof.
Weshowthat
Q
q

1
i
=0
~
˙
i


Q

2
k
(
E
).Let

2
G
L
(
E
)
.Byourpreviouscalcula-
tionsandwith
x
0
and
g
Q
asintheproofofproposition3.2.1,wededucethatthereissome
R
2
M
suchthat

(

Q
)=
R:x
0
(
g
Q
).Then

0
@
q

1
Y
i
=0
~
˙
i


Q

1
A
=
0
@
p

1
X
i
=0
˙
i
(
R
)
1
A
:x
0
(
g
Q
)=

Q
(4.4)
bylemma4.3.1.Now
Q
q

1
i
=0
~
˙
i


Q

isobviouslyby~
˙
andtherefore
Q
q

1
i
=0
~
˙
i


Q

2
k
(
E
)
:
Let
;˝
2
G
k
(
E
)
anddenote

=

0
~
˙
i
;˝
=
˝
0
~
˙
j
with

0
;˝
0
2
G
L
(
E
)
.Thenby
of

weseethat

k
(inf(
f
L
))(
;˝
)=e
0
@
(
i

1)
i
2
P

iQ;˙
i
0
@
"
˝
0


Q


Q
#
ˆ
P


˝
0
(

P
)

P

ˆ
Q
1
A
1
A
=e
0
@
(
i

1)
i
2
P

iQ;
"
˝
0


Q


Q

˝
0
(

P
)

P


i
#
ˆ
P


˝
0
(

P
)

P

ˆ
Q
1
A
=e
 
(
i

1)
i
2
P;


˝
0
(

P
)

P

ˆ
Q
!
e
0
@
iQ;
"
˝
0


Q


Q

˝
0
(

P
)

P


i
#
ˆ
P
1
A
=

˝
0
(

P
)

P


(
i

1)
i
2
 
˝
0
(

Q
)

Q
!

i

˝
0
(

P
)

P

i
2
=

˝
0
(

P
)

P

(
i
+1)
i
2
 
˝
0
(

Q
)

Q
!

i
50
Nowconsiderthemap
g
:
G
k
(
E
)
!
k
(
E
)


7!

0
 
i

1
Y
n
=0
~
˙
n
(

Q
)
!
;
(4.5)
where

=

0
~
˙
i
forsome

0
2
G
L
(
E
)
.Thetialof
g
canbecalculatedasfollows:If
i
+
j<q
,then
dg
(
;˝
)=

0

Q
i

1
n
=0
~
˙
n
(

Q
)



˝
0

Q
j

1
n
=0
~
˙
n
(

Q
)

(
˝
)
0

Q
i
+
j

1
n
=0
~
˙
n
(

Q
)

=

0

Q
i

1
n
=0
~
˙
n
(

Q
)



0
~
˙
i
˝
0


Q
j

1
n
=0
~
˙
n
(

Q
)



0
~
˙
i
˝
0
~
˙

i


Q
i
+
j

1
n
=0
~
˙
n
(

Q
)

=

0

Q
i

1
n
=0
~
˙
n
(

Q
)



0
~
˙
i
˝
0
~
˙

i


Q
i

1
n
=0
~
˙
n
(

Q
)

=
Q
i

1
n
=0
~
˙
n
(

Q
)

~
˙
i
˝
0
~
˙

i


Q
i

1
n
=0
~
˙
n
(

Q
)

=
i

1
Y
n
=0
~
˙
n
(

Q
)

~
˙
i
˝
0
~
˙

i
+
n

(

Q
)
=
i

1
Y
n
=0
~
˙

i
+
n
(

Q
)

~
˙
i
˝
0
~
˙

i
+
n


Q
)
=
i

1
Y
n
=0


P
˝
0
(

P
)


i
+
n

Q
˝
0
(

Q
)
=


P
˝
0
(

P
)


i
2
+
(
i

1)
i
2


Q
˝
0
(

Q
)

i
=


P
˝
0
(

P
)


(
i
+1)
i
2


Q
˝
0
(

Q
)

i
51
Asimilarcalculationshowsthatif
i
+
j

q
,then
dg
(
;˝
)=
0
@
q

1
Y
n
=0
~
˙
n
(

Q
)
1
A


P
˝
0
(

P
)


(
i
+1)
i
2


Q
˝
0
(

Q
)

i
Thestatementfollowsbysubtractingthistrivialcocycleandapplyingproposition2.2.18.
Proposition4.3.4.
r



inf(
f
L
)=


inf(
f
L
)
.
Proof.
Bythepreviouslemma,wegetfrom

(inf(
f
L
))theelement
(

0
˙
i
;˝
0
˙
j
)
7!
8
>
>
>
<
>
>
>
:
1
i
+
j<q
P
q

1
i
=0
(
˙
(
Q
)

(0))
i
+
j

q
in
H
2
(
G
k
;
Prin(
E
)),where

0
;˝
0
2
G
k
.Ontheotherhand,
r
(inf(
f
L
))canbepresentedby
thecocycle

0
˙
i
7!
i
X
m
=0
˙
m
(
Q
)

0
:
Thisliftstothemap

0
˙
i
7!
i
X
m
=0
(
˙
m
(
Q
))

(0)
2
Div
0
(
E
)
;
52
andadirectcomputationoftheboundarymapgives
(

0
˙
i
;˝
0
˙
j
)
7!
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
˙
i

P
j
m
=0
(
˙
m
(
Q
))

(0)



P
i
+
j
m
=0
(
˙
m
(
Q
))

(0)

+

P
i
m
=0
(
˙
m
(
Q
))

(0)

i
+
j<q
˙
i

P
j
m
=0
(
˙
m
(
Q
))

(0)



P
i
+
j

q
m
=0
(
˙
m
(
Q
))

(0)

+

P
i
m
=0
(
˙
m
(
Q
))

(0)

i
+
j

q
=
8
>
>
>
<
>
>
>
:
1
i
+
j<q

P
q

1
m
=0
(
˙
m
(
Q
))

(0)

i
+
j

q
;
whichcoincideswiththepreviouscalculation.
Corollary4.3.5.

inducesasplittothesequence
0
/
/
q
Br(
k
)
i
/
/
q
Br(
E
)
r
/
/
q
H
1

k;E

k

/
/
0
:
Proof.
Wededucefromthepreviouspropositionandproposition4.1.2that
r


=

.Fur-
thermore,

(ker(

))=0followsasintheproofofproposition4.1.3.
4.3.2TheImageoftheRestrictionMap
Wenowcalculatetheimageofthecomposition
˚

1

reswith
˚
asineq.(4.3).By[Ser79,
ChapterVII,Section5]theactionof
G
k
on
L

=
(
L

)
q

L

=
(
L

)
q
compatiblewith
˚
is
givenby
˙:
(
a;b
)=
˚

1
(
˙:˚
(
a;b
))=

˙

1
(
a
)
˙

1
(
b
)
;˙

1
(
b
)

:
53
Lemma4.3.6.
Undertheisomorphism
˚

1
,thedset
H
1
(
L;M
)
G
k
=G
L
correspondsto

a;
a
˙
(
a
)

:
˙
(
a
)
2

˙
2
(
a
)
a
mod(
L

)
q
˙
:
Proof.
Let(
a;b
)
2
L

=
(
L

)
q

L

=
(
L

)
q
bebytheaboveaction.Then
a

˙

1
(
a
)
˙

1
(
b
)
,
whichimpliesthat
b

a
˙
(
a
)
.Now
b

˙

1
(
b
)andthus
a
˙
(
a
)

˙

1
(
a
)
a
whichimpliesthat
a
2

˙
(
a
)
˙

1
(
a
)orequivalently
˙
(
a
)
2

˙
2
(
a
)
a
.
Lemma4.3.7.
f
2
H
1
(
L;M
)
G
k
=G
L
isintheimageoftherestrictionmapifandonlyif
f
(

q
)=0
forany

2
G
k
.
Proof.
Let

2
G
k
andsupposethat
f
isintheimageoftherestrictionmapwithpreimage
g
.Thenwecanwrite

=

0
˙
i
forsome

0
2
G
L
.Wecalculatedirectlyusinglemma4.3.1
that
f
(

q
)=
g
(

q
)=
q

1
X
i
=0

q
g
(

)=
q

1
X
i
=0
˙
iq
g
(

)=
q

1
X
i
=0
˙
i
g
(

)=0
:
Fortheconverse,assumethat
f
theconditionthat
f
(

q
)=0forany

2
G
k
.In
particular
f
(
˙
q
)=0.
g
2
H
1
(
k;M
)bysetting
g
(

)=
f
(

0
),where

=

0
˙
i
for

0
2
G
L
.Thisiswasforany
;˝
2
G
k
with

=

0
˙
i
;˝
=
˝
0
˙
j
and

0
;˝
0
2
G
L
wehave
that
g
(
˝
)=
g


0

˙
i
˝
0
˙

i

˙
i
+
j

=
g


0

g

˙
i
˝
0
˙

i

=
g


0

˙
i
g
(
˝
0
)=
g
(

)
g
(
˝
)
:
Wewillnowproveatechnicallemmathatwillbeusefultodeterminetheimageofthe
restriction.
Lemma4.3.8.
Let
k
beaofcharacteristicprimeto
q
containingaprimitive
q
-throotof
unity.Let
k
ˆ
L
ˆ
F
beatowerofextensionssothateachextensionisGaloisofdegree
q
.Let
a
2
L

suchthat
F
=
L
(
q
p
a
)
.Fixarepresentative
˙
2
G
k
thatgenerates
Gal(
L=k
)
.
54
Supposethatforevery

2
G
k
wehavethat

q

q
p
˙
i
(
a
)

=
q
p
˙
i
(
a
)
for
0

i<q
.Then
thereexistssome
b
2
k

suchthat
a

b
mod(
L

)
q
.
Proof.
Assumethat
a
62
k

.Fix
˙
2
G
k
suchthat
˙
generatesGal(
L=k
).DenotetheGalois
closureoftheextension
F
over
k
by
~
L
.ByGaloistheory
~
L
=
L

q
p
a;
q
p
˙
(
a
)
;:::;
q
p
˙
q

1
(
a
)

andGal(
~
L=L
)isisomorphicto(
Z
=q
Z
)
r
for
r
=1,or
r
=
q
.Weprovethelemmabycon-
tradiction.Soassumethat
r
=
q
.Let
˝
2
Gal(
~
L=L
)betheelementwith
˝
(
q
p
a
)=
ˆ
q
p
a
and
˝

˙
i
q
p
a

=
˙
i
q
p
a
for1

i<q
.Now(
˙˝
)(
q
p
a
)=(
ˆ˙
q
)(
q
p
a
)=
ˆ
(
q
p
a
),whichisa
contradictiontoourassumptions.Weconcludethat
F=k
isGaloisofdegree
q
2
.Wewantto
showthatGal(
F=k
)=
Z
=q
Z

Z
=q
Z
.
NextsupposethatGal(
F=k
)=
Z
=q
2
Z
anddenotethegeneratorby
˝
.Then
˝
q

L
,
as
˝
q
(
q
p
a
)=
q
p
a
impliesthat
˝
q
(
a
)=
a
and
L
=
k
(
a
).Hence
˝
2
Gal(
F=L
),whichimplies
that
˝
isoforder
q
,acontradiction.WeconcludethatGal(
F=k
)
˘
=
Z
=q
Z

Z
=q
Z
.Consider
the
F
h
˙
i
,whichisadegree
q
extensionof
k
.ByKummertheory,thereexists
anelement
b
2
k

sothat
F
h
˙
i
=
k

q
p
b

.Finally,
F
=
L

q
p
b

=
L
(
q
p
a
)andthusby
Kummertheory
a

b
mod(
L

)
q
.
Proposition4.3.9.
Theimageoftherestrictionmapcorrespondstotheset

k

=

(
L

)
q
\
k


f
1
gˆ
L

=
(
L

)
q

L

=
(
L

)
q
undertheisomorphism
˚

1
.
Proof.
Let(
a;b
)
2
L

=
(
L

)
q

L

=
(
L

)
q
beintheimageoftherestrictionmap.There
existssome
f
2
H
1
(
k;M
)sothat
˚

1

res(
f
)=(
a;b
).Then(
a;b
)isnecessarilyinthe
preimage
˚

1

H
1
(
L;M
)
G
k
=G
L

andbylemma4.3.6weseethat(
a;b
)


a;
a
˙
(
a
)

sothat
55
˙
(
a
)
2

˙
2
(
a
)
a
mod(
L

)
q
.Itremainstoshowthatwecanchoose
a
2
k

.
Byof
˚
andbylemma4.3.7,wegetthat

q
(
q
p
a
)=
q
p
a
and

q

q
q
a
˙
(
a
)

=
q
q
a
˙
(
a
)
forany

2
G
k
andforanychoiceof
q
th
rootof
a
˙
(
a
)
.Usingtheconditionthat
˙
(
a
)
2

˙
2
(
a
)
a
,wededucethat

p

q
p
˙
i
(
a
)

=
q
p
˙
i
(
a
)forany
i
.Thestatementfollows
fromlemma4.3.8.
Thefollowingtheoremsummarizestheresultsofthissection.
Theorem4.3.10.
Supposethat
[
G
k
:
G
L
]
isoforder
q
andassumethatthereissome
˙
2
G
k
with
˙
(
Q
)=
P

Q
suchthat
˙
generates
G
k
=G
L
.Additionallydenoteaprimitive
elementfortheextension
L=k
by
l
.
q
Br(
E
)=
q
Br(
k
)

I
and
I
hasgenerators
n

l
q
;n
Q

k
(
E
)
;
(
a;t
P
)
k
(
E
)
:
a
2
k

o
;
where
n
Q
2
k
(
E
)
with
n
q
Q
=
N
L
(
E
)
=k
(
E
)
(
t
Q
)
.
4.4
q
dividesthedegree
[
L
:
k
]
Supposeforthissectionthat
q
divides[
L
:
k
].Let
k
ˆ
L
0
ˆ
L
beanintermediateso
that
L=L
0
isaGaloisextensionofdegree
q
and
q
doesnotdividethedegree
L
0
=k
.After
renaming
P
and
Q
,wemayassumethatthereissomeelement
˙
2
G
k
sothat
˙
generates
Gal(
L=L
0
)and
˙
(
P
)=
P
and
˙
(
Q
)=
P

Q
.Furthermore,let
l
2
L
with
l
q
2
L
0
and
L
=
L
0
(
l
).Fix
t
P
;t
Q
2
L
(
E
)withdiv(
t
P
)=
q
(
P
)

q
(0)anddiv(
t
Q
)=
q
(
Q
)

q
(0).
Assumeadditionallythat
t
P

[
q
]
;t
Q

[
q
]
2

L
(
E
)


d
:
Furthermore,let

Q
2
k
(
E
)sothat
56

q
Q
=
t
Q
.
Althoughtheextension
L
0
=k
mightnotbeGalois,restrictionfollowedbycorestric-
tioncoincideswithmultiplicationby[
L
0
:
k
],whichisanisomorphism.Usingtheprevious
sectionwededucethefollowingresult.
Proposition4.4.1.
Undertheaboveassumptions,theBrauergroupdecomposesas
q
Br(
E
)=
q
Br(
k
)

I
andeveryelementin
I
canbeexpressedusingthegenerators
n
cor

l
q
;n
Q

L
0
(
E
)
;
cor(
a;t
P
)
L
0
(
E
)
:
a
2
L

o
:
Thiscompletesthedescriptionofthegeneratorsof
q
Br(
E
).
57
Chapter5
Relations
InthischapterwedescribetherelationsintheBrauergroup.Recallthatthatanelement
in
I
asbeforeistrivialifandonlyifitisintheimageofthecomposition
E
(
k
)
=
[
d
]
E
(
k
)

/
/
H
1
(
k;M
)

/
/
d
Br(
k
)
:
Asinthesectionbefore,wewillconsidervariouscasesdependingontherationalityofthe
d
-torsion
M
.
5.1
M
is
k
-rational
Assumeforthissectionthat
M
is
k
-rational.Recalltheisomorphism
˚
:

k

=

k


d



k

=

k


d

!
H
1
(
k;M
)
(
a;b
)
7!

c
a;b

fromeq.(4.3).Wedescribethecomposition
˚

1


.
58
Proposition5.1.1.
Let
R
2
E
(
k
)
=
[
d
]
E
(
k
)
andlet
t
P
;t
Q
2
k
(
E
)
with
div(
t
P
)=
d
(
P
)

d
(0)
,
div(
t
Q
)=
d
(
Q
)

d
(0)
.Assumethat
t
P

[
d
]
;t
Q

[
d
]
2
(
k
(
E
)

)
d
.Then
˚

1


(
R
)=
8
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
:
(1
;
1)
R
=0

t
Q
(
P
)
;
t
P
(
P

Q
)
t
P
(
Q
)

R
=
P

t
Q
(
P

Q
)
t
Q
(
P
)
;t
P
(
Q
)

R
=
Q

t
Q
(
R
)
;t
P
(
R
)

else
TheproofofthispropositionisinspiredbyacomputationoftheKummerpairingin
[Sil09,Ch.X,Theorem1.1].
Proof.
Let
R
2
E
(
k
)
=dE
(
k
)
nf
0
;P;Q
g
andsome
S
2
E
(
k
)with[
d
]
S
=
R
.Let
t
P
;t
Q
as
aboveand
g
P
;g
Q
2
k
(
E
)with
g
d
P
=
t
P

[
d
]and
g
d
Q
=
t
Q

[
d
].Sincethedivisorsof
g
P
and
g
Q
are
G
k
-invariant,wemaychoose
g
P
;g
Q
2
k
(
E
).Bytheof
˚
weseethat
for
˚
(
f
)=(
a;b
)forsomecocycle
H
1
(
k;M
)meansthat
e(
f
(

)
;P
)=


d
p
b

d
p
b
ande(
f
(

)
;Q
)=


d
p
a

d
p
a
:
TheWeilpairing
e(

(
S
)

S;P
)=
g
P
(

(
S
)

S

S
)
g
P
(
S
)
=
g
P
(

(
S
))
g
P
(
S
)
=

(
g
P
(
S
))
g
P
(
S
)
:
Additionallybyof
g
P
,weseethat
g
P
(
S
)
d
=
t
P

[
d
](
S
)=
t
P
(
R
).Asimilarresult
holdsfor
Q
aswell.Therefore
˚

1


(
R
)=

t
Q
(
R
)
;t
P
(
R
)

.Theotherresultsfollowby
bilinearityoftheWeilpairing.
59
Summarizingtheseresults,weconcludetheorem1.0.5.
Fromnowonassumethat
d
=
q
isanoddprimeand
L
isthesmallestGaloisextension
of
k
sothat
M
is
L
-rational.
5.2
[
L
:
k
]
iscoprimeto
q
Supposethroughoutthissectionthat
q
doesnotdividetheorder[
L
:
k
].Considerthe
followingcommutativediagram
E
(
k
)
=
[
q
]
E
(
k
)
res
/
/

k


E
(
L
)
=
[
q
]
E
(
L
)
cor
/
/

L


E
(
k
)
=
[
q
]
E
(
k
)

k


H
1
(
k;M
)
res
/
/
H
1
(
L;M
)
cor
/
/
H
1
(
k;M
)
;
wherethehorizontalcompositionscoincidewithmultiplicationby[
L
:
k
]andaretherefore
isomorphisms.Thus,theimageof

k
isalsogivenbytheimageofthecomposition

k

cor=
cor


L
.Usingthedescriptionoftheimageof

L
intheprevioussection,wededucethe
followingresult.
Proposition5.2.1.
Supposethat
[
L
:
k
]
isnotdivisibleby
q
.Fixtwogenerators
P
and
Q
of
M
andandlet
t
P
;t
Q
2
L
(
E
)
with
div(
t
P
)=
q
(
P
)

q
(0)
and
div(
t
Q
)=
q
(
Q
)

q
(0)
.
Assumeadditionallythat
t
P

[
q
]
;t
Q

[
q
]
2

L
(
E
)


d
:
Anelement
cor(
a;t
P
)
L
(
E
)

cor(
b;t
Q
)
L
(
E
)
in
I
istrivialifandonlyifitissimilartooneofthefollowing
60

cor

t
Q
(
P
)
;t
P

k
(
E
)

cor

t
P
(
P

Q
)
t
P
(
Q
)
;t
Q

k
(
E
)
,

cor

t
Q
(
P

Q
)
t
Q
(
P
)
;t
P

k
(
E
)

cor

t
P
(
Q
)
;t
Q

k
(
E
)
;
or

cor

t
Q
(
R
)
;t
P

k
(
E
)

cor

t
P
(
R
)
;t
Q

k
(
E
)
forsome
R
2
E
(
k
)
nf
0
;P;Q
g
.
Thefollowingobservationwillbeusefultocalculatethesecorestrictionsexplicitly.Con-
siderthefollowingcommutativediagram
E
(
k
)
/
/

k


E
(
L
)
/
/

L


E
(
k
)
/
/

k


E
(
L
)
/
/

L


E
(
k
)

k


H
1
(
k;M
)
res
/
/

k


H
1
(
L;M
)
cor
/
/

L


H
1
(
k;M
)
res
/
/

k


H
1
(
L;M
)
cor
/
/

L


H
1
(
k;M
)

k


q
Br
E
/
/
q
Br
E
L
/
/
q
Br
E
/
/
q
Br
E
L
/
/
q
Br
E
;
wherethecompositionofmorphismsalongarowgivemultiplicationby[
L
:
k
]
2
,whichisan
isomorphism.Furthermore,thecompositionres

corcoincideswiththeNormmap[NSW08,
Ch.1,Corollary1.5.7].Therefore,anelementin
I
istrivialifitliesintheimageofthe
compositioncor


L

N
L=k


L

res.Insection7.2,weseehowthisobservationcanbe
appliedtothecalculationoftherelationsin
I
.
5.3
[
L
:
k
]=
q
Throughoutthissectionsupposethat[
L
:
k
]=
q
andsomegenerator
˙
ofGal(
L=k
).Let
P;Q
2
M
sothat
˙
(
P
)=
P
and
˙
(
Q
)=
P

Q
.Furthermore,let
l
2
L
with
l
q
2
k
and
L
=
k
(
l
).Fix
t
P
;t
Q
2
L
(
E
)withdiv(
t
P
)=
q
(
P
)

q
(0)anddiv(
t
Q
)=
q
(
Q
)

q
(0).Assume
additionallythat
t
P

[
q
]
;t
Q

[
q
]
2

L
(
E
)


d
:
Fix~
˙
2
~
G
L=k
asinsection4.3.Furthermore,
61
let

Q
2
k
(
E
)sothat

q
Q
=
t
Q
anddenote
n
Q
=
Q
n

1
i
=0
~
˙
i
(

Q
).
Considerthecommutativediagramwithexactrowsandcolumns
0


0


0
/
/
E
(
k
)
\
[
q
]
E
(
L
)
[
q
]
E
(
k
)
inf
/
/

L=k


E
(
k
)
=
[
q
]
E
(
k
)
res
/
/

k


E
(
L
)
=
[
q
]
E
(
L
)

L


0
/
/
H
1
(Gal(
L=k
)
;M
)
inf
/
/
H
1
(
k;M
)
res
/
/
H
1
(
L;M
)
;
where

L=k
isthemapinducedby

k
.Itisimmediatethat

L=k
isinjective.Recall
that
H
1
(Gal(
L=k
)
;M
)iscyclicoforder
q
withgenerator
f
L
.Furthermore,wesawthat

L
(inf(
f
L
))=

l
q
;n
Q

k
(
E
)
(proposition4.3.3).Wededucethefollowingresult.
Proposition5.3.1.
TheBrauerclassof

l
q
;n
Q

k
(
E
)
istrivial,thatis,itisintheimageof
themap
Br(
k
)
!
Br(
E
)
,ifandonlyifthequotient
E
(
k
)
\
[
q
]
E
(
L
)
[
q
]
E
(
k
)
isnon-trivial.
Recallthatbyproposition4.3.9anyelementintheimageof

˚

1

res

canbewritten
as(
a;
1)forsome
a
2
k

.
Proposition5.3.2.
TheBrauerclassof
(
a;t
P
)
k
(
E
)
istrivialifandonlyifthereissome
R
2
E
(
k
)
=
[
q
]
E
(
k
)
sothat
˚

1
(
a;
1)=

L
(
R
)
.
5.4
q
divides
[
L
:
k
]
Supposethat
q
divides[
L
:
k
]andusethenotationusedinsection4.4.Recallthatevery
elementin
I
canbewrittenascor(
A
)forsome
A
2
q
Br(
E

Spec(
L
0
)).Suchanelementis
trivialifandonlyifitissimilarto

L
0


L
0
.Remarkthatsomecorestrictionsofelementsin
62
q
Br(
E
L
)maycoincideandwedonotaccountforthisinourdescription.
63
Chapter6
Conclusions{TheAlgorithm
Inthischapter,wesummarizetheresultsfromthepreviouschaptersandassemblethe
algorithmtocalculategeneratorsandrelationsoftheBrauergroup.Let
k
beaof
characteristictfrom2or3andlet
q
beanoddprime.Assumethat
q
iscoprimeto
thecharacteristicof
k
andthat
k
containsaprimitive
q
-throotofunity.Let
E
beanelliptic
curveover
k
.Denoteby
M
the
q
torsionof
E
(
k
).
TheBrauergroupof
E
decomposesas
q
Br(
E
)=
q
Br(
k
)

I
andgenerators
G
andrelations
R
of
I
canbecalculatedusingthefollowingalgorithm.
1.
DeterminethekernelofthenaturalGaloisrepresentation
:
G
k
!
End(
M
)=
GL
2

F
q

:
Denoteby
L
theofthiskernel.
2.
(a)
If
q
dividestheorderof
L=k
,someintermediate
L
0
sothat
L=L
0
isa
Galoisextensionofdegree
q
.Let
P
and
Q
beelementsin
M
sothatGal(
L=L
0
)
64
isgeneratedby
˙
with
˙
(
P
)=
P
and
˙
(
Q
)=
P

Q
.Set
G
L
0
=
n

l
q
;n
Q

L
0
(
E
)
;
(
a;t
P
)
L
0
(
E
)
:
a
2
L

o
;
where
t
P
2
L
0
(
E
)withdiv(
t
P
)=
q
(
P
)

q
(0)and
n
Q
2
L
0
(
E
)withdiv(
n
Q
)=
P
q

1
i
=0
˙
i
(
Q
)=
P
q

1
i
=0
(
iP
+
Q
)
:
Furthermore,let
t
Q
2
L
(
E
)withdiv(
t
Q
)=
q
(
Q
)

q
(0)and
n
q
Q
=
N
L
(
E
)
=k
(
E
)
(
t
Q
).
(b)
If
q
doesnotdividetheorderof
L=k
,somegenerators
P
and
Q
of
M
.Set
G
L
0
=
n
(
a;t
P
)
L
(
E
)
;

b;t
Q

L
(
E
)
:
a;b
2
L

o
;
where
t
P
;t
Q
2
L
(
E
)withdiv(
t
P
)=
q
(
P
)

q
(0)anddiv(
t
Q
)=
q
(
Q
)

q
(0).
3.
Set
R
L
=
8
>
>
>
>
<
>
>
>
>
:

t
Q
(
P
)
;t
P

L
(
E
)


t
P
(
P

Q
)
t
P
(
Q
)
;t
Q

L
(
E
)
;

t
Q
(
P

Q
)
t
Q
(
P
)
;t
P

L
(
E
)


t
P
(
Q
)
;t
Q

L
(
E
)
9
>
>
>
>
=
>
>
>
>
;
[
n

t
Q
(
R
)
;t
P

L
(
E
)


t
P
(
R
)
;t
Q

L
(
E
)
:
R
2
E
(
L
)
nf
P;Q
g
o
:
(a)
If
q
dividestheorderof
L=k
,let
R
L
0
;
res
=
n
(
a;t
P
)
L
0
(
E
)
:res(
a;t
P
)
L
0
(
E
)
2R
L
o
:
65
Further,ifthequotient
E
(
k
)
\
[
q
]
E
(
L
)
[
q
]
E
(
k
)
isnottriviallet
R
L
0
;
inf
=
n

l
q
;n
Q

L
0
(
E
)
o
:
Ifthequotientistrivial,let
R
L
0
;
inf
=
;
.Set
R
L
0
=
R
L
0
;
res
[R
L
0
;
inf
:
(b)
If
q
doesnotdividetheorderof
L=k
,let
L
=
L
0
and
R
L
0
=
R
L
.
4.
Set
G
=

cor(
A
):
A
2G
L
0

and
R
=

cor(
A
):
A
2R
L
0

:
Notethatthereareadditionalrelationsthatcomefromthefactthatthecorestrictionmap
isnotinjective.Theserelationsneedamorecarefultreatment.Seeforexamplesection7.2.
Adirectconsequenceofalgorithm7.2.2isthefollowing.
Corollary6.0.1.
Everyelementin
I
asabovecanbewrittenasatensorproductofatmost
2(
q

1)
2
(
q
+1)
symbolalgebras.
Remark6.0.2.
Notethatweassumethatthecharacteristicof
k
isentform
2
or
3
for
simplicityofthepresentationoftheellipticcurvesanditstorsionsubgroups.Thegeneral
resultsstillholdincharacteristicequaltotwoandthree.
66
Chapter7
Examples
Inthischapter,wecalculatethe
q
-torsionoftheBrauergroupforsomeellipticcurves
E
,
where
q
isanoddprime.Forcomputationalreasons,weonlyconsiderthecase
q
=3.The
algorithmdescribedinchapter6canbeusedtodetermine
q
Br(
E
)foranyoddprime
q
.As
before,wewillconsidervariouscasesdependingontheextension
L
,thatisthesmallest
Galoisextensionof
k
,sothat
M
is
L
-rational.
7.1
M
is
k
-rationaloveranumber
Let
k
=
Q
(
!
)
ˆ
C
,where
!
isaprimitivethirdrootofunity.In[Pal10]theauthordescribes
afamilyofellipticcurvessuchthat
M
is
Q
(
!
)-rational,forexample
E
givenbythe
equation
y
2
=
x
3
+16.Inthiscase,thethreetorsionof
E
isgeneratedby
P
=(0
;
4)and
Q
=(

4
;
8
!
+4)=(

4
;
4
p
3
i
).Furthermore,thetangentlinesat
P
and
Q
,respectively
aregivenby
t
P
=
y

4and
t
Q
=
y

6
2
!
+1
(
x
+4)

8
!

4=
y

4
p
3
ix

20
p
3
i:
Bythepreviousdiscussion
3
Br(
E
)=
3
Br(
k
)

I
andeveryelementin
I
canbewrittenasa
tensorproduct
(
a;y

4)
3
;k
(
E
)


b;y

4
p
3
ix

20
p
3
i

3
;k
(
E
)
(7.1)
67
forsome
a;b
2
k

:
Wecalculatewiththemagmacodeintheappendix,that
E
(
k
)=
M
andthereforealso
E
(
k
)
=
3
E
(
k
)=
M
.Byourpreviouscalculationsontherelations,atensor
productasineq.(7.1)istrivialifandonlyifitissimilartoanelementinthesubgroup
generatedby

4

20
p
3
i;t
P

k
(
E
)
and

6
19

8
19
p
3
i;t
P

k
(
E
)


4
p
3
i

4
;t
Q

k
(
E
)
:
7.2Degree
L=k
coprimeto
q
for
k
anumber
Weneedtodiscusscomputationalresultsforthecorestrictionofsymbolalgebras.
Lemma7.2.1.
Let
K
ˆ
F
beaextensionofover
k
.Then
cor
F=K
(
a;b
)=

a;N
F=K
(
b
)

forall
a
2
K

;b
2
F

.
Proof.
Seeforinstance[Ser79,page209].
Thefollowingalgorithmfrom[RT83,Section3]mayalsobeusedtocalculatethecore-
strictionexplicitly.Forapolynomial
p
(
t
)=
a
n
t
n
+
a
n

1
t
n

1
+

+
a
m
t
m
with
a
m
a
n
6
=0

p

(
t
)=
p
(
t
)
a
m
t
m
and
c
(
p
)=(

1)
n
a
n
.
Algorithm7.2.2.
Let
K
ˆ
F
beaextensionofover
k
andlet
a;b
2
F

.
Let
g
(
t
)
2
K
[
t
]
betheminimalpolynomialof
a
over
K
and
f
(
t
)
2
K
[
t
]
isthepolynomial
ofsmallestdegreesuchthat
N
F=K
(
a
)
(
b
)=
f
(
a
)
.asequence
g
0
;:::;g
m
ofnonzero
68
polynomialsbysetting
g
0
=
g;g
1
=
f;
andfor
i

1
let
g
i
+1
betheremainderofthedivision
of
g

i

1
by
g
i
aslongas
g
i
6
=0
.Then
cor
F=K
(
a;b
)
F
=

m
X
i
=
q

c
(
g

i

1
)
;c
(
g
i
)

K
:
Let
k
=
Q
(
!
)and
E
theellipticcurvegivenbytheequation
y
2
=
x
3
+
B;
where
B

2mod(
Q

)
3
and
B

1
;

3mod(
Q

)
2
.By[BP12,Theorem3.2andCorol-
lary3.3]wegetthat
L
=
k
(
p
B
).Let
˙
begivenby
˙
(
p
B
)=

p
B
.Thethreetorsionof
E
hasgenerators
P
and
Q
with
P
=(0
;
p
B
)and
Q
=

3
p

4
B;
p

3
B

:
Then
˙
(
P
)=2
P
and
˙
(
Q
)=2
Q
.Weneedtocalculate
cor
L
(
E
)
=k
(
E
)

(
a;t
P
)
L
(
E
)

(
b;t
Q
)
L
(
E
)

:
Recallthatbyremark4.2.2,itwillbeenoughtoconsider(
a;b
)aNormin
L

=
(
L

)
3

L

=
(
L

)
3
.For(
a;b
)
2
L

=
(
L

)
3

L

=
(
L

)
3
wehave
N
L=k
(
a;b
)=

a
˙
(
a
)
;
b
˙
(
b
)

,or
equivalentlywemayassumethat
N
L=k
(
a
)=1and
N
L=k
(
b
)=1and
a;b
2
L

n
k

.Note
that
t
P
=
y

p
B
and
t
Q
=
y

3
3
p

4
B
2
p

3
B
x

3
p

3
B:
Furthermore,
p

3
2
k
as
!
2
k
and
3
p
16
B
2
2
k

since
B

2mod(
Q

)
3
.
69
Let
a
=
a
1
p
B
+
a
2
with
a
1
6
=0and
N
L=k
(
a
)=1.Wenowemployalgorithm7.2.2to
calculatecor(
a;t
P
)
L
(
E
)
.Inthenotationofalgorithm7.2.2,wecalculate
g
0
=MinimalPolynomialof
a
over
k
=(
t

a
1
p
B

a
2
)(
t
+
a
1
p
B

a
2
)
=
t
2

2
a
2
t
+
a
2
2

Ba
2
1
=
t
2

2
a
2
t
+
N
L=k
(
a
)
=
t
2

2
a
2
t
+1
;
g

0
=
t
2

2
a
2
t
+1
;
g
1
=
y

a
2

t
a
1
=
1
a
1
t
+
y

a
2
a
1
g

1
=
1
a
1
y

a
2
t
+1
:
Theelement
g
2
istheremainderofthedivisionof
g

0
by
g
1
.Notethat

a
1
t

a
1
a
2

a
2
1
y

g
1
=
t
2

2
a
2
t
+
a
2
2

a
2
1
y
2
=
g

0

1+
a
2
2

a
2
1
y
2
andtherefore
g
2
=1

a
2
2
+
a
2
1
y
2
=1

a
2
2
+
a
2
1
x
3
+
a
2
1
B
=1

N
L=k
(
a
)+
a
2
1
x
3
=
a
2
1
x
3
:
70
Overall,thecorestrictionis
cor(
a;t
P
)=

1
;

1
a
1


1
k
(
E
)


1
a
2

a
1
y
;a
2
1
x
3


1
k
(
E
)
=

a
2

a
1
y;a
2
1
x
3

k
(
E
)
=

a
2

a
1
y;a
2
1

k
(
E
)
:
Finally,let
b
=
b
1
p
B
+
b
2
with
b
1
6
=0and
N
L=k
(
b
)=1.Usethenotationofalgo-
rithm7.2.2tocalculatecor(
b;t
Q
)
L
(
E
)
.Then
g
0
=
t
2

2
b
2
t
+1
;
g

0
=
t
2

2
b
2
t
+1
;
g
1
=

p
3
i
b
1
(
x
+1)
t
+
y
+
p
3
ib
2
b
1
(
x
+1)
:
Furthermore,notethat
 

b
1
p
3
i
(
x
+1)
t
+
b
1
b
2
p
3
i
(
x
+1)
+
yb
2
1
3(
x
+1)
2
!
g
1
=
t
2

2
b
2
t
+
b
2
2
+
b
2
1
3(
x
+1)
2
y
2
=
g

0

1+
b
2
2
+
b
2
1
3(
x
+1)
2
y
2
andtherefore
g
2
=1

b
2
2

b
2
1
3(
x
+1)
2
y
2
=1

b
2
2

b
2
1
(
x
3
+
B
)
3(
x
+1)
2
:
71
Wededucethat
cor(
b;t
Q
)
L
(
E
)
=
 
1
;
p
3
i
b
1
(
x
+1)
!

1
k
(
E
)

0
B
@
p
3
i
b
1
(
x
+1)
y
+
p
3
ib
2
b
1
(
x
+1)
;
1

b
2
2

b
2
1
(
x
3
+
B
)
3(
x
+1)
2
1
C
A

1
k
(
E
)
=
 
yb
1
p
3
i
(
x
+1)
+
b
2
;
1

b
2
2

b
2
1
(
x
3
+
B
)
3(
x
+1)
2
!
k
(
E
)
:
Overall,the3-torsionoftheBrauergroupdecomposesasfollows.
Proposition7.2.3.
Let
k
=
Q
(
!
)
andlet
E
beanellipticcurvegivenby
y
2
=
x
3
+
B;
where
B

2mod(
Q

)
3
and
B

1
;

3mod(
Q

)
2
.Thenthe
3
-torsionoftheBrauer
groupdecomposesas
3
Br(
E
)=
3
Br(
k
)

I
andeveryelementin
I
canbewrittenasatensorproduct

a
2

a
1
y;a
2
1

k
(
E
)

 
yb
1
p
3
i
(
x
+1)
+
b
2
;
1

b
2
2

b
2
1
(
x
3
+
B
)
3(
x
+1)
2
!
k
(
E
)
forsome
a
1
;a
2
;b
1
;b
2
2
k

with
a
1
;b
1
6
=0
,and
a
2
2

Ba
2
1
=
b
2
2

Bb
2
1
=1
.
Tocalculatetherelationsweneedtospecify
B
.Weconsiderthecase
B
=

1024.
Usingthecodeintheappendix,wecalculatethat
E
(
k
)=0.Thustherearenoadditional
relations.Notethatsomeelementsmightstillbecometrivialduetothefactthatthecore-
strictionmapisnotsurjective.
Considerthecase
B
=2.Weusethecodeintheappendixtoseethat
E
(
k
)
˘
=
Z
2
with
generators
R
=(

!;
1)=

1
2

p

3
2
;
1

and
S
=(

1
;

1).Inthiscase
P
=(0
;
p
2)
;Q
=
(

2
;
p

3
p
2)
;t
P
=
y

p
2
;
and
t
Q
=
y
+
p

3
p
2
x
+
p

3
p
2
:
Thereforebyadirect
72
computation
t
P
(
R
)=1

p
2
;
t
P
(
R
)
˙
(
t
P
(
R
))
=2
p
2

3
;
t
Q
(
R
)=1+
3
2
p

3
p
2+
3
2
p
2
;
t
Q
(
R
)
˙
(
t
Q
(
R
))



51+57
i
p
3

p
2

323+18
i
p
3
;
t
P
(
S
)=

1

p
2
;
t
P
(
S
)
˙
(
t
P
(
S
))
=

2
p
2

3
;
t
Q
(
S
)=

1
;
t
Q
(
S
)
˙
(
t
Q
(
S
))
=1
:
Considerthefollowingcommutativediagram
E
(
k
)
/
/

k


E
(
L
)
/
/

L


E
(
k
)
/
/

k


E
(
L
)
/
/

L


E
(
k
)

k


H
1
(
k;M
)
res
/
/

k


H
1
(
L;M
)
cor
/
/

L


H
1
(
k;M
)
res
/
/

k


H
1
(
L;M
)
cor
/
/

L


H
1
(
k;M
)

k


3
Br
E
/
/
3
Br
E
L
/
/
3
Br
E
/
/
3
Br
E
L
/
/
3
Br
E
;
wheretherowscomposetomultiplicationby4,whichistheidentityonthreetorsion.An
elementin
I
istrivialifitissimilartoanelementinthesubgroupgeneratedby

k


k
(
R
)
and

k


k
(
S
).Finally,

k


k
(
R
)=cor


L

N
L
(
E
)
=k
(
E
)


L

res(
R
)
73
andthereforebyourpreviouscalculationsoftheNormmapandcorestriction,wegetthat

k


k
(
R
)=cor

t
Q
(
R
)
˙
(
t
Q
(
R
))
;t
P

L
(
E
)

cor

t
P
(
R
)
˙
(
t
P
(
R
))
;t
Q

L
(
E
)
=


323+18
i
p
3



51+57
i
p
3

y;


51+57
i
p
3

2

k
(
E
)


2
y
p
3
i
(
x
+1)

3
;

8

4(
x
3
+2)
3(
x
+1)
2

k
(
E
)
andsimilarly

k


k
(
S
)=cor

t
P
(
S
)
˙
(
t
P
(
S
))
;t
Q

L
(
E
)
=


2
y
p
3
i
(
x
+1)

3
;

8

4(
x
3
+2)
3(
x
+1)
2

k
(
E
)
:
7.3Degree
L=k
=
q
for
k
anumber
Let
k
=
Q
(
!
),
!
=

1
2
+
i
p
3
2
andlet
E
betheellipticcurvegivenbytheequation
y
2
=
x
3
+4
:
Thenthethirddivisionpolynomialis
 
3
(
x
)=3
x
4
+48.Weusethistocalculatethat
generatorsofthethreetorsionaregivenby
P
=(0
;
2)and
Q
=(

2
3
p
2
;
2
i
p
3).Let
l
=
3
p
2.
Inourpreviousnotation
L
=
k

3
p
2

andtheGaloisgroupGal(
L=k
)isgeneratedby
˙
with
74
˙
(
Q
)=
P
+
Q
=(

!
2
3
p
2
;
2
i
p
3).Itcanbeseenthat
t
P
=
y

2
t
Q
=
y
+
i
p
3
3
p
4
x
+2
i
p
3
Usingchapter6ortheorem4.3.10wededucethattheBrauergroup
3
Br(
E
)=
3
Br(
k
)

I
and
I
isgeneratedby
ˆ

l
3
;n
Q

k
(
E
)
;
(
a;t
P
)
k
(
E
)
:
a
2
k

:
˙
Wewillnowdetermine
n
Q
explicitly.Notethatthelinethrough
Q
and
P
+
Q
hasdivisor
Q
+(
P

Q
)+(2
P

Q
).Furthermore,astraightforwardcalculationshowsthat
(
y

2
p
3
i
)
3
=
y
3

6
p
3
iy
2

36
y
+24
p
3
i
=
y
3
+6
p
3
iy
2

36
y

24
p
3
i

12
p
3
i

y
2

4

=
y
3
+6
p
3
iy
2

36
y

24
p
3
i

12
p
3
ix
3
=

y
+2
p
3
i

3
+4

i
p
3
x

3
=
N
L
(
E
)
=k
(
E
)
(
t
Q
)
Wesummarizeourcalculationsinthefollowingproposition.
Proposition7.3.1.
Let
k
=
Q
(
!
)
,
!
=

1
2
+
i
p
3
2
andlet
E
betheellipticcurvegivenby
theequation
y
2
=
x
3
+4
:
ThentheBrauergroupdecomposesas
3
Br(
E
)=
3
Br(
k
)

I
75
andeveryelementin
I
canbewrittenasatensorproductofthesymbolalgebras

2
;y

2
p
3
i

3
;k
(
E
)
and
(
a;y

2)
3
;k
(
E
)
forsome
a
2
k

.
Wecalculatewithmagma,that
E
(
k
)=
h
P
i
and
E
(
L
)
˘
=
Z
=
6
Z

Z
=
6
Z
.Therefore
E
(
k
)
=
3
E
(
k
)=
h
P
i
and
E
(
L
)
=
3
E
(
L
)=
M
andthequotient
E
(
k
)
\
[3]
E
(
L
)
[3]
E
(
k
)
istrivial.Therefore
thesymbolalgebra

2
;y

2
p
3
i

k
(
E
)
isnottrivial.
Finally,




res(
P
)=

2+
i
p
3
;y

2

L
(
E
)
andthereforeasymbolalgebra(
a;y

2)
k
(
E
)
istrivialifandonlyifitissimilartooneofthefollowing
ˆ
(1
;
1)
k
(
E
)
;

2+
i
p
3
;y

2

k
(
E
)
;


8+8
i
p
3
;y

2

k
(
E
)
˙
7.4Positiverankoveranumber
Let
!
beaprimitivethirdrootofunityin
C
,i.e.
!
=

1
2
+
i
p
3
2
.Set
k
=
Q
(
!
)andlet
E
betheellipticcurvebytheequation
y
2
=
x
3

48
:
Thethirddivisionpolynomialassociatedto
E
is
 
3
(
x
)=3
x
3

576
x
=3
x
3

2
6
3
2
x:
76
Wecalculatedirectlythatthethreetorsion
M
of
E
(
k
)isgeneratedby
P
=(0
;
8
!
+4)=

0
;
4
i
p
3

and
Q
=

4
3
p
3
;
12

.Furthermore,weseebydirectcomputationthat
P

Q
=

!
2
4
3
p
3
;
12

.Nowusingthecodeintheappendixwecalculatethat
E
(
k
)
˘
=
Z
2

Z
=
3
Z
and
E
(
L
)
˘
=
Z
2

Z
=
3
Z

Z
=
3
Z
.Furthermore,generatorsof
E
(
L
)are
8
>
>
<
>
>
:
P;Q;R
=(

4
!

4
;

4)=


2
i
p
3
;

4

;
S
=

(
!
+1)

3
p
3

2
+
!
3
p
3+3
;
(
!

1)

3
p
3

2
+(3
!
+6)
3
p
3+6
!
+3

9
>
>
=
>
>
;
Weconcludethatthequotient
E
(
k
)
\
[3]
E
(
L
)
[3]
E
(
L
)
isnontrivial.Finally,thepolynomials
t
P
and
t
Q
are
t
P
=
y

8
!

4=
y

4
i
p
3
t
Q
=
y

1
6

3
p
3

2
x

12

2
3
ByouralgorithmthethreetorsionoftheBrauergroupdecomposesas
3
Br(
E
)=
3
Br
k

I
andeveryelementin
I
canbewrittenasa(
a;t
P
)
k
(
E
)
with
a
2
k

.Furthermore,an
elementin
I
istrivialifandonlyifitsrestrictionto
L
issimilartoanelementinthe
77
subgroupgeneratedby
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
:

4
i
p
3

12

2
3
;t
P

L
(
E
)
;

1
134

5

9
i
p
3

;t
P

L
(
E
)


12

4
i
p
3
;t
Q

L
(
E
)
;
 
p
3
i
3

3
p
3

2

16

2
3
;t
P
!
L
(
E
)



4

4
i
p
3
;t
Q

L
(
E
)
;

(
!

3)

3
p
3

2
+(
!
+4)
3
p
3+
11
2
!

20
3
;t
P

L
(
E
)


(
!

1)

3
p
3

2
+(3
!
+6)
3
p
3

2
!

1
;t
Q

L
(
E
)
9
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
;
7.5Overalocal
Denoteby
Q
7
the7-adicItiseasytoseethatthe
Q
7
containsaprimitivethird
rootofunity
!
.Let
E
betheellipticcurve
E
:
y
2
=
x
3
+16
over
k
.Considerthereduction
~
E
of
E
modulo7.Then
~
E
isanon-singularcurveandusing
thecodeintheappendixweseethat
~
E
(
F
7
)=
Z
=
3
Z

Z
=
3
Z
=
f
0
;
(0
;
3)
;
(0
;
4)
;
(3
;
1)
;
(3
;
6)
;
(5
;
1)
;
(5
;
6)
;
(6
;
1)
;
(6
;
6)
g
:
Denoteby
^
E
theformalgroupassociatedto
E
andconsiderthegroup
^
E
(7
Z
7
).By[Sil09,IV
Theorem6.4],thereisanisomorphism
^
E
(7
Z
7
)
!
^
G
a
(7
Z
7
),where
G
a
denotestheadditive
group.By[Sil09,IV.3andVII.2]thereisanexactsequence
0
/
/
^
G
a
(7
Z
7
)
/
/
E
(
Q
7
)
/
/
~
E
(
F
7
)
/
/
0
:
78
Furthermore,by[Sil09,VII.3Proposition3.1]thereductionmap
3
E
(
Q
7
)
!
~
E
(
F
7
)isin-
jective.Thus
E
has
k
-rational3-torsion.Since3isaunitin
Z
7
wefurtherdeducethat
E
(
Q
7
)
=
[3]
E
(
Q
7
)=
~
E
(
F
)
=
[3]
~
E
(
F
)=
M:
Finally,
Q

7
=

Q

7

3
˘
=

Z

7

(7
Z
7
)

=

Z

7

(7
Z
7
)

3
˘
=
Z
=
3
Z

Z
=
3
Z
:
Therefore,
H
1
(
k;M
)
˘
=

Q

7
=

Q

7

3

2
˘
=
(
Z
=
3
Z
)
4
.Bythealgorithmandusing[Gro68b,
Corollaire2.3],the3-torsionoftheBrauergroupdecomposesasfollows
3
Br(
E
)
˘
=
3
Br(
Q
7
)

(
Z
=
3
Z
)
2
=
3
(
Q
=
Z
)

(
Z
=
3
Z
)
2
=(
Z
=
3
Z
)
3
:
Remark7.5.1.
Theabovecomputationsalsoshowthat
3
Br

~
E

=
3
Br(
F
7
)=0
.
7.6Overa
Let
k
=
F
5
(
!
)betheextensionofthewitheelementsgivenbyattachingathird
rootofunity
!
.Let
E
betheellipticcurvegivenbytheequation
y
2
=
x
3
+1
:
Thethreetorsionof
E
(
k
)isalso
k
-rationalwithgenerators
P
=(0
;
4)and
Q
=(1
;
3
!
+4).
Furthermore,weusethecodeintheappendixtocalculatethat
E
(
k
)
˘
=
Z
=
6
Z

Z
=
6
Z
.A
directcalculationgivesthat(
k

)
3
=
F

5
[
(1+2
!
)
F

5
Therefore,thequotient
k

=
(
k

)
3
isisomorphicto
Z
=
3
Z
withdistinctrepresentatives
f
1
;!;!
+1
g
.Weconcludethat

is
surjective,andtherefore
3
Br(
E
)=
3
Br(
k
).Since
F
5
isa
C
1
andextensionsof
79
C
1
are
C
1
aswell[GS17,Lemma6.2.4,page161],wededucethatBr(
k
)=1and
3
Br(
E
)istrivial.
7.7Degree
[
L
:
k
]
divisibleby
q
Let
k
=
Q
(
!
),where
!
isaprimitivethirdrootofunity.Considertheellipticcurve
E
given
bytheequation
y
2
=
x
3
+
x
+1
:
Thethreetorsion
M
of
E
(
k
)isgeneratedby
P
=(
x
1
;y
1
)and
Q
=(
x
2
;y
2
)with
x
1
=

1
2
v
u
u
u
t
3
p


8
3

8
p
3
q

3
p


4
+
q

3
p


4
2
p
3
;
x
2
=
1
2
v
u
u
u
t
3
p


8
3

8
p
3
q

3
p


4
+
q

3
p


4
2
p
3
;
y
1
=
q
x
3
1
+
x
1
+1
;
y
2
=
q
x
3
2
+
x
2
+1
where=

496isthediscriminantof
E
(see[Pal10,Section3]).Denote
P
+
Q
=(
x
3
;y
3
)
and2
P
+
Q
=(
x
4
;y
4
).By[Pal10,Theorem4.1(1)],the
L
is
k
(
x
1
;x
2
;y
1
)=
k
(
x
2

x
1
;y
1
)andtheGaloisgroupof
L
over
k
isisomorphicto
SL
2
(
F
3
).Considerthe
subgroup
P
generatedby
0
B
@
11
01
1
C
A
anddenoteitsby
L
0
.Bytheproofofthe
primitiveelementtheorem,anelement
l
2
L
with
l
3
2
L
0
and
L
=
L
0
(
l
)is
l
=
x
2
+
!x
3
+
!
2
x
4
:
80
Aswehaveseenbefore,
n
Q
isgivenbytheequationofthelinethrough
Q
and
P
+
Q
,that
is
n
Q
=
y

y
2

y
3

y
2
x
3

x
2
(
x

x
2
)
:
Bythemainalgorithmwededucethat
3
Br(
E
)=
3
Br(
k
)

I
andeveryelementin
I
canbewrittenasatensorproduct
cor
L
0
(
E
)
=k
(
E
)


x
2
+
!x
3
+
!
2
x
4

3
j
;y

y
2

y
3

y
2
x
3

x
2
(
x

x
2
)

L
0
(
E
)

cor
L
0
(
E
)
=k
(
E
)
 
a;y

y
1

3
x
2
1
+1
2
y
1
(
x

x
1
)
!
L
0
(
E
)
forsome
a
2
L
0
andsome
j
2f
0
;
1
;
2
g
.Itremainstocalculatecor
L
0
(
E
)
;k
(
E
)
ofthesealgebras,
whichcanbecomputedforspvaluesof
a
and
j
usingalgorithm7.2.2.Wecantherefore
writeeveryelementin
I
asatensorproductofatmost16symbolalgebrasover
k
(
E
).
81
APPENDIX
82
Appendix
MagmaCode
Thisappendixcontainsthemagma-codeusedinchapter7.
Codeforsection7.1
K<w>:=CyclotomicField(3);
E:=EllipticCurve([L|0,16]);
AbelianGroup(E);
Generators(E);
Codeforsection7.2
Thisisthecodefor
B
=

1024:
K<w>:=CyclotomicField(3);
E:=EllipticCurve([L|0,-1024]);
AbelianGroup(E);
Thisisthecodefor
B
=2:
K<w>:=CyclotomicField(3);
E:=EllipticCurve([L|0,2]);
83
AbelianGroup(E);
Generators(E);
Codeforsection7.3
Wecalculatethe
k
-rationalpointsof
E
byusing:
K<w>:=CyclotomicField(3);
E:=EllipticCurve([L|0,4]);
AbelianGroup(E);
Generators(E);
Nowwecalculatethe
L
-rationalpointsof
E
with:
K<w>:=CyclotomicField(3);
R<y>:=PolynomialRing(K);
f:=y^2-2;
L:=ext<K|f>;
E:=EllipticCurve([L|0,4]);
Generators(E);
Codeforsection7.4
K<w>:=CyclotomicField(3);
E:=EllipticCurve([K|0,-48]);
AbelianGroup(E);
Generators(E);
84
R<y>:=PolynomialRing(K);
f:=y^3-3;
L:=ext<K|f>;
E:=EllipticCurve([L|0,-48]);
AbelianGroup(E);
Generators(E);
Codeforsection7.5
F:=FiniteField(7);
E:=EllipticCurve([F|0,16]);
AbelianGroup(E);
Points(E);
Codeforsection7.6
F:=FiniteField(5);
R<w>:=PolynomialRing(F);
f:=w^2+w+1;
L:=ext<F|f>;
E:=EllipticCurve([L|0,1]);
AbelianGroup(E);
Points(E);
85
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