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DATEDUE DAIEDUE DATEDUE 6/07 p:/ClRC/DateDue.indd-p.1 Galois Structure of Modular Forms of Even Weight By Erhan Gii'rel A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Mathematics 2007 ABSTRACT Galois Structure of Modular Forms of Even Weight By Erhan Gilrel We calculate the equivariant Euler characteristics of an even power of the canonical sheaf on modular curves over Z with a tame action of a finite abelian group. As a consequence, we obtain information on the Galois module structure of “twisted” modular forms of even weight having Fourier coefficients in a ring of algebraic integers To my family. iii ACKNOWLEDGMENTS I wish to express my sincere gratitude to my advisor, Professor George Pappas, for his invaluable guidance. It would be impossible for me to finish this disserta- tion without the uncountable number of hours he spent sharing his knowledge and discussing various ideas throughout the study. Perhaps the best way to express my thanks is, in turn, to treat my students with the same kindness and respect in the future. I am obliged to Professors Christel Rotthaus, Michael Shapiro, Jonathan 1. Hall and Gregory Pearlstein, in my thesis committee. Thank you for your detailed com- ments and time. I would also like to thank Professor Rajesh Kulkarni for all his help and valuable communications. I thank all the people who helped me during this study. iv TABLE OF CONTENTS Introduction ................................. 1 1 Definitions and Preliminaries ...................... 7 1.1 Tame covers of schemes .......................... 7 1.2 Modular Forms and Diamond Operators ................ 8 1.3 Tate Curves ................................ 10 1.4 Coarse Moduli Scheme of Elliptic Curves ................ 11 2 Coarse Moduli Schemes X1,Xo and X H ................ 14 3 Lattices of cusp forms ......................... 16 4 Galois structure of modular forms ................... 22 5 Proof of Main Theorem ........................ 26 5.0.1 Main Theorem of [CPTI] ..................... 29 5.0.2 Lower bound for 6 ........................ 36 BIBLIOGRAPHY .............................. 42 Introduction The normal basis theorem implies that if N / K is a finite Galois extension of number fields with Galois group G, then N is a free K [Cl-module of rank one. In particular, N is a free Q[G]-module. Let 0N and OK be the ring of integers of N and K respectively. Then we can ask for the analogous statement, namely, “IS ON a free module over the group ring Z[G]?” The first observation regarding this question belongs to E. Noether. Theorem 0.1 (E. Noether) Let N / K be a finite Galois extension of number fields with Galois group G. Then the ring of integers, ON is a projective Z[G]—module if and only if N / K is at most tamely ramified. When N / K is tamely ramified, the obstruction to ON to be a stably free Z[G]— module is the class (ON) in the class group Cl(Z[G]). Frbhlich’s conjecture, proved by M.Taylor in [T], gives an interesting description for this class: Theorem 0.2 ( M. Taylor) We have the following equality, (ON) = WN/K (1) in Cl(Z[G]). Here WN/K is the “root number class”; the class WN/K has order two and is given by the signs of the e-constants in the functional equation of the Artin L-functions of symplectic representations of G. It was natural to try to extend Frohlich’s conjecture by relating the e-constants with the Galois modules attached to a group action on an arithmetic scheme. How- ever, the right formulation of the generalized conjecture was not clear until the work of Chinburg and of Chinburg, Erez, Pappas and Taylor ([CEPT], [CPTD . Let 7r : X —+ Y = X / G be a geometric tame G-cover of projective flat regular schemes over Z. In [CEPT], the authors define an equivariant deRham Euler characteristic class x(X, G) in Cl(Z[G]) using equivariant Euler characteristics of differential sheaves. When X = Spec(ON) this generalizes the class of the ring of integers (ON). They also define a root number class WX/y (similar to WN/K) and introduced a ramification class R X/y which depends on the e—constants of the branch locus of the covering 7r. The definition of these classes was motivated by the prior work of Chinburg [C] who considered the same constructions for covers of varieties over a finite field. Under some additional technical assumptions on X and Y they show Theorem 0.3 ([CPTj) We have X(X, G) = WX/Y + RX/Y (2) in Cl(Z[G]). This generalizes Frohlich’s conjecture to higher dimensional varieties over Z. It turns out that one can consider more general equivariant projective Euler char- acteristics: Suppose that X is a scheme projective and flat over Z which supports a tame action of the finite group G. For any coherent sheaf .7: on X which supports a G-action that is compatible with the action of G on X one can define following Chinburg [C] the equivariant projective Euler characteristics ")2? (X, .77) E Cl(Z[G]). The calculation of these Euler characteristic often connects to other fundamental problems in Number Theory. A recent method, developed by Chinburg, Pappas and Taylor in [CPTl], shows how to calculate the Euler characteristic of coherent sheaves on projective flat schemes over Z on which a finite abelian group acts tamely. Unlike other techniques, this one does not neglect any torsion information if the base scheme has dimension less than 5. Roughly speaking, the idea in their paper was that Euler characteristic should differ by computable terms from classes in Grothendieck groups which have “n—cubic” structures. This idea was motivated by previous works of Pappas in [P] and [P1]. In this recent paper, a precise formula is given for the Euler characteristic. Furethermore, they determined the structure of the lattice of weight 2 cusp forms for I‘] (p) which have integral Fourier expansions as a module for the action of the finite group of diamond Hecke operators. This is done by calculating the equivariant Euler characteristic 7” (X, Ox) where X is a certain integral model of the modular curve X1(p). In this thesis, we calculate the equivariant Euler characteristic of k-th power of the “twisted” canonical sheaf over an integral module of the modular curve X 1 (p) (here some twists are allowed along a fibral divisor at p for some technical reasons). Moreover, we find a lower bound to the degree of the twist which guarantees that the first cohomology group vanishes. Consequently, the structure of the lattice of “twisted” cusp forms of weight 2k and Nebentypus character can be obtained as a module for the diamond Hecke operators. Here twist means that we allow the Fourier coefficients to have denominator a certain (bounded) power of the uniformizer over p (see below). More properly, let p E 1 mod 24 be a prime and F = (Z/pZ)* / {21:1}. Suppose X : I‘ —> p, C Z[(,.]“ is a character of prime order r|(p —— 1) with r > 3. Let 52k,5(I‘1(p),p‘5kZ[Cr])x be the Z[C,.]-module of “twisted” cusp forms of weight 21:, level p and of Nebentypus character x (for some technical reasons some twists to the dualizing sheaf on the modular curve is allowed and this 6 represents the order of the pole that we allow along the twist). In addition, we ask that the Fourier expansion an ' . Weizmnz where the coefficrents 7121 an belong to Z[(,.]. The locally free Z[(,.]-module 82k,5(I‘1(p),p‘5’°Z[C,.])X is of rank n(x) = (2k — 11):!) — 25). For a E (Z/rZ)* let {a} be the unique integer in the range of these modular forms is of the form F (2) = 0 < {a} < r having residue class a, and let on E Gal(Q((,.)/Q) be the automorphism for which 00(9) = (TM. Define to, : (Z/rZ)* —+ Z; to be the Teichmuller character. The ring Z (resp. Z,) is embedded into the pro-finite completion Z = H, prime Z, of Z diagonally (resp. via the factor I = r). Then a modified quadratic Stickelberger element of Z[Gal(Q(C,.)/Q)] can be defined by 02: Z %}¥({a}2—w. 2 + r, S2k,5(l“1(p),p‘6kZ[Cr-l)x 2’ ZlC.-]"(’"“1 ea 21 as Z[Cr] -modules. This extends the corresponding theorem of [CPTI] to higher weight cusp forms. CHAPTER 1 Definitions and Preliminaries This chapter contains basic definitions and facts of tame covers of schemes, modular forms, Tate curves and moduli schemes of elliptic curves. 1.1 Tame covers of schemes Let us recall the definition from [C]. Definition 1.1 Let Y be a normal scheme of finite type over R and let D be a closed subset of Y which is of codimension at least one. A morphism f : X ——> Y is tamely ramified covering of Y relative to D if the followings hold: a. f is finite, b. f is etale over Y — D, 0. Every irreducible component of X dominates an irreducible component of Y. d. X is normal, e. Let y on D have codimension one in Y and let a: be a point of X over y, then Oxrt/Oy‘y is tamely ramified extension of DVR’s. Definition 1.2 Let f : X ——> Y and D be as in the previous definition and let G be a finite group. Then f : X ——> Y is tame G—cover relative to D if X x (Y — D) H Y - D is a G—torsor when G is regarded as a constant group scheme over Y ~— D. Definition 1.3 The G-action on X is called tame if for every closed point :1: E X, order of inertia subgroup I1c C G is relatively prime to the charactetistic of the residue field k(.'r). Definition 1.4 Let f : X —+ Y be a tame G—cover. A quasi-coherent OX—G-module F is quasi-coherent Ox-module having an action G which is compatible with the action of G on OX. i.e. suppose at E X, g E G and let 1:9 be the image of a: under 9. The action of g on OX and F gives homomorphism of stalks OX‘xg H 0x4. and F39 i—+ Fx; both of these homomorphism is denoted by qt, and ¢(am) = qb(a)qb(m) for all a E OX,3:9 and m 6 Fry. 1.2 Modular Forms and Diamond Operators Definition 1.5 Let k be an integer. We say a function f is modular of weight 2k if it is meromorphic on the upper half plane and 00 also satisfying following condition (:12) = (cz+d)2’°f(z) (1.1) a b for all 6 SL2(Z). c d Definition 1.6 A modular function is called as modular form if it is holomorphic everywhere including 00. Definition 1.7 A modular form is called as cusp form if it is zero at 00. A modular form of weight 2k can also be written as a series, f(2) = Zanq" (1.2) n=0 27riz where q = e and verifies the identity f(1/Z) = We) (1.3) So, a0 = 0 when f is a cusp form. Let .7) denote upper half plane {2 E CIS‘z > O} on which we have S L2(Z) action as follows: a b az+b z: . d cz+d (1.4) When we extend the upper half plane by adding cusps P1(Q)(= QU{oo}) to f)" we can extend this action on cusps using the same fractional transformation. We have two subgroups of SL2(Z), namely F0(N) and I‘1(N) defined as: I‘O(N) := { a b E SL2(Z)| c E 0 mod N} (1.5) d a b F1(N):={ ESL2(Z)| c_=_—0 modN, aEdEI modN} (1.6) c d One can easily see that F1(N) is a normal subgroup of F0(N) and when we take the quotient we get, F0(N)/F1(N) ’5 (Z/NZ)* (1- ‘J v H d (1.8) We can define space of modular forms (cusp forms) of weight 21: and level N by a b restricting to F1(N) and we can denote it by M2k(I‘1(N)) (Sgk(F1(N))). c d There is a F0(N) action on SQk(F1(N)) by a b __ ~ _2k az+b Since the action of F 1(N ) is trivial, we can define an action of (Z/ N Z)* using the isomorphism 1.7 as follows: b V < d > f :2 f (1.10) This operator < d > is called as Diamond Operator. 1 .3 Tate Curves Let R be a noetherian local ring which is complete with respect to ideal qR. The Tate curve Eq proper smooth scheme over R[q‘1] defined (as in [CSS]) by following equation on an affine chart :5 74 0: y2+xy=r3+a4(q)1:+a6(q) (1.11) where 0° n3q" 0° (5n3 + 7n5)q” . = —5 E . — — E 1.12‘ The following series gives us parametrization of the curve. — oo Hg" 00 (In Ell“) = ; “(filly + 2; (1 _qnqn)2 (1 14) They induce homomorphism :c u , u ifu ¢ Z; qu‘lquz -> EARICI‘II), UH ( ( ) y< )) q 0, iquqz, This map is bijective when R is a complete discrete valuation ring. 1.4 Coarse Moduli Scheme of Elliptic Curves A moduli problem is roughly given by two ingredients. First, a class of objects together with a notion of being family of such objeCts over a scheme B. Second, an equivalence relation ~ on the set of S (B) of all such families over B. We can define a moduli functor F from the category of schemes to that of sets by F (B) = S (B) / ~ for our moduli problem. The functor F is representable if there is a scheme on and isomorphism i/J between F and the functor M or(o, 931). If that is the case, then we say that am is fine moduli spcheme for the moduli problem F. When a fine moduli exists, every family over B is the pullback'of universal family Q: via a unique map of B to mt. This allows us to translate information about the geometry of families of our moduli problem to information about geometry of the moduli space an itself. Unfortunately, most of the time it is not possible to have a fine moduli scheme. Definition 1.8 A scheme {Di and a natural transformation 2/29); from the functor F to the functor of points M or(0, an) of am is a coarse moduli scheme for the functor F if i. The map 'l/Jspecar) : F (Spec(lF)) —> 931(IF) is a bijection for every algebraically 11 closed field IF. ii. Given another scheme Dfl’ and a natural transformation pm from F to M or(0,9fl’ ), there is a unique morphism (,b : 931 —> Dfl’ such that the associ- ated transformation (I) : M or(o, 931) —+ ll/Ior(-, 91W) satisfies Il’m = (I) 0 Il'wt Often the moduli functor is represented by a more general type of object, a moduli stack. In our case, we will use the moduli stack ”1‘10” that classifies triples (E, C, 7) where E ——> S is a generalized elliptic curve i.e. n-gons are allowed (see [DR], or [CES]), G a locally free rank p subgroup of the smooth locus E8m and 7 : (Z/pZ)3 ——> CD a “generator” (in the sense of [KM, Ch. 1]) of the Cartier dual of C; we require that C intersects every irreducible component of every geometric fiber of E ———> 5'. (Notice that a group scheme embedding [1,, H E"m gives data C and 7; in fact, if p is invertible on 8, giving C together with 'y as above exactly amounts to giving a group scheme embedding pp <——) Em.) We denote by X1 = X 1 (p) the corresponding coarse moduli scheme over Spec (Z). The group (Z/pZ)* acts on ml‘dp) via (a modp) - (E, C, 7) = (E, ng o a”). (1.15) (When p is invertible, this action sends the corresponding j : up H E8m to the composition j o (z i—> z“) : up c—r Esm.) This produces a faithful I‘ = (Z/pZ)*/{:i:1}- action on X1. When H is a subgroup of I‘ we let X H be the quotient X1 / H, and set X0 = X 1‘. The singularity structure of X H is explicitly given in [CBS] depending on the order of H. When they analyze non-regular points they use deformation theory. In our case we used it as follows: Let 72,, be the formal deformation ring of the point 3 = (E, Z/pZ C E, 0) in the moduli stack 911130,). Then 72,, supports an action of Aut(E) >< (Z/pZ)* . Let H’ be the inverse image of H under the surjection 12 (Z / pZ)* —+ F, and let 3’ be the image of s on X H. The completion of the local ring is beam. 2 (Raw (1.16) as ring with G = (Z/pZ)* / H ’-action. It tells us that we can get the deformation of coarse moduli scheme X 1/ H from the deformation of the moduli stack 911nm. As it is mentioned in [CES], checking the regularity along the cusps can be done by using the Tate curve. We will have similar calculations using the Tate curve, therefore it is better to mention here what is the place of the Tate Curve in the Moduli scheme. The Tate curve Cm/qZ over Spec (Z[[q]]) together with the embedding up C gm/qZ (see [DR, VII]) gives a morphism r : Spec (Z[[q]]) ——> X H. We call the support of the corresponding section Spec (Z) —> Spec (Z[[q]]) —> X H the 00 cusp. Over (C, provided we trivialize up( Spec (Z) is a flat projective curve, X H is normal Cohen- Macaulay and XH[1/p] —) Spec (Z[1/p]) is smooth ( where XH[1/p] = XH <8); Z[1/p]). The special fiber of X H over p has two irreducible components Do”; and D51 distinguished by the fact that D50 intersects the cuspidal section 00; these have multiplicities 1 and (p — 1) / (2 - #H) respectively. b. The scheme X H has at most two non-regular points which are rational singu- 14 ii. larities and lie on DS’ - (Dg’ 0 Dig). Their exact number depends on #H mod 6: In particular, if 6 divides #H then there are no such points and X H is regular. In particular, when H = {1} there are two non-regular points on X1. There is a morphism b : X i —+ X1 which is a rational resolution of those two singular points and a morphism c : X i —-> X1 which is a sequence of blow-downs of exceptional curves such that X1 is regular and all the geometric fibers of X1 -—> Spec(Z) are integral. Let U = X1 — Dél} C X1. Then U —+ Spec (Z) is smooth, b and c are isomorphisms on b‘1(U) and X1 — c(b‘1(U)) has dimension 0. . The special fiber of X0 over p is reduced with simple normal crossings. Each of the two irreducible components Doo = D; and D0 = D}; are isomorphic to P}; and D0 - D00 = g0 +1 = (p— 1)/12. Assume that 6 divides the order #H. The morphism 7rH : X H -—> X0 is a tame G = I‘ / H cover of regular projective curves and 7rH[1/p] : XH[1/p] ——+ X0[1/p] is a G-torsor. The morphism 7r” is totally ramified over the generic point of D0, and unrami- fied over the generic point of D00. The irreducible components DS’ and D0}: of X H ®z F1, are the (reduced) inverse images of Do and Doc under my The char- acter X 06! giving the action of G on the cotangent space of the codimension 1 generic point of Db, equals w‘2'#H , where w : (Z/pZ)* ——> F; is the Teichmuller (identity) character. Proof: Check Theorem 4.2 and Theorem 4.3 in [CPTl]. 15 CHAPTER 3 Lattices of cusp forms If R is a subring of (C we will denote by Sgk(F1(p), R) the R—module of cusp forms F(z) 2 Zn?! anew“ of weight 2k for the congruence subgroup F1(p) C PSL2(Z) whose Fourier coefficients on belong to R. (These are the Fourier coefficients “at the cusp 00”). For simplicity, we will also denote “twisted modular forms” by an . — where an in p6}: 32k,5(l"1(p),p“”°R) when its Fourier coefficients are in this form R. Particularly, if 6 = 0, we assume 52k,0(I‘1(p), R) = 32k(F1(p), R). Proposition 3.1 There are F-equivariant isomorphisms 3....(r.(p).p-6'°Z) 2 H°(X1.w§f/z) (3.1) where the F—action on 52k,5(I‘1(p), p‘5kZ) is via the diamond operators and w x1 /z(6D;o) denotes the twisted canonical (dualizing) sheaf of X1 —+ Spec (Z) along the divisor 60:0. El PROOF. Let G(q) = %—q" E 52k,5(I‘1(p),p'5kZ) with q = 62”” and consider n21 G(q)(dq/q)®k as a regular differential over Spec (Z[[q]]). A standard argument using the Kodaira-Spencer map shows that G(q)(dq/q)®’c extends to a regular differential over X1[1/p] (cf. [Ma, 11 §4]). This extension must also be regular in an open neighborhood of the section at 00. Hence there is an open subset U’ of the set U C X1 16 defined in Theorem 2.1 (b) such that G(q)(dq/q)®k is regular on U’ and U - U’ is a finite set of closed points. We obtain an injective F-equivariant homomorphism : 52k,,(r (p ),p 5m) —-> H°(U', wU,'/Z(k6D1 )). (3.2) The equalities, HO(U’,w wU,/Z(k(5D1 ))=H0(X1,wX1/Z(k6Dl ))= H0(X1,wxf /Z(kal )), (3.3) follow from the fact that b : X; ——> X1 and c : X i ——> X1 are rational morphisms which are isomorphisms on b"1(U’) and that X1 — c(b‘1(U’)) has codimension 2 in X1 and the following lemma, which proves that O’th cohomology of the k’th power of the dualizing sheaf is preserved under blow-up of two Singular points on X1 and blow-down of a —1 curve. The surjectivity of ‘1) follows from pulling back elements of H0(X1,r.oX:c /Z(k6D1 )) via 7' : Spec (Z[[q]]) -—> U and using the Kodaira—Spencer isomorphism. We keep the same notation as in Theorem 2.1. Lemma 3.1 Let X1 i» X1 be the blow-up of the surface X1 at some point Q. Let wgl(6D;o) and w x1 (6Déo) be the twisted dualizing sheaves respectively. Then, H°(X1,w§§f(k61§;o)) = H°(X,,w,;f (1:61)1 )) (3.4) PROOF. As it is explained in [pg.380, CES], X1 has two singular points corresponding to j = 0 and j = 1728 which we call them OE and Q1: respectively. When we blow-up QB we get one exceptional curve E such that E2 = —2. Similarly, when we blow-up Qp we get another exceptional curve F such that F 2 = —3. Both of these curves are isomorphic to P1. 17 CASE 1. (j = 0) and (j = 1728) Since we will follow the same routine, we consider both cases together. First, we will show that QE and QF are rational singularities in the sense of Artin [pg.268, Ar]: Recall that the point Q is a rational singularity if the stalk of R1 f...(9)g1 at Q is zero for every desingularization X1 —f+ X1. We will Show that every singular point Q with multiplicity oz (i.e. when Q is blown-up we get an exceptional curve C, isomorphic to P1, whose self intersection is —a < —1) is a rational singularity. Let F i = [iLnHi(Cn, 00") where 0,, is closed subscheme of X1 defined by I" and I is the ideal sheaf of the exceptional curve C. We have, 0 —-> I"/I"+1 —> 00.,“ —+ 00,, —> 0 (3.5) We also know that I/I2 2 (90(0) then l'”/IZ'"+1 2 SKI/1'2) 2 00(an). Since C is just 1P", H'(C’, 00(an)) = 0 for i > 0 and n > 0. By writing the long exact sequence, we will get the following for i > 0 Hi(C, 00,“) = H‘(C, 0a,). (3.6) When n = 1, 00, = 00, therefore Hi(C,OCn) = O for i > 0. So, Fi = 0 for i > 0. Since it is supported just at Q, F i = 0 for i > 0. So, when we take o = 2 and or = 3 we prove that QE and Qp are rational singularities respectively. If the point Q with multiplicity a > 1 is a rational singularity, then coil 2 f ‘wxl by proposition 5.1 in [Ar] . Remark 3.2 Since QE is a double point singularity, the canonical sheaf of the X1 is locally around QE 8. line bundle and we could get the equality to X, = f *w x1 by a calculation which uses the adjunction formula. However, it is not true for the point Qp; the canonical sheaf of X1 is no longer a line bundle in a neighborhood of QF. 18 We need to show that H0021, r(w;‘3f(k60;.))) 2 H°(X..w§i°(k60;.)). (3.7) Using the projection formula, we get R‘f.(f‘(w§f(kao;o))) 2 defacing) a 127.0 ,2. (3.8) By taking i = 0, we get Rof.(f‘(w§f(k50éo))) 2 wfflkwio) (39) because, (9X, = f*0X1 which simply follows from the fact that X1 is normal and f is birational. Now, using the Leray spectral sequence, Him, ij.(f*(w§f(k60;.)))) => H‘+J‘(X1,r(wif) (3.10) and by choosing i = j = 0 we get the desired result. CASE 2. (j # 0,1728) Let X1 —f+ X1 be the blow-up of the surface X1 at a regular point Q. We have an exact sequence 0 —> 0231 —-* 0X1(kE) -* 0kg(kE) -—> 0 . (3.11) By tensoring the sequence by f * (a)??? (k6DAO)), we obtain an exact sequence 0 —> f*(w§f(k6D;O)) 8) OX, —> f*(w§’f(k6D;o)) <8) OX1(kE) ——> 1' —> f*(w?&f(k603,o)) ® 01.13MB) —» 0 (3.12) 19 Since 213(61):.) '—‘-’ f*(wx.(5D§o))(E) (3.13) we have wngfllDéo) 2 f*(w§3f(k50;0))(kE) . (3.14) By restricting to [CE we get wifawtnw 2 r(w?2:°i.E 2 claws) (3.15) Let’s try to show that H°(E, Okg(kE)) is trivial. We have 0 —> 1”“‘1/1"c —-> Oil/I“ -—* OX/Ik"1—+ 0 (3.16) where I is the ideal sheaf of E. After twisting the sequence by the divisor (ICE) and since E is just 1P”, 1""1/1" 2 03(l) therefore degree in the each summand becomes negative. By induction on k we obtain, H°(X1,0k3(kE)) is trivial. Using the above calculation on the following cohomology long exact sequence, 0 2 H003 Nahum») —» matey/cans» —> We 0.3km) 2 - -- (3.17) We conclude, H°(X1,f*(w§:°(k60;o))) 2 H°(X1,w%°(k6D;o)). Now, the only thing that we need to show is that, H°(X1, f*(w§f(k6D;o))) 2 H°(X1,w§f(k6D;C)). Again, using the Projection Formula, Rif.(f*(w§'zf(k60;.))> 2 22(21):.) 2 R‘wx, - (3.18) The standard argument from Hartshorne (pg. 387, Prop.3.4) gives Rf LOX-1 = 0 20 if i > 0, and ROLOXI = 0x, for i = 0. Now, using the Leray spectral sequence, Hive. Barrera/c623)» => Hi+j(X1.f*(w§f(kéDéo))) (3.19) and by choosing i = j = 0 we get the desired result. Proposition 3.1 gives us explicit relation between R-module of the “twisted” cusp forms of weight 2k and the global sections of the k’th power of the dualizing sheaf. If we follow the same argument as in the proof of the proposition, we can get the similar relation for cusp forms. This is given in the following: Corollary 3.3 There are F-equivariant isomorphisms sump), Z) 2 H°(X1.w§?f)z)- (3.20) 21 CHAPTER 4 Galois structure of modular forms Let’s start this section by defining the module of the “twisted” modular forms of weight 2k on X H which will be called as 82k,5(I‘H(p),p"5’°Z). We can define it as follows: 523,5(FH(P),P_6kZ) 3: 52k.5(F1(P)1P—6kZ)H We will try to calculate it here. Let u : X1 —+ X H be the quotient morphism. Since u is a finite morphism, Rip... = 0 for j > 0. Now, using the Leray spectral sequence, H’(XH. Rju*((w§?f(k50lo)))) => Hi+j(X1,w3”Ef(k5Déo)) (4-1) fori=0,j=0weget: H°(XH, M*w§f(k50;c)) 2 H°(X1,w§%f(k60;.)). (42) Therefore shamans—“6%) = H°(X1.w§~?f(k60;.))” (4.3) 22 and H°(X1,w§f(k50éo))" ’1 H°(XH.M*(W§f(k5Dio)))” 1' 2’ H°(XH. (u*(wféf(k50io)))”)- (4-4) Since 111wa (6Dfo)) and tax, (6Déo) are line bundles, one of them can be written as a twist of the other. This twist- is supported along the ramification locus since the line bundles are isomorphic on the complement of the ramification locus. N 0w we can write wx.(5Dio) ’2 u*(wxh(5Dfo))(Rl) (4-5) where R1 is supported on the ramification locus. Also, when p E 1 mod 24 then the ramification locus of the map 71' : X1 —+ X0 is D3, j = 0 and j = 1728. Their rami— fication degrees are ”3:1, 2 and 3 respectively. A local calculation like in [pg.74 Ma], shows that R1 = (23125 +{73i‘ + 2W1 where 3:6? and m‘ are closure of the generic point of each lines j = 0 and j = 1728 on X1. When we take the k-th power of the sheaves we get overhung.) 2 u‘(w?éf;(k6D£’o))(le) . (4.6) After taking the H —invariants, we obtain (wry/c6033)” 2 (u*(w§f,(k600’i)(kR‘))” . (4.7) We know that (u‘wimww := (wit (1.3053,) eox, on)” (4.8) H 23 (42%(14605’0) @019, 0x1)” =wx k(k5DH)®ox,, 0x. =wx,, ”4519”) (4-9) Therefore, (u. 2wxg,(k60”>®o. (403.4431))” (410) Notice that OX,(kR1) is allowing poles of at most order 19 times along ml, at most 2k along m and k(L——— '2’) along D3,. The group H acts on the sheaf By writing the sheaf as a direct sum of the eigenspaces with respect to the different characters of H we can see that it is a direct sum of an invertible sheaf and its powers. When we take H -invariants, only the powers that correspond to multiples of ramification degree remain. This happens when 2, 3, or 9% divides k. Therefore, we can write explicitly, (llll—“ll—l where [t] means the rational number t is rounded to the next smaller integer number. Thus, (u*(wx, (k513i. ))l” W ll ll *llw) (4.12) Now, we will state a key proposition that allows us to understand how we can relate the lattice of modular forms and Cl(Z[(,]). This is based on results of [Hi] (see [CPTn) 24 Proposition 4.1 Let x : F —+ Z[Cr]* be a 1-dimensional character of prime order r 2 5 with kernel H. Let G = P / H and suppose M is a finitely generated torsion-free Z[G]-module. Define M X to be the Z[(,]-module (M 8) Z[Cr]x'")G. a. There is a unique homomorphism e; : G0(Z[G]) —-> C1(Z[C,.]) such that for all M as above, either MX = {0} and e;(([M]) = O or MX is isomorphic to Z[C,,’ 635.1 for some integer s 2 0 and a Z[C,]-ideal 5.1 in the ideal class e;( [M ]) b. There is a unique isomorphism tx : K0(Z[G]) -—> Z®C1(Z[Cr]) such that tx([P]) = (rankz[G](P),ex([P])) if P is a projective Z[G]-module, where {—P] is the image of P in Cl(Z[G]) and ex : C1(Z[G]) —* C1(Z[C,.]) is the unique homomorphism such that eX([P—]) = e;(( f ([P])) for all projective P, where \ f : K0(Z[G]) —> G0(Z[G]) is the forgetful homomorphism. CHAPTER 5 Proof of Main Theorem We now compute the image of 7” (X H, (11.4wa (lcr’iD;Q)))H ) under the injective homo— morphism O 2 O3 : C1(Z[G]) ——> CZ(G, 3), which is defined in [CPTl], by applying their main result and using the isomorphism (4.11). This result allows us to calculate the equivariant Euler characteristic of a sheaf if there is a tame cover and if the sheaf can be written as a pullback from the quotient. In our case, let my : X H —-> X0 be our cover. Since the index of H in I‘ is the prime r 2 5, the order of H is divisible by 6. By Theorem (2.1) 7m is ramified only at the fiber over p. By (4.11) we already get the sheaf ()u,,(wXl (kriD1 )))H in terms of ka (kaH ) twisted by a certain divisor. Also, a)?“ (k6Dg) can be written as pull back of the sheaf w§:(k6Doo) with some twist. Therefore, (11,.(wX’“(kciD1 )))H can be written as pull back of the sheaf w§§(k5Doo) with some twist. Let’s try to see the relation between them. Making the similar local calculations in [pg.74 Ma], we can say (11?: (deH)— —— 7”,on ”(kciDH + k(r—— 1)D(§1). And hence, (aiwiflkéDQDH 2 7th (12513” + RH + n03?) (5.1) Here, we denote by RH the divisor [g] {j = 0}H + [231'] {j = 1728}H on XH , which 26 can be identified as pull back of the divisor R0 = [g] {j = 0} + [2-3’5] {j = 1728} on X0. We also note 7] = k(r —— 1) + [W]. We have this fundamental sequence ; 0 —> OXH(—77D51)—> OX” ——> (9an —+ 0 (5.2) and tensoring by ngwfflkéDg + RH + nDé’ ) we get, 0 —+ hyweflhaogm’i) —» n3w§§(k6D£+RH+nDé’) 2 hywxflchDHmHmDH MM. —» 0. (5.3) Since (114wa (deéoD) 2 nHwXKkaH + R” + 71D”), then our sequence becomes, 0 —> rng’chiDl + R”) (h.(w§3f(kcso;o)))H —» c —+ 0 (5.4) where C = 7r;,(u.X0(k6DH + R” + nDo ))[nDu is also supported on D”. The sequence implies the following relation between equivariant Euler character- istics, YP(XH1(H*(wX1(k6Dc1>o)))H):XP(XH17erx:(k6DH+RH))+XP(XHvC) (5-5) On the right hand side, the first equivariant Euler characteristic will be calculated easily using the main theorem of [CPTl], and the second one will be calculated using the adjunction formula as follows: We have the following exact sequence ; 0 —+ I'M/1" —> OxH/l" —+ (OM/1"—1 —» 0 (5.6) where I is the ideal sheaf of 03,". By induction on 77 and tensoring each sequence by 27 wgwfiydeg + R” ) we get the following sum for the equivariant Euler characteristic of C n—l YP(XH,C) = 7P(Xn,lfii1w;®2§(k50§o + R”)®(1"/1"+1)l|pg!). (5-7) q=0 Let us denote by N5”, the conormal bundle of D5. Then (5.7) gives 17—1 for”, C) = 27PM”, [nawrmwé’o + RH)®(N1\351®Q)HD§)- (5.8) q=0 Let us revisit the calculation of the Euler characteristic XP (X H,C). Since the group G acts trivially on D5! , the Euler characteristic 7” (X H, C) is just the numerical Euler characteristic times the class of the character of [X0] of the group G. Here the class of a character [X0] is defined as follows: As we know [X0] is a homomorphism from G to field Q. We can extend this homomorphism to Z[G] the kernel of this homomorphism, which is an ideal in Z[G] gives a class in Cl(Z[G’]). This class is referred as a class of a character. The numerical Euler characteristic of the sheaf 7r},w§§(k6Dfo + R”) <8) (Ngéf®q)]|DOH on 135’ for any q can be calculated by using Riemann-Roch. We obtain XP(XH:l7r;iw§:(k6D<1>o + RH) ®(N1\$gl®qll|0{,*) = = deg(7rFJW§f(k5D£ + R") ®(N}$51®q)|ng) + 1 - 9(XH) (59) where degree of the sheaf in the formula above can be calculated by adjunction for— 28 mula: we get, degmwfiykw; + R") e (Ngfqnpg) = _ _kr(p- 1) k(7‘- 1)(p-1) — 12 + 2k 127‘ + kr6(p - 1) k 2k (p — 1) + 12 +([2]+[3])r qn 12 . (5.10) So, we can write the Euler characteristic as, n—l fume) = 23mm, m + We. 6,r))1x31 (5.11) :0 for integers m(k 7‘) = -n(p_1) (5.12) ’ 12 and _ kr(p —— 1) k(r -—-1)(p — 1) kr6(p — 1) k 2k n(k,6,r) —— 12 +21: 127‘ + 12 +( 2 + 3 )7‘ (5.13) depending 011 k,5 and 7". Now let’s turn to the calculation of XP (X H, wfiwfig (deg + RH)). Since we will use the main result of [CPTI] for this calculation, it is better to recall the theorem. For the convenience of the reader we will report the main theorem and some of the arguments for our calculation from [CPTl]. 5.0.1 Main Theorem of [CPTI] Let R be the ring of integers of a number field K. Let 7r : X —» Y be a G-cover, which is tame, i.e. for every closed point x E X, order of inertia subgroup 1,; C G is relatively prime to the charactetistic of the residue field k(:1:), and also domestic, i.e the residue field characteristic of each point of Y which ramifies in 7r : X —> Y is 29 relatively prime to the order of the group G. Denote by S the finite set of rational primes such that the cover 7r : X —> Y is only ramified at points above S. By our assumption assumption, p E S implies p [#G. Denote by SK the set of places of K that lie above 8. Suppose g is a locally free coherent Oy—sheaf on Y and consider the G-sheaf .7: = «*g on X. Consider the injective homomorphism 6) = 9d” : C1(R[G]) = Pic(R[G]) —> CR(G;d + 2) where CR(G; d + 2) is the isomophism classes of the objects which has cubic structure. For a finite place 12 of K, we denote by w” a uniformizer of the completion R4, and fix an algebraic closure 1?” of its fraction field K1,. Also any finite idele (a1), 6 A;.K[Gd+2] gives the element (flv(Rv[Gd+2]av fl K[Gd+2]), 1) of CR(G;d + 2). Now let 1) 6 SK (then (1), #G) = 1) and denote by R1, the complete discrete valuation ring R1, C K, obtained by adjoining to R, a primitive root of unity of order equal to #G. Then 17),, is also a uniformizer for R1,.Let us consider the cover 7r1, : X (8);; R1, ——> Y (813 R1, obtained from 7r by base change. Since R1, has residue field characteristic prime to #G and contains a primitive #G-th root of unity. There is an isomorphism Kled+2l* :* HomGaKRu/K.)(Ch(Gd+2)vsR1?) (5-14) given by evaluating characters of Gd”. Also if (I) = (151 <8) - - - 18> ¢d+2 is a character of G‘”2 given by a (1+ 2-tuple (gm),- of 1-dimensional R'v-valued characters of G, we have GD(¢) = ($1 — 1) - - - (¢d+2 — 1) E Ch(Gd+2)v. The function Tug defined on R—valued characters is defined in [CPTl] by an explicit formula and its definition implies that, for all a E Ga1(Ku/Kv) and 11’ E Ch(G)v, we have T MW) = new”). 30 Hence, the map (1) +—> wJT"'G(e(¢)) gives a function in HomGal(Rv/Kv)(Ch(Gd+2)v, K3). Theorem 5.1 With the above assumptions and notations, 9(2 ' XP(X,J"")) = (flv(Ru[Gd+2l/\v F1 KlGd+2l), 1)» where (A1,)v E A}.K[Gd+2l is the (unique) finite idele which is such that 1, if ’U ¢ SK ; ¢(’\v) = (5.15) __ . D wv2Tv,Q(e (¢)), if’U 6 SK , for all Kv-valued characters ()5 of Gd”. If the “usual” Euler characteristic x(Y, Q) = Zi(—1)irankR[Hi(Y, 9)] is even, then we can eliminate both occurrences of the factor 2 from the statement: @0213 (X, f)) is then given by the idele (11,). with ¢(,\;,) = 1 if v e SK, hug) :- he; “9‘90“” if v E S K. D The field (2,, already contains a primitive p — 1-st root of unity. Hence, we may take R110) = Zp. We find that 8(7P(Xy,wf,w§1f(k60£ + R”))) E Gz(G; 3) is given by the idele (b1), 6 A}.Q[G3] which is 1 at all places 22 75 (p) and is such that (X (g) 95 ® ¢)(b(p)) = p_T((X—l)(¢—1)(¢—l)) (5.16) with T : Ch(G)p —-> Q the function associated to the cover X H ®z 2,, —-> X0 (82 L, in the main theorem . For a E Z/TZ let {a} be the unique integer in the range 0 s {a} < 7‘ having residue class (1. Using the eqn (3.15) in [CPTl], T becomes + (1 —2k(1+6))g—(w1200) TM”) = p — 1 . _g(l/)1D0)2 12 2 ) +(1— 2k<1+ Myth 00) (5.17) 31 = 2.1221. @1522. _ (1 — 2k(1+ 6))%) + (1 — 2k(1+ 6)){:}". (5.18) where 1,!) = )6“, X0 = coma-12 and w : (Z/pZ)* ——> Z; is the Teichmuller character. For a E Z/T‘Z define wr(a) = 0 if a = 0, and otherwise let wr(a) E Z. C Z be the Teichmuller character associated to 7'. Define W) = J” 1‘21 (5(a)?) and 13(5) = (1 — 2k(1+ 5)) {‘2’ (5.19) 27‘2 where 1/) = x5“ as above. We extend 2,!) ——> 1110,12) to a function on the character ring Ch(G)p by additivity. Since p E 1 mod 24 and rll—2‘f, we can define fl = (5,), with ,8. E Z @Z Z,[G]* by 1’ if v 74 (P); 1pm”) 2 (5.20) p_T(¢)+T1(¢)+T2(‘/’), if U = (p) Since Cl(Z[G]) is a torsion group, ,3 defines a unique class [B] in Cl(Z[G]). We now show rpm. whwi’éflkwé + 12“» = 161. (5.21) Define D = [fl] —5(‘P(XH,7r;,w}9}§(k6Df° +RH)), and let R = Zifz': land R = Z if z’ = 2. From (5.19) one has rill-(1p) E R and rill-(2,0) E a mod TR. It follows that for all triples (x, (15,112) elements of C'h(G)p, Ti(x¢z/J — x¢ — (In!) — x1!) + x + a3 + w —— 1) lies in R. Hence there are elements a,- = (ai,v)v E [1,, finite(R ®z Qv[G3]*) for which 1» if v 75 (P); (X Q?) ab 69 whee-.5) = (5.22) pr.-(x¢w—x¢—¢w—xw+x+¢+uv—1), if 1, = (p) 32 We now conclude from (5.16), (5.17) and (5.20) that 1® G(D) = 61 + Cg (5.23) in Z (812 CZ(G; 3), where c,- is the class associated to the element a,. Let us first show c2 = 0. (5.24) For this it will suffice to show that there is a cubic element A E Q[G3]* such that Aag is a unit idele of Q[G3]. Fix a primitive p—th root of unity (p E W, and let 7(2) = Z vex; jE(Z/p)‘ be the usual Gauss sum associated to w. Let 7' be the unique extension of the map 1/2 —> 701)) to a. homomorphism from BC to Q7. We let 7(3) be the element of Hom(Rca,©7) which sends (x, (15,1/1) to T(x¢w—x¢-¢w—xw+x+¢+¢-1) From the behavior of Gauss sums under automorphisms of 7Q, and the factorization of the ideals they generate (c.f. [La, §IV.3]), it follows that 7(3) is Gal(@/Q)-equivariant, and corresponds to an element A E Q[G3]* of the required kind. This shows (5.24). Turning now to Cl, let 0(3) be the automorphism of G which sends g E G to g3 for s E (Z/TZ)*. By (CPTl) the action of Aut(G) on Zp[G]* corresponds to the action of Aut(G) 011 f E H0m(Ch(G)p, Q?) defined by (0(8)(f))(X) = f (0(3)"1(X)) = f (X3) for X E Ch(G)p. From the definition of the T1 in (5.19) and the multiplicativity of the Teichmuller character we have (0(3)T1)(1p) = w.,.(s)2T1(2/2). It follows that the 33 element 0(3) 2 0(3) — wr(s)2 sends 01 to the identity function, so a(3)c1 = 0 (5.25) Because 9 is Aut(G)-equivariant, we can now conclude from (5.23), (5.24) and (5.25) that 1®(9(a(3)-D))=0 in Zeazcz(0;3). (5.26) If A -—> B is an injection of abelian groups, and A is finite, then A = Z®ZA ——> Z®ZB is injective, as one sees by reducing to the case in which A ——> B is the inclusion n‘lZ/Z —-> Q/Z for some n 2 1. So (5.26) and the injectivity of 9 implies 02(3) - D = 0 in C1(Z[G]). (5.27) Similarly, since r2T1(¢) is in Z, C Z and C1(Z[G]) is a torsion group, we see from the injectivity of 9 that D is in the r-Sylow subgroup of Cl(Z[G]). We now use the fact that Cl(Z[G]) is isomorphic to Cl(Z[C,]). Define C, to be the group of classes c in the r-Sylow subgroup of Cl(Q(C,.)) for which (0(3) —-w,.(s)j)(c) = 0 for all s E (Z/r)*. We have shown 01 corresponds to a class in Cg. By the Spiegelungsatz (c.f. [Wa, Theorem 109]), 02 = 0 if C_1 = 0. Herbrand’s Theorem ([Wa, Thm. 6.17)) shows that if G_1 aé 0, then the Bernoulli number B,_(,-2) = 82 .1. 6 and r 2 5, so we have is congruent to 0 mod 71.. This is impossible since 82 = shown 5.21. To complete the proof of our theorem, we will find —€x(7P(XH1(#a(w§f(k5055)))")) ah... calculating «Morse» and adding with our result 5.21. By choosing a suitable element of A = Gal(Q(Cr)/Q) to apply, 34 we can reduce to the case in which 111:1.) x=X0=w r . (5.28) With these conventions from the definition of ex in Proposition 4.1 we have -€x(lX0l) == [PX] and hence —ex(1x31>= 10:105.», if(a,r)=1 . am) = 0 otherwise. (5.29) So, when we apply —ex to 7P (X 3,6) we will get; n—l — €x(5C_P(XH,C)) ——- Zora/arm + n>a;1 . [em] = :0 q n—1 n—1 = m(k.r)q 5,-1 - [Px.1+ Zn(k.6,r)o;‘-1Px.1 (5.30) q=0 q=0 From the definition of all terms, we get the following equality, —ex(7”(xe, (a.(w§f>>”>> = 02172010 _ p52). _ 1) -01179...]—1611-1a.1—16011a.1 where 01 = Z {0.}01‘,‘1 E Z[A], (5.31) aE(Z/r)* [91] = Z m(k.r)qaq‘1 (532) 0>)1—1H1(Xa,Mash/«60:01)» (5.34) in Cl[Z(G)]. If we arrange so that the first cohomology group vanishes then, we obtain a precise formula for the twisted cusp forms. Details are given in the following corollary. Corollary 5.1 There is (50 such that for every 6 > 60, we have the following. Suppose m C Z[Cr] is an ideal with ideal class 92 - [’PXO] — [01] - ['PXO] — [00] . [Pm]. Then, S2k.6(rl(p)1ZlCrllx ’1 Z[C1~]"(""1 EB ‘21 (535) as Z[CJ-modules. PROOF. With the notations of the theorem, recall that Sgk,5(I‘1(p), p‘6kZ[(r])x is the Z[CJ-submodule of 82k,5(1"](p), p‘akZKTJ) consisting of twisted cusp forms of weight 2117 and of Nebentypus character X whose n’th Fourier coefficients at 00 are in the 36 form of 5E7,— where r in Z[Cr]. Proposition 3.1 and its proof together with the fact that formation of the canonical sheaf commutes with the base change Z ——+ Z[CT] implies that Sgk,5(I‘1(p),Z[C,.]) c: 52k,6(I‘1(p),Z) ®z Z[Cr]. Propositions 3.1 and ?? now give an isomorphism of (torsion free) Z[CJ-modules 321,.(r1(p).p‘6’°Z[51)x 2 H°(XH, (max, (k619i. )))” )X The projective class XP(XH,(u.(le (MD1 )))”) E K0(Z[G]) has the property that f (XP (X11, (#a(W§'f(k5Dée)))” )) = = 010(th(#:1015210'951?1 ))l")l - [H1(XH,(ua(wxf(kéDlo )))”)l (536) where f : K0(Z[G]) —+ G0(Z[G]) is the forgetful homomorphism. If P is a projective Z[G]-module, then Q ®z P is a free Q[G]-module, so rankz[g](P) = rankz(P)/r. Therefore, using Riemann-Roch we get raakzmHWXH, (Max: (1553,, m”) — rankzelnwxa, (#a(w§?f(k50éo)))”) = where g(XH) is the genus of X”. Because the generic fiber of X H ——> X0 is étale of degree 7‘, by the Hurwitz Theorem, we can say (9(XH) — 1)/r— -- g(X0) — 1 and we know that g(X0)= M hence, n(X) = (2k —11)ép— 25) + [k(p(;:I)2r)] + [g] + {—2313} . (5.38) Let YP(XH, (u.(wX1‘(k6D1 )))”) be the image of XP(XH, (,u..(w§f(k6Déo)))H) in Cl(Z[G]). If we prove that H1(XH, (p.(w§f(k6D10)))”) vanishes when 6 > 50 for 37 some 60, we can easily conclude from (5.36) and 4.1 that there is an isomorphism of Z[CJ-modules Sea,a(1‘1(p).Z1<.1)x 2 215.1110“ 6921 (5.39) where fl is a Z[C,]-ideal having ideal class —eX(XP (X H, (12100331c (k6D10)))H )) The only thing left is to prove first cohomology group is trivial when 6 > 60 for some 60. This is done in following lemma. Lemma 5.1 H1(XH, (,u...(wX1"(lcc5D1 )))H ) is trivial when 6 > 60 for some 60. PROOF. If we can show that H1(XH, ((1l...(wX1‘(k6Dl )))”)V) is torsion free (which is necessary condition for duality), then the result will follow by duality as follows: H1(XH) (u.(w}‘?1°(k6D§o)))") = Homz(H°(XH,(0541531215131 ll)” )V), Z) (540) Here H°(XH,((11..(wX1°(k(5D1 )))H )V) is trivial because of the degree of the sheaf is negative. To Show that H1(X H, ((1l..((.oX1°(kc5Dl )))H)V) is torsion free it is enough to check that H°((XH)3, ((11...(wX1c (lrcch1 )))H )V) it is trivial on each fiber B. If the fiber 3 7é p, then it is just P1 and degree of the sheaf is negative implies result. Otherwise (fl = p), we have two component namely, D5! and D011, one of them is totally ramified and the other one is unramified. Let 3 be a global section of our sheaf, then its restriction to D111, is zero since D11, is reduced. Let’s call W for the non-reduced component. So, Wm“: D” and 0W 2 005; as N as N‘g’2 a; - - - 39 NW“ (5.41) where N = OXH(—D5’)|D€. Recall the following equation, (111(wa (MDl )))H 2 #11on k(k6DH + R” +77DH) (5.42) 38 Therefore, 3 is given by r-tuple of sections 3,- of the sheaf N®‘((1 — k)KH . Dg’ -— 56001.05! — 171351.051 — R” 03,") We know that NW = 0Xu(-TD§)|D,§I = Oxymgllpg then deg(N®") = 05.19;: = (pl—21) which implies We also know that KH-(Dfo+ng’) =0=>KH-Do’1, = —rKH-D{,’ Adjunction formula gives, (KH+D£IO)'D:!0 = 290:1. ‘2 and Hurwitz formula gives, — 1 —— 1 290g) - 2 = 7(2ng — 2) + (T 112]) ) both together imply that -— 1 — 1 KH-Dfoz—Dfo-Dg—2r+(r 1gp ) 39 (5.43) (5.44) (5.45) (5.46) (5.47) (5.48) (5.49) (5.50) Also, Dfo-(Do’f,+rD{,’)=0:>D£,-D11,=—rD{,’-Do’g=2.717112LL) (5.51) So, _15_Np-U (r-Dw—1Y_ Ky Doo — 12 + 12 27‘ (5.52) and -1) (T—1)(p-1) K . H = __(£_____ _ .1. - H D” 12 121' ' 2 (5 53) If we plug all these into the degree calculation of our sheaf we get; 10—4) kflp—D 5—1) U-JXp-D 121‘ 12 + (1 k)( 12 127' + 2)+ k(p—1—2'r) 11—1 116 2 +1 (15—1) 1(127') ,2 3 Tw-J) kflp-D 0-1) 0—1) NW < — —— — — — . _ 12 12 +(2k 1) 12 + 12? +2 3 (5 54) We want to find a lower bound to 60 which guarantees that this term is negative, we say 71’" 1) WP 1) (p- 1) (1U— 1) 10k 12 _ 12 + (2k — l) 12 + 127, + 2 “ T S 0 (5.55) 1 24 4O 6>2+—(r—1+—+(p_1))—(p_1) (5.56) and remember that 'r > 3 and 247" divides p — 1, therefore 7" + 2 is going to be enough for 60 . 40 BIBLIOGRAPHY 41 BIBLIOGRAPHY [Ar] M. Artin: Lipman’s proof of resolution of singularities for surfaces, Arithmetic Geometry. Editors: G. Cornell, J. H. Silverman, Springer, New York, 1986. [BFQ] P. Baum, W. Fulton, G. Quart: Lefschetz-Rjemann-Roch for singular varieties. Acta Math. 143 (1979), no. 3-4, 193—211. [C] T. 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