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On Selmer Groups of Geometric Galois Representations Tom Weston On Selmer Groups of Geometric Galois Representations Tom Weston Department of Mathematics, Harvard University, Cambridge, Mass- chusetts 02140 E-mail address: [email protected] iii Dedicated to the memory of Annalee Henderson and to Arnold Ross Contents Introduction ix Acknowledgements xii Notation and terminology xv Fields xv Characters xv Galois modules xv Schemes xv Sheaves xvi Cohomology xvi K-theory xvi Part 1. Selmer groups and deformation theory 1 Chapter 1. Local cohomology groups 3 1. Local finite/singular structures 3 2. Functorialities 4 3. Local exact sequences 5 4. Examples of local structures 6 5. Ordinary representations 7 6. Cartier dual structures 8 7. Local structures for archimedean fields 9 Chapter 2. Global cohomology groups 11 1. Selmer groups 11 2. Functorialities 13 3. The global exact sequence 13 4. A finiteness theorem for Selmer groups 14 5. The Kolyvagin pairing 16 6. Shafarevich-Tate groups 18 7. The Bockstein pairing 20 Chapter 3. Annihilation theorems for Selmer groups 21 1. Partial geometric Euler systems 21 2. The key lemmas 22 3. The annihilation theorem 25 4. Right non-degeneracy of the Bockstein pairing 27 5. A δ-vanishing result 28 Chapter 4. Flach systems 31 1. Minimally ramified deformations 31 v vi CONTENTS 2. Tangent spaces and Selmer groups 34 3. Good primes 36 4. Flach systems 38 5. Cohesive Flach systems 39 6. Cohesive Flach systems of Eichler-Shimura type 40 Chapter 5. Flach systems of Eichler-Shimura type 43 1. The map on differentials 43 2. The Tate pairing 45 3. A special case 47 4. A matrix computation 52 5. Computation of Ξ in the non-diagonal case 53 6. Computation of Ξ in the general case 54 Part 2. Construction of cohesive Flach systems 55 Chapter 6. The Flach map 57 1. The coniveau spectral sequence in ´etalecohomology 57 2. The localization sequence 59 3. Grothendieck's purity conjecture 61 4. The coniveau spectral sequence in K-theory 62 5. Definition of the Flach map 64 6. Functoriality and passage to the limit 66 7. Functoriality II 67 Chapter 7. Local analysis of the Flach map 71 1. Overview 71 2. Local behavior I 72 3. Local behavior II 72 4. Local behavior III 74 5. The divisor map 75 6. The cycle map 77 7. Relations with Galois cohomology 78 8. Functoriality and passage to the limit 78 9. Example : Schemes over global fields 79 10. Local behavior at places over l 80 Chapter 8. Flach classes for correspondences 83 1. Algebraic correspondences 83 2. Correspondences and operations on ´etalecohomology 84 3. Composition of correspondences 86 4. Marked varieties 88 5. Divisors and compositions 89 6. The Leibniz relation 90 7. Algebras of correspondences 92 8. Derivations in the self-adjoint case 93 9. Local diagrams in the self-adjoint case 94 10. Derivations in the general case 95 11. Untwistings and cycle classes 97 12. Derivations modulo η 97 CONTENTS vii Chapter 9. Construction of geometric Euler systems 99 1. Divisorial liftings of cycles 99 2. Construction of partial Euler systems 100 3. Partial Euler systems on products 101 4. Construction of Flach systems in the self-adjoint case 102 5. Construction of Flach systems in the general case 105 6. Construction of cohesive Flach systems 106 Part 3. Examples 107 Chapter 10. The modular curve X0(N) 109 1. The geometry of X0(N) 109 2. The modular unit ∆ 113 3. The divisor of fp in positive characteristic 115 4. The cohesive Flach system 116 Chapter 11. The modular curve X1(N) 117 1. The geometry of X1(N) 117 2. Admissible markings 120 3. The cohesive Flach system 121 Chapter 12. Kuga-Sato varieties 123 1. The geometry of Kuga-Sato varieties 123 2. Admissible markings 125 3. The cohesive Flach system 126 4. Applications 126 Appendix 129 Appendix A. Edge maps of spectral sequences 131 1. Notation for filtered complexes 131 2. Edge maps 132 3. Edge maps in spectral sequences of filtered complexes 133 4. Edge maps in Grothendieck spectral sequences I 134 5. Edge maps in Grothendieck spectral sequences II 135 6. Boundary maps of exact sequences of filtered complexes 136 7. Boundary maps of Grothendieck spectral sequences 137 8. Edge maps of exact couples 138 Appendix B. Gorenstein linear algebra 141 1. Definitions 141 2. Gorenstein traces and congruence elements 142 3. Gorenstein duality 143 4. Gorenstein pairings 144 5. Skew-symmetric Gorenstein pairings 145 6. Bilateral derivations 146 7. Torsion modules 148 Bibliography 151 Introduction Fix a squarefree integer N and let f be a rational weight 2 newform for Γ0(N). Let H be the l-adic representation associated to f for some l 7; H is a free ≥ Zl-module of rank 2. Let QS denote the maximal extension of Q unramified away from Nl and set GQS = Gal(QS=Q). Flach proved the following theorem regarding the deformation theory of H. (See [Wes00, Appendix A] for the proof that the set of primes satisfying Flach's conditions has density 1; this set can be given quite explicitly.) Theorem 0.1 ([Fla92]). Fix f as above and assume that f does not have complex multiplication. Then the set of primes l such that the universal deforma- tion ring of the residual representation ρ : GQ AutF (H=lH) is isomorphic to S ! l Zl[[T1;T2;T3]] has density 1. This result was extended to the case of newforms defined over arbitrary number fields in [Fla95] and [Maz]. In this case the l-adic representation H is free of rank 2 over a certain completion A of a Hecke algebra; this ring A is a reduced, finite, flat, local, Gorenstein Zl-algebra and contains a Hecke operator Tp for every prime p. Let k denote the residue field of A. Mazur observed that Flach's construction can be used to obtain results on the Taylor-Wiles deformation problem for almost all l 7 not dividing N. ≥ Theorem 0.2 ([Maz]). Let f be a newform of weight 2 for Γ0(N). Let l 7 be a prime not dividing N and let H be the l-adic representation associated to≥f. Assume that the natural map GQ Autk(H A k) is surjective. Let R be the ! ⊗ minimally ramified universal deformation ring for H A k. Then R is a finite A-algebra and the natural map R A induces an isomorphism⊗ of differentials ! ΩR R A ΩA. ⊗ ! In this thesis I extend the results of Flach and Mazur to the case of (most) newforms of weight at least 2 for Γ1(N). I also show that the \geometric Euler system" used to prove these results has a very rich algebraic structure and that the isomorphism it yields in deformation theory is essentially canonical. The main result in this context can be phrased as follows. Theorem 0.3 (Theorem XII.3.1 and Theorem IV.6.2). Let f be a newform of weight k 2 for Γ1(N). Let l max 7; k be a prime not dividing N and let H be the l-adic≥ representation (over≥ an appropriatef g completion A of a Hecke algebra) associated to f. Assume that f can be \cleanly realized" in the cohomology of the universal elliptic curve with level N-structure (see Chapter XII for precise condi- 0 tions). Set T = End H(1) and assume that the natural map GQ Autk(H A k) A ! ⊗ is surjective. Let R be the minimally ramified universal deformation ring for H A k (see Section IV.1). Then R is a finite A-algebra and the natural map R A⊗in- ! duces an isomorphism of differentials ΩR R A ΩA. Furthermore, the inverse of ⊗ ! ix x INTRODUCTION this isomorphism is characterized by the fact that the identification 1 ΩA ΩR R A = HomZ H (Q;T ∗[η]); Ql=Zl ! ⊗ ∼ l f (see Section IV.2) identifies the differential of Tp A with 12 times the image under the Bockstein pairing 2 − 1 1 H (Q; T/ηT ) HomZ H (Q;T ∗[η]); Ql=Zl f ! l f 1 of the cohomology class cp Hf (Q; T/ηT ) obtained via the Flach construction. 1 2 Here H (Q; ) is a Selmer group (see Section II.1), T ∗ is the Cartier dual of T , f · and η is the congruence element (see Appendix B.2) for the Gorenstein Zl-algebra A. However, it is natural to hope that the method of proof of these results is as interesting as the results themselves. For this reason we proceed in as much generality as we can. Let X be a nonsingular algebraic variety over a global field 2m F and let H be a quotient of the ´etalecohomology group H (XFs ; Zl(m + 1)) for some m. I give a \general" method (contingent on the existence of appropriate 0 geometric data on X X) for the production of geometric Euler systems for EndA H; these in turn yield× corresponding annihilators of certain Selmer groups. These annihilators yield results on the deformation theory of the Galois representation H=lH. Somewhat more generally one can hope to use appropriate geometric data on X itself to control the Selmer group of H; this could then possibly be related to the Bloch-Kato conjectures. The required geometric data (in the deformation theory case) does exist for modular curves and Kuga-Sato varieties. It seems likely that it exists in the case of Hilbert modular surfaces as well. One can give \explicit" conditions for the existence of this data in general, although at this point these are not particularly useful.
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