Arithmetic of Euler Systems August 2015 Centro de Ciencias Pedro Pascual Benasque, Spain Organizers X. Guitart and M. Masdeu Scientific Advisers D. Loeffler and S. Zerbes Foreword The main aim of the workshop was to give an introduction to the recent developments in the area, in particular the work of Bertolini-Darmon-Rotger, Lei-Loeffler-Zerbes and Kings-Loeffler- Zerbes on Euler systems for Rankin convolutions, at a level accessible to graduate students and other younger researchers. The instructional part of the workshop consisted of 12 lectures giving an account of the above works, together with a selection of more advanced talks on related areas of current research. Students and postdocs volunteered to give talks themselves in the introductory lecture series; which were allocated and coordinated by David Loeffler and Sarah Zerbes, the scientific advisers. iii Acknowledgements The organizers wish to thank the participants for all the work they put in preparing the talks and typesetting the first version of these notes. We wish to thank also the Centro de Ciencias Pedro Pascual for its hospitality and for providing us with exceptionally inspiring setting. Finally, the organizers wish to thank Joan-Carles Lario for his help and guidance, without which this workshop wouldn't have been possible. iv v Lectures 1 Introduction to Modular Curves 2 Yukako Kezuka 2 Hida Theory 5 Chris Williams 3 Siegel units and Eisenstein classes 15 Antonio Cauchi 4 Definition of the global classes 26 Guhan Venkat 5 Compatibility in p-adic families 29 Francesc Fit´e 6 Norm compatibility relations 37 Vivek Pal 7 p-adic Hodge theory and Bloch{Kato theory 47 Bruno Joyal 8 p-adic Eichler{Shimura Isomorphisms 57 Yitao Wu 9 (Modified) syntomic and FP cohomology 61 Lennart Gehrmann 10 Evaluation of the regulators 66 Netan Dogra 11 Proofs of the explicit reciprocity laws 77 Giovanni Rosso 12 Applications: bounding Selmer and Sha 84 Jack Lamplugh vi Arithmetic of Euler Systems 1 Introduction to Modular Curves Talk by Yukako Kezuka [email protected] Notes by Alex Torzewski [email protected] Let H = fτ 2 C: Im(τ) > 0g denote the complex upper half plane and Γ ⊆ SL2(Z) be a conguence subgroup. By the modular curve associated to Γ we refer to the quotient ΓnH where the action of SL2(Z) on H is given by linear fractional transformations. For τ 2 H, let Λτ = Z + Zτ and Eτ be the elliptic curve C=Λτ . If Γ = Γ1(N), then Γ1(N)nH parametrises elliptic curves with a marked point of exact order N. Explicitly there is a bijection Γ1(N) · τ $ [Eτ ;P ]; 1 between cosets and equivalence classes of pairs [Eτ ;P ] where we may assume P = N +Λτ 2 Eτ [N]. We will rephrase the definition of a pair [Eτ ;P ] so that it makes sense with C replaced by any scheme S. Definition 1.1. Let S be any scheme. Then an elliptic curve over S is a scheme E with a proper flat morphism π : E ! S whose fibres are smooth genus 1 curves with a choice of section O : S ! E. Definition 1.2. Let Y1(N) be the smooth Z[1=N]-scheme representing the functor on Z[1=N]- schemes 8 9 < Isomorphism classes of pairs (E,P) = F : S 7! where E is an elliptic curve =S and : : P 2 E(S) a point of exact order N. ; By \Y1(N) represents F", we mean F(·) is isormorphic to Hom(·;Y1(N)) as functors. If S = C there is a natural bijection φ:Γ1(N)nH ! Y1(N)(C) which is an analytic isomorphism and (Y1(N); φ) is a model for Γ1(N)nH. Remark 1.3. [LLZ14, 2.1.4] The cusp Γ1(N) · 1 of Y1(N)(C) is usually not defined over Q[[q]] but 2πi rather over Q(µN )[[q]] since it corresponds to the pair (Gm=qZ; ζN ) where ζN = e N . This leads to the following alternative definition of Y1(N). A choice of P 2 E(S) of exact order N amounts to giving a closed immersion ι:(Z=NZ)S ! E of group schemes, where (Z=NZ)S denotes the constant group scheme of Z=NZ over S. Similarly we can use a model for Y1(N)(C) which parameterises pairs (E; ι), where ι:(µN )S ,! E is a closed immersion. The corresponding smooth scheme is denoted Yµ(N), and thus we obtain a model (Yµ(N); φN ) for Y1(N)(C). Here φµ is defined by τ 7! (Eτ ; ιτ ) for τ 2 H and ιτ denotes the 1 embedding defined by ι(ζN ) = N + Λτ . The cusp Γ1(N) is defined over Q[[q]] with respect to this model. Definition 1.4. Let L ≥ 3 and let Y (L) be the smooth Z[1=L]-scheme representing the functor sending, E is an elliptic curve =S S 7! (E; e1; e2)= ∼ : : and e1; e2 generate E[L]. There is a left action of GL2(Z=LZ)={±1g on Y (L) given by the action on e1; e2 given by, 0 e1 a b e1 0 = · : e2 c d e2 2 Yukako Kezuka Introduction to Modular Curves Remark 1.5. If S = C we have an isomorphism of analytic spaces × ∼ (Z=LZ) × Γ(L)nH ! Y (L)(C) a 0 (a; τ) 7! · ν(τ); 0 1 where ν is the canonical map ν : H! Y (L)(C); τ 7! (Eτ ; τ=L; 1=L). This in particular tells us that Y (L) is not geometrically connected. Definition 1.6. Let L ≥ 3, M; N ≥ 1 and M; N j L. Set Y (M; N) = GM;N nY (L) where a b (a; b) ≡ (1; 0) (mod M); G = 2 GL ( =L ): : M;N c d 2 Z Z (c; d) ≡ (0; 1) (mod N): If M + N ≥ 5, Y (M; N) represents the functor of triples (E; e1; e2) where e1 has order M, e2 order N and together they generate a subgroup of order MN. In order to define Hecke operators on K2(Y (M; N)) and ther duals, we need also: Definition 1.7. Let A ≥ 1;L ≥ 3, M and N s.t. M j L and AN j L. Define Y (M; N(A)) to be the quotient of Y (L) by the subgroup GM;N(A) ≤ GL2(Z=LZ) given by a b a ≡ 1 (mod M); b ≡ 0 (mod M) G = 2 GL ( =L ): : M;N(A) c d 2 Z Z c ≡ 0 (mod NA); d ≡ 1 (mod N): Similarly define Y (M(A);N) using GM(A);N given by a b a ≡ 1 (mod M); b ≡ 0 (mod MA) G = 2 GL ( =L ): : M(A);N c d 2 Z Z c ≡ 0 (mod N); d ≡ 1 (mod N): The Z[1=L]-scheme Y (M; N(A)) (resp. Y (M(A);N) represents the functor which sends 8 9 < C is a cyclic subgroup of order NA (resp. MA), = s 7! (E; e1; e2;C)= ∼ : e2 2 C, (resp. e1 2 C) and the product he1iC : : (resp. he2iC) as subgroups of E is a direct sum: ; There is an isomorphism φA : Y (M; N(A)) ! Y (M(A);N); 0 0 0 0 (E; e1; e2;C) 7! (E ; e1; e2;C ): 0 0 Given by letting E be the quotient of E by NC, a cyclic subgroup of order A. Then e1 is defined 0 to be the image of e1 which by the disjointness of he1i and C is necessarily of order M. Define e2 −1 0 to be the image of A e2 in E . There is necessarily such a point as C is cyclic of order NA and 0 0 −1 0 e2 is independent of this choice. Lastly, set C to be the image of A he1i in E . This is a cyclic subgroup of E0 of order MA. 1 Hecke Operators 0 1 The Hecke operators T (n) on K2(Y (M; N)) and their duals T (n) on H (Y (M; N)(C); Z) for n ≥ 1, (n; M) = 1 are defined as follows: • For n = 1, T (1) = T 0(1) = id, • For n = p; p - M let π1 : Y (M; N(p)) ! Y (M; N); π2 : Y (M(p);N) ! Y (M; N) be the −1 ∗ ∗ 0 projections defined by forgetting C. Then set T (p) = (π2)∗ ◦ (φp ) ◦ (π1) and T (p) to be ∗ ∗ (π1)∗ ◦ (φp) ◦ (π2) . • For n = pe; p - M (and e ≥ 2) we set, 8 e >T (p) if p j N < !∗ T (pe) = 1 0 : T (p)T (pe−1) + T (pe−2) p p if p - N :> 0 p For T 0(pe) the formula is identical in T 0. 3 Yukako Kezuka Introduction to Modular Curves Q e(p) • If n = p p , where e(p) ≥ 0 and p ranges over all prime numbers not dividing M, we define T (n) and T 0(n) multiplicatively using the above definitions. Theorem 1.8. If p is a prime p - MN, then Y(M,N) has a smooth model over Zp. Proof. Pick L 2 N such that M; N jL. Since Y (M; N) is a quotient of Y (L) = Y (L; L) it is enough to check that Y (L) has a smooth model over Z[1=L], so if we take K = lcm(M; N) then L is invertible in Zp. The functional criterion for smoothness shows that Y1(L) is smooth over Z[1=L] if and only if for all local Z[1=L]-algebras A and nilpotent ideals I, the map Y (L)(A) ! Y (L)(A0) is surjective, where A0 = A=I.