p-adic regulators and p-adic families of modular forms Relatore: Prof. Massimo Bertolini Coordinatore: Prof. Lambertus van Geemen

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1 UNIVERSITÀ DEGLI STUDI DI MILANO Facoltà di Scienze Matematiche, Fisiche e Naturali Dipartimento di Matematica Federigo Enriques Dottorato di Ricerca in Matematica XXV CICLO p-adic regulators and p-adic families of modular forms MAT/02, MAT/03 Relatore: Prof. Massimo Bertolini Coordinatore: Prof. Lambertus van Geemen Dottorando: Rodolfo Venerucci Anno Accademico 2011/2012

2 p-adic regulators and p-adic families of modular forms Rodolfo Venerucci

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4 Abstract The theme of this Thesis is Iwasawa theory of Hida p-adic analytic families of modular forms. Our main goal is to describe special values of Hida s p-adic L-functions in the context of a p-adic Birch and Swinnerton-Dyer conjecture for the weight variable. Let E/Q be an elliptic curve with ordinary reduction at a prime p, corresponding to a weight two newform f. The exceptional zero formulas of Bertolini-Darmon [BD07] and Greenberg-Stevens [GS93] establish deep relations between the arithmetic of E and the behaviour at k, s = 2, 1 of the Mazur-Kitagawa two-variable p-adic L-function L p f, k, s attached to the Hida family f containing f. The goal of Part 1 is to give an interpretation of these formulas in the framework of a p-adic Birch and Swinnerton-Dyer conjecture for L p f, k, s. The main problem consists in the construction of p-adic regulators encoding the arithmetic of the special L-values of the classical specializations of f. We address this problem by appealing to Nekovář s theory of Selmer complexes [Nek06]. More precisely, the key ingredient in the definition of the p-adic regulator is the p-adic weight pairing, defined on the extended Mordell-Weil group of E/Q. This pairing comes from Nekovář duality for a suitable big Selmer complex attached to Hida s universal ordinary deformation Λf of the p-adic Tate module of E. In Part 2 we consider the algebraic side of the matter, i.e. we study the special values of algebraic Hida s p-adic L-functions. These are defined as characteristic ideals of big Selmer groups or complexes attached to Λf and Z p -power extensions of number fields. Making use of Mazur-Rubin theory of organizing modules and of a general theory of abstract height pairings that we develop in Appendix C, we deduce various several-variable algebraic p-adic Birch and Swinnerton-Dyer formulas, generalizing well known results of Schneider, Perrin-Riou, Jones, Nekovář et. al. Via the Main Conjectures of Iwasawa theory for GL 2, recently proved thanks to the work of Kato-Rohrlich, Bertolini-Darmon, Vatsal, Ochiai, Skinner-Urban et. al., these formulas provide strong evidence in support of the conjectures proposed in Part 1. In Part 3 we study the Mazur-Tate-Teitelbaum conjecture for elliptic curves E/Q with split multiplicative reduction at p i.e. in the presence of an exceptional zero for the cyclotomic p-adic L-function in the sense of [MTT86]. Making use of Nekovář s theory we prove exceptional zero formulas for the p-adic L-functions arising from norm-compatible systems of cohomology classes via Perrin-Riou-Coleman big logarithm. Applying this formulas to Kato s Euler system for the p-adic Tate module of E, we are able to reprove the main result of [GS93], and to relate the second derivative of the cyclotomic p-adic L-function of E to the cyclotomic p-adic regulator. Besides providing evidence in support of the Mazur-Tate-Teitelbaum conjecture, our result suggests an analogue in the multiplicative setting of a conjecture of Perrin-Riou, relating Beilinson-Kato elements to Heegner points. More detailed descriptions of the results are given in the introduction to each part of the Thesis. iii

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6 Acknowledgements It is a pleasure to express my sincere gratitude to my supervisor Prof. Massimo Bertolini. Since my undergraduate studies he has guided me through the exiting field of number theory and has constantly encouraged and motivated my work. Every meeting with him has been a source of ideas and enthusiasm; this Thesis surely originated from and grew up through these meetings. The possibility to work with him has certainly been a great fortune and pleasure. I d like also to thank Prof. Jan Nekovář for his EGA I for Iwasawa Theory [Nek06]. His work has been fundamental for the results presented here. We hope that this Thesis could represent a little new step in the development of his beautiful theory. Besides mathematics: thank you Lissi for your unconditional love and support during these years. Thanks to my family and to the monkeys. v

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8 Contents Abstract Acknowledgements iii v Part 1. p-adic regulators and Hida p-adic L-functions 3 Introduction 4 1. Kummer theory 6 2. Hida theory 8 3. Nekovář duality The p-adic weight pairing The regulator term A p-adic Birch and Swinnerton-Dyer conjecture Results on Conjecture Part 2. Organizing modules for Hida families Introduction Hida theory Selmer complexes in Hida theory Organizing modules and p-adic L-functions Cyclotomic Iwasawa theory 70 Part 3. A note on Kato zeta elements and exceptional zero formulas 83 Introduction The Coleman map Nekovář s extended height An exceptional Rubin s style formula Kato s Euler zeta elements Proofs 103 Appendix A. A short course in Nekovář s theory 105 Appendix B. Iwasawa theory 115 Appendix C. Abstract height pairings 127 Appendix. Bibliography 135 1

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10 Part 1 p-adic regulators and Hida p-adic L-functions

11 Introduction Hida theory of p-adic analytic families of modular forms has proved to be a powerful tool in the study of the arithmetic of p-ordinary elliptic curves. The proof by Greenberg and Stevens of the Mazur-Tate- Teitelbaum exceptional zero conjecture is a well-known example. As another remarkable example, recently Bertolini and Darmon [BD07] proved a p-adic Gross-Zagier formula allowing us to produce in some cases rational points on elliptic curves from derivatives of certain Hida p-adic L-functions. In this note we relate the analytic results of [BD07] to Nekovář algebraic theory of Selmer complexes [Nek06]. This leads us to propose a p-adic Birch and Swinnerton-Dyer conjecture for the weight variable, placing the results of [BD07] in a more general and natural setting cfr. [BD07, pag. 375, Rem. 8]. Let E be an elliptic curve defined over Q of conductor N E = pm, having a prime p 5 of multiplicative reduction, and let f E be the newform of weight 2 on Γ 0 N E attached to E/Q. Hida theory allows us to consider f E = f 2 as the weight 2 element in a p-adic analytic family of modular forms f = {f k } k UE Z 2. Here k runs through the even integers in a p-adic disc U E Z p centered at 2, and f k is the p-stabilisation of a normalized eigenform of weight k and level Γ 1 M. For every k U E Z 2, we denote by α p k := a p f k the p-th Fourier coefficient of f k. Fix a quadratic character χ, of conductor coprime with N E. We denote by L p f, χ, k, s the Mazur- Kitagawa two-variable p-adic L-function, attached to f E, χ and the choice of Shimura periods Ω k C k U E Z 2. It is a C p -valued p-adic analytic function defined on U E Z p, interpolating the critical values of the Hecke L-series Lf k, χ, s of f k twisted by χ see [BD07] or Section 2.2. We consider the restriction L p f, χ, k, k/2 : U E C p of L p f, χ, k, s to the central critical line s = k/2. It satisfies the following interpolation properties: for every k U E 1 L p f, χ, k, k/2 = 1 χpα p k 1 p k/2 1 Lf k, χ, k/2, where = denotes equality up to a non-zero scalar. The Euler factor 1 χpα p k 1 p k/2 1 appearing in 1 is zero precisely when k = 2 and 2 χp = α p 2. In this case L p f, χ, k, s has an exceptional zero at k, s = 2, 1, meaning that L p f, χ, 2, 1 = 0 independently on whether LE/Q, s vanishes or not at s = 1. In this note we are especially interested in this exceptional zero situation, and we assume for the rest of this introduction that 2 is satisfied. The arithmetic of the data f, p, χ strongly depends on the sign signe, χ {±1} appearing in the functional equation satisfied by LE, χ, s := Lf E, χ, s. If signe, χ = +1, then L p f, χ, k, k/2 0 vanishes identically see Section 2.2. This is the situation considered for χ = 1 by Greenberg and Stevens in [GS93]. If 3 signe, χ = 1, a conjecture of Greenberg predicts that L p f, χ, k, k/2 is not identically zero, i.e. that Lf k, χ, k/2 0 for almost all k U E. Assume for the rest of the introduction that 3 is satisfied. The assumptions 2 and 3 imply that L p f, χ, k, k/2 vanishes to order at least 2 at k = 2. For the second derivative, we have the following result. Write K χ /Q for the quadratic field attached to χ resp., K χ := Q, if χ 1 resp., χ = 1. There is a global point P χ EK χ χ and a rational number l Q such that 4 d 2 dk 2 L pf, χ, k, k/2 k=2 = l log E P χ 2, where log E : EQ p G a Q p is the formal group logarithm on E/Q p. The point P χ is a Heegner point, coming from an appropriate Shimura curve parametrisation of E/Q, and it is of infinite order if and only if L E, χ, 1 0. This result has been proved by Bertolini and Darmon in [BD07] assuming an extra hypothesis subsequently removed by Mok in [Mok11].

12 INTRODUCTION 5 As remarked by the authors in [BD07], it would be worthwhile to understand 4 in the framework of a Birch and Swinnerton-Dyer conjecture for the Hida p-adic L-function L p f, χ, k, k/2. This faces us with the problem of constructing a regulator term compatible with 4. The aim of this note is to show that we can indeed construct such a regulator via Nekovář duality for the Selmer complex attached to a suitable Hida big Galois representation, interpolating the Deligne representations of the elements of f see Sec More precisely, Nekovář s construction of abstract Cassels-Tate pairings recalled in Sec. 3 produces a cohomologically-defined p-adic weight pairing, Nek K χ,p : E K χ Q p E K χ Q p Q p, where E K χ is the extended Mordell-Weil group of E/K χ. We can think of, Nek K χ,p as an analogue in this context of the canonical p-adic height considered in [MTT86], [BD96] and [PR92], and as a p-adic variant of the classical Neron-Tate height, with the essential difference that, Nek K χ,p is alternating. Write EK χ χ and E K χ χ for the subgroups of EK χ and E K χ on which GalK χ /Q acts via χ. Under our assumptions 2 and 3 rank Z E K χ χ = rank Z EK χ χ + 1. More precisely, E K χ χ modulo torsion is generated by a basis of EK χ χ modulo torsion and a suitable Tate period q χ E K χ χ see Sec This is the algebraic manifestation of the presence of an exceptional zero for the p-adic L-function. In Section 4.4 see also the proof of Prop. 5.6 we prove the following explicit formula : Theorem 0.1. For every P EK χ χ we have 5 q χ, P Nek K χ,p = c log EP, where c = 1 if χ 1 and c = 1/2 if χ = 1. Using 5, we can rephrase 4 in the following way, emphasizing the analogy with the classical Gross- Zagier formula: there is a scalar l Q such that d 2 2 dk 2 L pf, χ, k, k/2 k=2 = l q χ, P χ Nek K χ,p. This result, combined with Nekovář theory, suggests a close relation between the dominant term in the Taylor expansion of L p f, χ, k, k/2 at k = 2 and the determinant of, Nek K χ,p. This leads us to propose in Sec a p-adic Birch and Swinnerton-Dyer conjecture for the weight variable, in the spirit of the conjectures formulated in [MTT86] and [BD96]. More generally: let E/Q be an elliptic curve of conductor N E ordinary at a prime p 5, and χ a primitive quadratic character of conductor coprime with p N E. The constructions of f and L p f, χ, k, s generalize to this setting and we can consider cfr. Sec. 2.2 the generic part L gen p f, χ, k of the restriction of L p f, χ, k, s to the central critical line s = k/2. This is a p-adic analytic function on U E, which is conjecturally not identically zero. Conjecture 6.1 relates the leading term of L gen p f, χ, k to a p-adic regulator, defined in terms of, Nek K χ,p and the Mazur-Tate-Teitelbaum p-adic cyclotomic height. We finally mention the work of Delbourgo, related to the subject of this note. In [Del08, Ch. 10], a two-variable big Selmer group is attached to the cyclotomic and Hida deformation of the p-adic Tate module of E/Q. Assuming that E/Q p does not have split multiplicative reduction, the leading term of its characteristic power series is expressed in term of a certain p-adic regulator. Moreover, a main conjecture is formulated, relating this power series to the Mazur-Kitagawa p-adic L-function. It is likely that analogues of the results resp., conjectures of [Del08, Ch. 10] can be proved resp., formulated also in the exceptional case, in terms of the cyclotomic deformation of the big Selmer complex H f 2 Q, T P defined in Sec. 3 and the p-adic regulator of Sec. 5. This Iwasawa theoretic point of view may also serve as a motivation for Conjecture 6.1, in the same way as the main conjecture of Iwasawa theory [Gre94b], together with

13 6 the algebraic p-adic BSD formulas of Schneider et al. see for example [BD95] motivate the p-adic Birch and Swinnerton-Dyer conjectures of [MTT86] and [BD96]. Notations. The following notations will be used throughout this note: - E/Q is an elliptic curve defined over Q of conductor N E ; - f E = n 1 a ne q n S 2 Γ 0 N E, Z is the newform attached to E/Q by the modularity theorem; - p 5 is a rational prime of ordinary i.e. good ordinary or multiplicative reduction for E; - K/Q is a number field of discriminant d K ; we write D K = d K ; - S f {v p N E D K } is a finite set of finite primes of K; - K = Q resp. K v = Q l, S f v l is a fixed algebraic closure of Q resp. of the completion K v of K at v S f ; - G K,S := GalK S /K is the Galois group of the maximal algebraic extension K S K of K which is unramified outside S f {v }; - ρ v : K K v for v S f is a fixed embedding which extends K K v ; - ρ v : G v := GalK v /K v G K or G Ql G Q if v l is the morphism attached to ρ v ; - if M is a Z[G K,S ]-module, we write M v for the Z[G v ]-module M, on which G v acts via ρ v; - if L is a field, we write T p L := lim µ p nl; - if M is a Z p [GalL/L]-module, M1 := M L 1 := M Zp T p L with diagonal GalL/L- action. 1. Kummer theory In this section we recall some results from Kummer theory. Given a profinite group G and a finite dimensional Q p -vector space M, endowed with a continuous Q p -linear action of G for the p-adic topology on M, H L, M = H C contg, M denotes the continuous cohomology group of G L with values in M, as defined in [Tat76] or [Jan88]. If G L := GalL/L is the absolute Galois group of a field L, we use the notation H L, M for H G L, M The multiplicative group. In this paragraph L/Q p is a local field and π L is a uniformizer in the maximal order O L of L. By Hilbert Satz 90, the connecting morphism attached to the short exact sequence of discrete G L - modules 0 µ p n L pn L 0, defines an isomorphism L /L pn H 1 L, µ p n. Taking the inverse limit n and extending scalars to Q p, we obtain the Kummer isomorphism L Q p H 1 L, Q p 1. Given x L we write γx L or simply γ x if L is fixed for the image of x 1 under the Kummer map. By local Tate duality, we have a perfect duality, L := inv L x y : H 1 L, Q p 1 H 1 L, Q p H 2 L, Q p 1 Q p, where is induced by multiplication Q p 1 Q p Q p 1 and inv L is the invariant map of local class field theory. Note that H 1 L, Q p = Hom cts G ab L, Q p G ab := G/[G : G] for the closure [G : G] of the commutator subgroup of G. Recall also the reciprocity map [Ser67]: rec L : L G ab L, normalized in such a way that rec L π L 1 Fr L is an arithmetic Frobenius in G ab L. We write also rec p := rec Qp. Proposition 1.1. a For every q L and χ H 1 L, Q p we have γ q, χ L = χ rec L q. b Let χ cy : G ab Q p Z p be the p-adic cyclotomic character. For every q = p ordpq u Q p χ cy rec p q = u Z p.

14 1. KUMMER THEORY 7 Proof. a Follows by [Ser67, Sec. 2.3] see also [Nek06, Sec ]. b Recalling our normalization of rec p, this follows by [Ser67, Sec. 3.1] Elliptic curves. Let us consider the elliptic curve E/Q fixed above. Denote by Ta p E := lim EQ[pn ] the Tate module of E/Q and by V p E := Ta p E Zp Q p. As S f contains every prime of bad reduction for E/Q, Ta p E is a continuous G K,S -module [Sil86, Ch. VII]. For every v S f, the embedding ρ v induces an isomorphism of G v -modules again denoted ρ v ρ v : EQ[p n ] EK v v [p n ]. Taking limits we obtain an isomorphism of Q p [G v ]-modules ρ v : V p E v Vv E := lim EK v[p n ] Q p. When the context makes it clear, we write simply V p E for the G v -module V p E v. Recall the injective global Kummer map EK Q p = lim EK/pn EK Q lim κn p lim H1 G K,S, E[p n ] Q p H 1 G K,S, V p E, where κ n is the usual Kummer map on EK/p n ant the last isomorphism is obtained extending scalars to Q p from H 1 G K,S, Ta p E lim H 1 G K,S, Ta p E/p n see for example [Nek06, Lemma 4.2.2] or [Jan88]. For every P EK we write γ P for the image of P 1 under this map. Replacing G K,S by G v v S f we obtain also a local Kummer map EK v Q p H 1 K v, V v E. Given P v EK v, we write again γ Pv for the image of P v 1 and we consider EK v Q as a subspace of H 1 K v, V v E. We have res v γ P = ρ 1 v γρvp : here we have written by abuse of notation again ρ 1 v for the map induced in cohomology by the isomorphism ρ 1 v : V v E V p E v and res v : H 1 G K,S, V p E H 1 K v, V p E := H 1 G v, V p E v for the restriction map induced in cohomology by the morphism of pairs ρ v, id. When there is no risk of confusion, we identify V p E v with V v E and res v with ρ v res v. Furthermore, we consider EK v Q p also as a submodule of H 1 K v, V p E under ρ 1 v : H 1 K v, V v E H 1 K v, V p E. Writing Sel Qp E; K H 1 G K,S, V p E for the Selmer group defined by the local conditions EK v Q v for v S f, we have a short exact sequence 6 0 EK Q p Sel Qp E; K Ta p XE/K Q p 0, where XE/K is the Tate-Shafarevich group of E/K. As shown by R. Greenberg, we can also describe this Selmer group in terms of the ordinary filtration on the Galois representation Ta p E, in the following way. If E/Q has good ordinary reduction we have a short exact sequence of Q p [G Qp ]-modules 7 0 V p E + i+ v p V p E v v V p E 0, with dim Qp V p E ± = 1. Here V p E := Ta p E p Q p resp., V p E + := Ta p Ê Q p is the p-adic Tate module of the reduction E p /F p of E/Q p resp., of the formal group Ê of E/Q p [Sil86, Ch. VII] with Q p -coefficients. The map p v is induced by ρ v and the reduction map EQ p E p F p. In particular V p E is unramified at p. If E/Q p has multiplicative reduction, Tate s p-adic analytic uniformisation gives us a group isomorphism 8 Φ T ate : Q p /q Z E EQp, where q E pz p is Tate p-adic period of E/Q p [Sil94, Ch. V]. We have Φ T ate x g = χg Φ T ate x g for every g G Qp, where χ : G Qp {±1} is the quadratic unramified character resp., the trivial character if E/Q p has non-split resp., split multiplicative reduction [Sil94, Ch. V]. As q E pz p, we have an exact sequence of G Qp -modules 0 µ p nχ Φ T ate EQ p [p n ] P T ate Z/p n Z χ 0. Taking the inverse limit on n and extending scalars to Q p, we obtain the fundamental exact sequence of Q p [G Qp ]-modules 9 0 Q p1 V i+ v p p E v v Q p 0.

15 8 Here Q p := Q p χ and i + v resp. p v is induced by the limit of ρ 1 v Φ T ate resp. P T ate ρ v. We will write V p E + := Q p1 and V p E := Q p, so that we obtain 7 also in this case. Define, for every v p, Hf 1K v, V p E H 1 K v, V p E as the image of H 1 K v, V p E + under the map induced in cohomology by i + v, and put Hf 1K v, V p E = 0 for S f v p. These local conditions define a Selmer group Hf 1K, V pe H 1 G K,S, V p E independent on the choice of S f {v p N E }. The following Lemma is proved in [Gre97, Sec. 2] see also [Nek06, Lemma ]. Lemma 1.2. EK v Q p = H 1 f K v, V p E for every v S f. In particular Sel Qp E; K = H 1 f K, V pe. Remark 1.3. Suppose that E/K v has split multiplicative reduction, i.e. Q p = Q p as G v -modules: a if Φ T ate P = P EK v, for a P Kv, then Φ T ate γ ep = γ P we write again Φ T ate : Q p 1 V v E for the map induced by Φ T ate. This follows from the definitions and proves the first assertion of the preceding Lemma in this case. b Let v : Q p H 1 K v, Q p 1 be the connecting morphism attached to 9. A short inspections reveals that v 1 = γ qe := γq Kv E. In other words, under the identifications of elements in H 1 K v, Q p 1 with continuous extensions classes in Ext 1 Q p[g v]q p, Q p 1, γ qe corresponds to the class of Products. Let v S f be a prime which divide p and let, W : Ta p E Ta p E T p Q resp.,, v : T p E T p E T p Q p be the Weil pairing on Ta p E resp., on T p E := lim EQ p [p n ]. We use the opposite sign convetion to that of [Sil86], so that our Weil pairing is minus that defined in [Sil86, Ch. III]. It is a perfect, alternating and G K,S -equivariant resp., G Qp -equivariant Z p -bilinear form. If E/Q p has multiplicative reduction and writing again ρ v : T p Q v Tp Q p for the isomorphism of Z p [G v ]-modules induced by ρ v, we have 10 ρ v x, y W = ρ v x, ρ v y v ; Φ T ate α, β v = α P T ate β for every x, y Ta p E, α T p Q p and β T p E here is multiplication, once we identify T p Q p with Z p as Z p -modules. The first equality follows from the definition of the Weil pairing, while the second can be proved using the description of principal divisors on E qe := Q p /q Z E in terms of p-adic theta functions see for example [Tat95]. Write W : C contg v, V p E v C contg v, V p E v C contg v, T p Q v Q p for the cup-product induced on cochains by, W. If E/Q p is multiplicative, it follows by 10 that 11 ρ v y W i + v x = p v y x C 2 contk v, Q p 1 for every x C 1 contk v, V p E + and y C 1 contk v, V p E. In 11 is the cup-product pairing induced by multiplication V p E + V p E Q p 1 and we have written again ρ v for the isomorphism induced on cochains by ρ v Q p. 2. Hida theory In this section we recall some fundamental results of Hida Theory. We use [NP00], [Nek06, Sec. 12.7] and [BD07] as main references Hida families. Let N := N E /p ordpn E be the tame conductor of E and fix an embedding ρ p : Q Q p. Writing ψ for the trivial character modulo N E, let X 2 a p EX + ψpp = X α p X β p, with α p, β p Q. Since E is ordinary at p, we have α p, β p Z p under ρ p and we can assume α p Z p and β p pz p. We define the p-stabilization f 0 E S 2Γ 0 Np, Z p of f E by 12 f 0 Ez := f E z + β p f E pz.

16 2. HIDA THEORY 9 In particular f 0 E = f E if E/Q p has multiplicative reduction. As follows by [Hid85, Lemma 3.3], f 0 E is the unique normalized eigenform on Γ 0 Np such that a l f 0 E = a le for every prime l p. Moreover a p f 0 E = α p Z p. Consider Hida universal ordinary Hecke algebra of tame conductor N h ord = h ord N := lim Here h ord 2,r := e ord hγ 1 Np r Z Z p, where hγ 1 Np r End Z S 2 Γ 1 Np r, Z is the algebra generated by the Hecke operators T l, for every prime l and the diamond operator a, for every a Z/Np r Z, and e ord := lim n Tp n! is Hida s ordinary projector. We will write also U p for T p. We have a morphism r : Z p [Z/Np r Z ] h ord 2,r. Putting Γ := 1 + pz p and taking the inverse limit r, we obtain the diamond morphism with the normalization of [NP00, 1.4] r 1 h ord 2,r. : Λ := Z p [[Γ]] Z p [[Z N,p]] h ord, where Z N,p := Z p Z/NZ. gives h ord a structure of Λ-algebra. By [Hid86a], h ord is a free Λ-module of finite rank. It follows that h ord = m j h ord,m j decomposes as a finite direct sum of its completions at maximal ideals m j. As α p Z p, f E or better fe 0 gives rise to a morphism of Z p-algebras 13 η fe : h ord defined sending Z N,p to 1 and T l to a l fe 0 for every prime l. η f E factorizes through h ord h ord,m for a unique maximal ideal m = m j. Write P := Kerη fe Spech ord,m: by [Hid86a, Cor. 1.4] see also [Nek06, ] the localization of h ord,m at P is a discrete valuation ring, unramified over Λ p, where p = γ 1 for a topological generator γ of Γ. Then h ord,m contains a unique minimal prime P min s.t. η fe factorizes through the local domain R = R E := h ord,m/p min. We will write from now on R = R E and P := P/P min SpecR. The localization R P is a discrete valuation ring, unramified over Λ p. Fix a topological generator γ of Γ e.g. γ = 1 + p Γ and the corresponding uniformizer of R P ϖ := γ 1 R P. We write again T l and U p for the image of the Hecke operators in R. As η E takes values in Z p i.e. E is defined over Q the residue field FracR/P = R P /ϖr P of R P is identified with Q p. With the terminology of [Hid86a], R := F racr is the primitive local component to which f E belongs. h ord,m is the Hida family attached to f E and R is the branch of the Hida family in which f E lives. This terminology is justified by the following analytic interpretations of the results above, given in [GS93] see also the next section. Let A Q p [[w 2]] be the ring of formal power series in w 2 converging for w in some p-adic neighborhood of 2. The ring A is endowed with a structure of Λ-algebra, defined as follows: let f X be the isomorphism Λ Z p [[X]] determined by f γ X = X +1. We associate to Λ the analytic function on Z p given by w f γ w 2 1. Since A is Henselian and since the augmentation ideal γ 1 Λ is unramified in R P, there exists a unique morphism of Λ-algebras Z p 14 η f : R P A such that η f r w=2 = η fe r for every r R. Define, for every positive integer n, α n w := η f T n A, where T n is the n-th Hecke operator, defined in terms of the T l s by the usual relations [Shi71, Ch. III]. As R is finite over Λ, there exists a p-adic neighborhood 2 U such that α n w A U for every n N, where A U A is the ring of analytic functions on U. Consider the formal q-expansion f := n 1 α n w q n A U [[q]]. For every even integer k U Z 2, the weight k-specialization f k := n 1 α nk q n is the q-expansion of a normalized eigenform on Γ 1 Np and f 2 = f 0 E. If k 2 mod p 1, then f k has trivial character, i.e. it is a normalized eigenform on Γ 0 Np. As follows by [Hid86a, Cor. 1.3], f k is new at the primes dividing

17 10 the tame level N and is not p-new for k > 2. More precisely, for every k > 2 such that k 2 mod p 1 there exists a unique newform f # k on Γ 0N such that a l f # k = a lf k for every prime l p. With the terminology used above, f k = f # k 0 is the p-stabilization of f # k and they satisfy a relations analogous to 12 for k = 2. Let Z := {z Z 2 : z 2 mod p 1} and f # 2 := f E. We call {f # k } k U Z the Hida family attached to E/Q p-adic L-functions. For more details on the results and constructions recalled in this section, we refer the reader to [BD07, Sec. 1]. Let χ : Z/mZ {±1} be a quadratic Dirichlet character, of conductor m coprime with p, with χ 1 =: w, and let τ χ = m j=1 χje2πij/m be the associated Gauss sum. For every k U Z, the complex L-function Lf # k, χ, s := n 1 χna nf # k n s defined for Res > k + 1/2 extends to an entire function on C. Recall our convention: f E = f # 2. For every integer 1 j k 1, we define the algebraic part of Lf # k, χ, j by 15 L f # k, χ, j := j 1!τ χ 2πi j 1 Lf # k, χ, j. Ω k Here we fix, for every k U Z, Shimura periods Ω ± C as in [BD07, Prop. 1.1], [Shi77a, Sec. 1], f # k and we write Ω k := Ω signw. If χ 1 = 1 j 1 w f #, 15 belongs to the field generated by the Fourier k coefficient of f # k, and we consider it as an element of Q p C p under the embedding ρ p : Q Q p fixed in the preceding section, where C p is the completion of Q p. Remark 2.1. In [BD07], the periods are chosen in such a way that Ω + f # k Ω f # k = f # k, f # k is the Petersson scalar product of f # k with itself. We do not impose this normalization here, as we will fix Ω 2 later in a convenient way cfr. Sec Up to shrinking U if necessary, Sec. 1 of [BD07] constructs a C p -valued function L p f, χ, k, s : U Z p C p which interpolates the Mazur-Tate-Teitelbaum p-adic L-functions L p f # k, χ, s attached in [MTT86] to the elements of f and the periods Ω k. More precisely L p f, χ, k, s satisfies the following properties: 1. L p f, χ, k, s is locally analytic on each variable; 2. for every k U Z and every odd integer 1 j k 1, there exists a scalar λk C p such that 16 L p f, χ, k, j = λk1 χpα p k 1 p j 1 1 χpα p k 1 p k j 1 ɛk L f # k, χ, j, where ɛ k = 1 if f k f # k 3. λ2 = 1. and ɛ k = 0 otherwise; Remark 2.2. Using the terminology of [BD07], L p f, χ, k, s is determined by the choice of an ordinary, Γ 0 N-equivariant modular symbol µ with values in the space of measures on Z 2 p the set of primitive vectors in Z p Z p, interpolating the classical modular symbol attached to f k in weight k see [BD07, Sec. 1.3]. The existence of such a modular symbol follows from [GS93, Th. 5.13], and the scalars {λk} come from the interpolation process [BD07, Th. 1.5]. Once we have fixed the periods {Ω k } as above depending on χ 1 = w, µ, and then L p f, χ, k, s, is unique up to multiplication by a nowhere vanishing analytic function α on U, satisfying α2 = 1. Here we fix such a µ and call L p f, χ, k, s the Mazur-Kitagawa p-adic L-function attached to χ. Note that taking j = k/2 in 16 we obtain for k 2 mod 2p 1 17 L p f, χ, k, k/2 = λk1 χpα p k 1 p k/2 1 β L f # k, χ, k/2,

18 2. HIDA THEORY 11 where β = 2 if f k f # k and β = 1 otherwise i.e. if k = 2 and E/Q p is multiplicative. This shows that, if χp = α p, L p f, χ, k, s has an exceptional zero at k, s = 2, 1, i.e. L p f, χ, 2, 1 = 0 independently on whether LE/Q, s vanishes or not at s = 1. Write w k for the sign in the functional equation satisfied by the Hecke L-series Lf # k, s, for k U Z. It is known [NP00, Sec ] that w k =: w gen is constant for every k > 2, and that w gen if p N E ; w 2 = signe, Q = α p w gen if p N E. Recalling that f # k constant sign k > 2 is a newform on Γ 0N, by [Shi71, Th. 3.66] we see that Lf # k, χ, s has 18 signf, χ := χ N w gen in its functional equation, for every k > 2. Note that signf, χ is opposite to the sign see again loc. cit. signe, χ = χ N E w 2 of Lf E, χ, s if and only if χp = α p, i.e. if and only if L p f, χ, k, s has an exceptional zero. Moreover this happens precisely when the twist E χ /Q of E/Q has split multiplicative reduction at p [MTT86]. As follows by 18 and the interpolation formula 17, L p f, χ, k, k/2 0 vanishes identically if signf, χ = 1. We define the generic part of the restriction of L p f, χ, k, s to the central critical line by L gen p f, χ, k := L pf, χ, k, s s k/2 egenχ ; e gen χ := s=k/2 0 if signf, χ = +1; 1 if signf, χ = 1. It is a C p -valued, p-adic analytic function on U. The terminology is justified by Greenberg conjecture, predicting that L gen p f, χ, k is not identically zero see Sec. 7 for results in this direction. When χ = χ triv is the trivial character, we write simply L p f, k, s, L gen p f, k = L gen p f /Q, k and e gen for the objects attached to χ triv. Let K be a quadratic field such that D K, p = 1 and let ɛ K : Z/D K Z {±} be the associated quadratic character. Putting e gen K := e gen + e gen ɛ K, we define the Hida p-adic L-function of E/K by p f /K, k := L pf, k, s L p f, ɛ K, k, s s k/2 egenk = L gen p f, k L gen p f, ɛ K, k. s=k/2 L gen 2.3. Big Galois representations. Let Q Np Q be the maximal algebraic extension of Q which is unramified outside p N, and let G := GalQ Np /Q. In this section we recall briefly how we can construct a self-dual big Galois representation of G which interpolates V p E in weight two and, more generally, a suitable self-dual twist of the Deligne representation of f # k in weight k U Z. For more details and references, see [Nek06, Sec. 12.7] or [NP00]. As explained in [Nek06, Sec ] there exists a continuous R P [G]-module T R P, free of rank two over R P, such that: for every prime l Np 19 trace Frl T R P = T l ; det Frl T R P = l l, where Frl G Q is an arithmetic Frobenius and : Z p [[Z N,p ]] hord R is the diamond morphism. The term continuous means that T R P ad R[G] Mod is an admissible R[G]-modulo, as defined in [Nek06, Sec. 3.2] see also the next section. The representation T R P can be constructed as follows [Hid86a],[NP00]. Let X r := X 1 Np r /Q be the modular curve over Q as defined, for example, in [Roh97] and J r := Pic 0 X r. The Hecke algebra hγ 1 Np r acts on J r via algebraic correspondences and this action commutes with that of G Q. Let π 1 : X r+1 X r be the morphism defined by the inclusion Γ 1 Np r+1 Γ 1 Np r and π 1 : J r Q p J r+1 Q p be the map induced by contravariant functoriality. Write J :=

19 12 lim J π 1 rq p and J ord = e ord J for its ordinary part. By a fundamental theorem of Hida, J ord is an h ord [G Q ]-module whose Pontrjagin dual is free of finite rank over Λ. Define with the notations of Sec. 2.1 Ta ord := Hom Zp J ord, µ p h ord hord,m; Ta R := Ta ord h ord,m R. With these notations, T R P := Ta R R R P is the localization of Ta R at P. As J ord is unramified at every rational place l Np, the same is true for T R P i.e. it is a R P [G]-module. The identity 19 is a manifestation of the Eichler-Shimura congruence relation [Roh97, page 72]. To obtain a self-dual representation, we consider a suitable twist of T R P. More precisely define the character Ψ Q = χ cy 1/2 : G Q R as follows: let χ cy : G Q GalQµ p /Q Z p be the p-adic cyclotomic character and κ : Z p 1 + pz p the projection on principal units. For every g G Q we put Ψ Q g := κ χ cy g 1/2 as p 2 every element of Γ = 1+pZ p has a unique square root in Γ. Note that, writing χ cy,n : G Q GalQµ Np /Q Z N,p, Ψ Qg 2 = χ cy,n g. This follows by the fact that f E or fe 0 has trivial character, i.e. the character of R is trivial with the terminology of [Hid86a]. We can finally define T R := Ta R R Ψ Q ; T P := T R R R P. As Ψ Q 1 mod ϖ and V p E is irreducible, the Eichler-Shimura relation 19, combined with the Chebotarev density theorem and the definition of P as the kernel of 13 gives us an isomorphism of Q p [G]-modules 20 T P k=2 := T P R P /ϖ V p E. Furthermore, again by 19 and the discussion above the determinant of T P is the p-adic cyclotomic character. In [NP00, Sec. 1.6] it is shown how these properties imply the existence of an R P -bilinear, alternating and G-equivariant form 21 π := π RP : T P RP T P R P 1 := R P T p Q which induces an isomorphism of R P [G]-modules adjπ : T P Hom RP T P, R P 1 =: T P 1. As remarked in [Nek06, Section ], the geometric construction of 21 given in [NP00] was done earlier by Ohta; see the reference given in loc. cit. Write mod ϖ for the compositions T P T P k=2 V p E and R P R P /ϖ Q p. Multiplying π by a unit in R if necessary, we can assume, as we do from P now on, that 22 πx y mod ϖ = x mod ϖ, y mod ϖ W for every x, y T P. This follows by the facts that π and, W are perfect and alternating Ramification at p. Fix a prime v S f which divide p. Recall our fixed embedding ρ v : Q Q p, and let I Qp G Qp G Q be the corresponding inertia and decomposition groups. As described in [Nek06, Sec ], [NP00] or [MT90], the restriction T P v of T P to G Qp is reducible. More precisely: write Ψ = Ψ Qp for the restriction of Ψ Q to G Qp. There exists a short exact sequence of R P [G Qp ]-modules 23 0 T P + i+ v p T P v v T P 0, with T P ± free of rank one over R P. Furthermore G Qp acts on T P + resp., T P via the character χ cy Ψ 1 resp., Ψ φ R, where φ 1 R φ R : G Qp G Qp /I Qp R

20 3. NEKOVÁŘ DUALITY 13 is the unramified character which sends an arithmetic Frobenius Frp at p to the p-th Hecke operator U p. In other words, if we fix a splitting of R P -modules T P T P + T P R 2, the action of G P Q p on T P v is described by the matrix χ cy Ψ 1 φ 1 R : G Qp GL 2 R P. 0 Ψ φ R Putting T P ± k=2 := T P ± R P /ϖ, 20 extends to an isomorphism of short exact sequences of Q p [G Qp ]-modules 24 0 T P + k=2 T P k=2 v T P k=2 0 V p E + V p E v V p E 0, where the bottom row is the exact sequence 7 or 9. Write T P 1 ± := Hom RP T P, R P 1. As π is alternating, adjπ induces an isomorphism of short exact sequences of R P [G Qp ]-modules 25 0 T P + T P v adjπ T P 0 T P 1 + T P v1 T P 1 0. Using the notations of [Nek06, 6.8], this means that T P + π T P +, after replacing R P with its dualizing complex ω RP := [F racr F racr/r P ] in 21. If w S f is another prime dividing p, then there exists σ w G Q and α w G Qp s.t. ρ w = α w ρ v σ w. Putting i + w := σ 1 w i + v α 1 w and p w := α w p v σ w, we obtain an exact sequence of R P [G Qp ]-modules 0 0 and analogues of 24 and T P + i+ w T P w p w T P 0 3. Nekovář duality In this section we introduce Nekovář s Selmer complexes attached to the big ordinary representation T P, and the abstract Cassels-Tate pairing in this setting. Every notation or sign convention regarding complexes which is not explicitly defined is as in [Nek06, Ch. 1] Selmer complexes. Let T P be the big Galois representation considered in the preceding section. T P ad R[G K,S ] Mod is an admissible R[G K,S]-module as defined in [Nek06, Sec. 3.2] and we can define, for G {G K,S ; G v, v S f }, the complex [Nek06, Def ] C contg, T P := lim C contg, T α ; T α ST P here T α ST P if T α T P is an R[G]-submodule such that a T α is a finitely generated R-module and b the action of G is continuous for the profinite topology on G and the m R -adic topology on T α m R is the maximal ideal of the local ring R = R E. For T α ST P, C contg, T α = lim C contg, T α /m n R T α is the usual complex defined in degree n by the set C n contg, T α of continuous maps G n T α. To be precise: if v S f, then T P v ad R[G v] Mod is an admissible R[G v]-module and C contg v, T P := C contg v, T P v. We write also C contk v, T P for C contg v, T P. As T P = T R R P and T R is finite over R, it follows from [Nek06, Prop ] that the natural morphism of complexes C contg, T R R R P C cont G, T P is an isomorphism and C contg, T R has the usual meaning.

21 14 We have, for every v S f, a natural restriction map res v : C contg K,S, T P C contk v, T P induced by the morphism of pairs ρ v, id. By the results of the preceding section, we also have the admissible R[G Qp ]-module T P ± and we define as above the complex C contk v, T P ± := C contg v, T P ±, for every v p. In the same way we can consider the continuous Q p [G K,S ]-module V p E and the Q p [G Qp ]-modules V p E ±. In this case C contg, V p E C contg, Ta p E Q p and C contk v, V p E ± for v p are the usual complexes of continuous cochains for the p-adic topology. Let X {T P, V p E}. We define, as in [Nek06, ], [NP00], local conditions for v S f by C contk v, X + if v p; U v + X := 0 if v p and the corresponding Nekovář Selmer complex C f G K,S, X := Cone C contg K,S, X U v + X res S f i + S f v Sf C contk v, X [ 1]. v S f Here res Sf := v S f res v, i + S f := v S f i + v and i + v : U v + X C contk v, X by abuse of notation is the map induced by the inclusion of G v -modules i + v : X + X v i.e. zero if v p. Write R X = R P resp. Q p for X = T P resp. X = V p E, RΓ f G K,S, X for the image of C f G K,S, X in the derived category DR X := D RX Mod of complexes of R X -modules and H f G K,S, X := H C f G K,S, X for the cohomology of RΓ f G K,S, X. We collect in the following propositions some important facts we will use below. Proposition 3.1. a RΓ f G K,S, X D b ft R X i.e. has bounded cohomology of finite type over R X. Furthermore it is independent up to isomorphism on the choice of the finite set S f. We write RΓ f K, X = RΓ f G K,S, X and H f K, X := H f G K,S, X. b there exists an exact triangle in D b R P inducing short exact sequences RΓ f K, T P ϖ RΓ f K, T P RΓ f K, V p E 0 H q f K, T P/ϖ H q f K, V pe i P H q+1 f K, T P[ϖ] 0. c H 1 f K, T P is a free R P -module. Proof. All these statements are special cases of [Nek06, Prop ]. For future reference, we recall how to prove b. Let G {G K,S, G v, v S f } and =, +. Combining the exact sequences when defined of complexes of R P -modules [Nek06, Prop ] 26 0 C contg, T P ϖ C contg, T P C contg, V p E 0, associated to the specialization maps T P T P w=2 V p E, we obtain a short exact sequence 27 0 C f G K,S, T P ϖ C f G K,S, T P C f G K,S, V p E 0, which defines the exact triangle above. i P is then the connecting morphism attached to 27.

22 3. NEKOVÁŘ DUALITY 15 Write H G, := H C contg, and C contk v, X := C contk v, X for S f v p. Noting that Cone C contk v, X + i+ v C contk v, X C contk v, X in the derived category, we obtain an exact triangle in D b R X v S f C contk v, X [ 1] RΓ f K, X C contg K,S, X. We note that H 0 K v, X = 0 unless v p, X = V p E and E/K v has split multiplicative reduction. For X = V p E this follows easily by the discussion in Sec The result for X = T P follows easily from this and Sec We then obtain in cohomology a short exact sequence of R X -modules 28 0 H 0 K v, X H f 1 K, X Hf 1 K, X 0, v S sp f where S sp f := {v p : E/K v has split multiplicative reduction} and Hf 1K, X H1 G K,S, X is the Selmer group attached to the local conditions i + v H 1 U v + X H 1 K v, X. Specializing 28 to X = V p E, we obtain by Lemma 1.2 an exact sequence 29 0 v S sp f Q p ι H 1 f K, V p E Sel Qp E; K Class field theory. Let M be an R-module, considered as an admissible R[G K,S ]-module with trivial G K,S -action. Define K M := Cone τ 2 C contg K,S, M1 res S f τ 2 C contk v, M1 [ 1], v S f where M1 := M Zp T p Q and τ 2 X is the good filtration of X in degree two [Nek06, page 33]. Note that, for v S f, C contk v, M1 := C contg v, M1 v. By class field theory [Nek06, Sec ] H q K M = 0 for every q 3 and the sum of the invariant maps of local class field theory induces an isomorphism of M-modules inv Sf M : H 3 K M M, which is functorial in M. We can describe inv Sf explicitly as follows. First of all, we have for every v S f an isomorphism inv v M : H 2 K v, M1 M, obtained as the composition H 2 G v, M1 v H 2 G v, M Zp T p Kv M. Here the first isomorphism is induced by id ρ v, and the second is defined taking limits by the invariant map of local class field theory as in [Nek06, 5.2]. Let x = x, y v KM 3 be a 3-cocycle, for x C3 contg K,S, M1 and y v v S f CcontK 2 v, M1 to be precise we should write [y v ] for the second component in x, where [y v ] denotes the class of y v modulo the image of δ : CcontK 1 v, M1 CcontK 2 v, M1. Since x is a cocycle we have 0 = d KM x = δx, δy v + res v x, where δ is the differential in C cont,. As H 3 G K,S, M1 = 0 [Mil04, Ch. I], there exists ϑ C 2 contg K,S, M1 such that δϑ = x, so [x] = [0, y v + res v ϑ v ] H 3 K M. We have inv Sf M[x] = v S f inv v M [y v + res v ϑ]. The facts that this expression does not depend on the choice of ϑ and that inv Sf is an isomorphism is essentially a restatement of the fundamental exact sequence of global class field theory for more details, see [Nek06, Ch. 5], in particular the exact sequence

23 16 We can consider Q p as an R-module, identifying it with the residue field R P /ϖ of R P see Sec By [Nek06, Prop ], C contg, Q p 1 is then identified with C contg, R P /ϖ1. By the functoriality of inv Sf we have 30 inv Sf R P x mod ϖ = inv Sf Q p x mod ϖ for every x K RP and mod ϖ : R P R P /ϖ Q p. We write from now on K for K RP Products. The morphism π : T P RP T P R P 1 induces, for G {G K,S, G v }, truncated cup-products π : C contg, T P RP C contg, T P C contg, R P 1 τ 2 τ 2 C contg, R P 1. The first map is the composition of the cup product C contg, T P C contg, T P C contg, T P T P defined by the usual formulas on cochains [Nek06, Sec ] with the map induced by π. When the context is clear, we write π also for the usual non-truncated cup-product. We will write C f T P := C f G K,S, T P and x n, x + n, x n 1 C f n G K,S, T P for an n-cochain, where x n CcontG n K,S, T P, x + n = x + n,v v p v p Cn contk v, T P + and x n 1 = x n 1,v v Sf v S f Ccont n 1 K v, T P. Given α = α v v Sf, β = β v v Sf v S f C contk v, T P, we write 31 α π β := v Sf α v π β v. Let r, s R. A simple direct computation [Nek06, Prop ] shows that the formula x n, x + n, x n 1 π,r y m, y m, + y m 1 := x n π y m, x n 1 π r res Sf y m + 1 r i + S f y m n 1 r res Sf x n + r i + S f x + n π y m 1 defines a morphism of complexes of R P -modules π,r : C f T P RP C f T P K. Moreover the formula k r,s x n, x + n, x n 1 y m, y + m, y m 1 = 0, 1 n r s x n 1 π y m 1 defines a homotopy k r,s : π,r π,s Generalized Cassels-Tate pairings. Define R := FracR P and R P := [R P i R], concentrated in degrees [0, 1]. The morphism v RP i, i : R P RP R P = [R P R R id,id R] R P defined by the identity resp. the projection on the first factor in degree zero resp. one is a quasiisomorphism. Write C f T P := C f G K,S, T P and let r R. We define a morphism of complexes 32 C f T P R RP P RP C f T P RP R P K RP R P. by the composition C f T P R s23 RP P RP C f T P RP R P C f T P RP id v RP C f T P RP C f T P RP R P π,r id K RP R P, C f T P RP RP RP R P with π,r as in preceding section and s 23 a b c d := 1 degbdegc a c b d. cup-product 32 induces in cohomology a morphism of R P -modules 33 π,2,2 : H 2 C f T P RP R P RP H 2 C f T P RP R P H 4 K RP R P, which is independent on the choice of r R. The

24 3. NEKOVÁŘ DUALITY 17 Let Z be a complex of R P -modules with cohomology of finite type over R P. The cohomology sequence of the exact triangle in DR P splits into short exact sequences of R P -modules 34 Z i Z RP R Z RP R P [1] 0 H q 1 Z RP R/R P H q Z RP R P H q Z RP T or 0, where M RP T or := Ker M M RP R. Taking Z = K and q = 4 we obtain from Section 0.4 an isomorphism 35 H 4 K RP R P H 3 K RP R/R P R/RP, where the last map is given by inv Sf id. Note that every term in 34 is a torsion R P -module and the first is P-divisible. Taking Z = C f T P and q = 2, it follows that the cup product in 33 factorizes through the projection H 2 Z RP R P H 2 Z RP T or. Composing π,2,2 with 35 we then obtain an R P -bilinear form 36 CT : H 2 f K, T P RP T or H 2 f K, T P RP T or R/R P. We have the following fundamental: Theorem 3.2. CT is non-degenerate and alternating. Proof. This is a special case of [Nek06, Prop ] or [Nek06, Th ] Behaviour under Galois conjugation. We assume in this paragraph that K/Q is a Galois extension. Let Sf 0 be the set of rational primes dividing p N E D K. Fix for every prime l Sf 0 an embedding ρ l : Q Q l, inducing ρ l : G Ql G Q. We also fix elements σ j,l G Q σ 1,l := id which represent the coset space G K \G Q /ρ l G Q l, and assume that {ρ l σ 1 j } j = {ρ v } v l where ρ v is the embedding fixed at the beginning of this note. As described in [Nek06, 8.8], GalK/Q acts on H q f K, X, for X {T P, V pe} and q 0. More precisely: for every g G Q, we can define a morphism of complexes Ad f g := Adg, Ad + g, F g; m g : C f G K,S, X C f G K,S, X as follows. First of all, Adg : C contg K,S, X C contg K,S, X denotes the usual action of g by Galois conjugation, i.e. the map induced by the morphism of pairs x gx x X, σ g 1 σg σ G K between G K, X and itself. In a similar way, Ad + g resp., F g = F g l l S 0 denotes the action of g by f Galois conjugation on the semilocal complex v p C contk v, X + resp., l Sf 0 v l C contk v, X v. We have F g i + S f = i + S f Ad + g and there exists a homotopy m g = m g X : res Sf Adg F g res Sf, which is functorial in X see [Nek06, ] for an explicit description of the homotopy m g. It follows that the formula 37 Ad f gx n, x + n, x n 1 := Adσx n, Ad + gx + n, F gx n 1 + m g x n defines a morphism of complexes C f G K,S, X C f G K,S, X. By [Nek06, Lemma ] this map induces in cohomology the action of GalK/Q on H q f K, X alluded to above. We denote by xg or gx the action of g GalK/Q on x H q f K, X. In [Nek06, Prop ] or loc.cit., formula it is proved that CT is GalK/Q-equivariant, i.e. 38 gx CT gy = x CT y for every x, y H 2 f K, T P R P T or and g GalK/Q. This follows essentially by the Galois invariance of the local invariants. For more details on the constructions above, we refer the reader to [Nek06, Ch. VIII], especially to paragraphs , 8.6 and 8.8.

25 18 4. The p-adic weight pairing In this Section we apply the constructions recalled above to define the p-adic weight pairing, Nek K,p on the extended Mordell-Weil group E K of E/K. For every prime v p of K at which E has split multiplicative reduction, we have a Tate period q v E K EK. Given P EK, we can compute q v, P Nek K,p explicitly in terms of the formal group logarithm on E/Q cfr. Cor As explained in the introduction see also Sec. 7, this computation is the key for relating the algebraic constructions of Nekovář to the analytic results of Bertolini and Darmon The extended Mordell-Weil group. Let S sp f by abuse of notation Φ T ate : K p EK v v S sp f and let K p := v S sp f K v. We write again for the direct sum of the Tate parametrisations 8. Following [MTT86] and [BD96], we define the extended Mordell-Weil group { E K := P, P P EK, P K p and ΦT ate P } = ρ v P v S sp. f Given v S sp f, we write q v := 0, 1,..., q E,..., 1 E K with q E as v-component. We have a short exact sequence 39 0 Z E K EK 0, v S sp f where the first map sends the v-th generator to q v and the second is projection. If S sp f E K := EK. We have a natural map 40 i E : E K H 1 f K, V p E, =, define defined in the following manner. Let P, P E K, with P = P v v S sp. K f p Since resv γ P = i + v γ epv see Remark 1.3, for every representatives γp 0 C1 contg K,S, V p E and γ 0 Pv e CcontK 1 v, Q p 1 of γ P and γ epv respectively, there exists a unique ε 0 v C 0 contk v, V p E such that res v γ 0 P = i+ v γ 0 e Pv δε 0 v, where δ is the differential in C contk v, V p E ε 0 v is unique since H 0 K v, V p E = 0, by [Sil86, page 118]. In the same way, for every v / S sp f, there exists a unique γ v H 1 U v + V p E s.t. i + v γ v = res v γ P. For v p resp. v p this follows from H 0 K v, V p E = 0 resp. U v + V p E := 0 and Lemma 1.2. In particular, for every representative γv 0 U v + V p E of γ v, we can find a unique ε 0 v CcontK 0 v, V p E such that res v γp 0 = i+ v γv 0 δε 0 v. Recalling the definition of the differential in the Selmer complex C f G K,S, V p E, P, P 0 := γp 0, γ 0 Pv e v S sp + γ 0 f v v / S sp, ε0 f v v Sf C f 1 G K,S, V p E is a 1-cocycle. Furthermore it is easily seen that a different choice γ 1 P = γ0 P + δϑ P, γ 1 e Pv = γ 0 e Pv + δϑ v and γ 1 v = γ 0 v + δϑ v of representatives leads to the 1-cocycle P, P 1 = P, P 0 + d ec f ϑ P, ϑ v, 0. We can then define in 40 i E P, P as the image in cohomology of P, P 0. Lemma 4.1. Let i E : E K Q p H f 1K, V pe be the map induced by 40. Then i E and is an isomorphism provided that XE/K p is finite. Proof. This follows easily from the exact sequences 29 and 6. is injective