HEEGNER CYCLES AND HIGHER WEIGHT SPECIALIZATIONS OF BIG HEEGNER POINTS

Transcription

1 HEEGNER CYCLES AND HIGHER WEIGHT SPECIALIZATIONS OF BIG HEEGNER POINTS FRANCESC CASTELLA Abstract. Let f be a p-ordinary Hida family of tame level N, and let K be an imaginary quadratic field satisfying the Heegner hypothesis relative to N. By taking a compatible sequence of twisted Kummer images of CM points over the tower of modular curves of level Γ 0 (N) Γ 1 (p s ), Howard has constructed a canonical class Z in the cohomology of a self-dual twist of the big Galois representation associated to f. If a p-ordinary eigenform f on Γ 0 (N) of weight k > 2 is the specialization of f at, one thus obtains from Z a higher weight generalization of the Kummer images of Heegner points. In this paper, we relate the classes Z to the étale Abel-Jacobi images of Heegner cycles when p splits in K. Contents 1. Introduction 1 2. Preliminaries p-adic modular forms Comparison isomorphisms Coleman p-adic integration Generalised Heegner cycles Big logarithm map The big Heegner point Weight two specializations Higher weight specializations 26 References Introduction Fix a prime p > 3 and an integer N > 4 such that p Nφ(N). Let f o = n>0 a n q n S k (X 0 (N)) be a p-ordinary newform of even weight k = 2r 2 and trivial nebentypus. Thus f o is an eigenvector for all the Hecke operators T n with associated eigenvalues a n, and a p is a p-adic unit for a choice of embeddings ι : Q C and ι p : Q Q p that will remain fixed throughout this paper. Also let O denote the ring of integers of a (sufficiently large) finite extension L/Q p containing all the a n. Date: April 26,

2 2 F. CASTELLA For s > 0, let X s be the compactified modular curve of level and consider the tower Γ s := Γ 0 (N) Γ 1 (p s ), X s α Xs 1 with respect to the degeneracy maps described on the non-cuspidal moduli by (E, α E, π E ) (E, α E, p π E ), where α E denotes a cyclic N-isogeny on the elliptic curve E, and π E a point of E of exact order p s. The group (Z/p s Z) acts on X s via the diamond operators d : (E, α E, π E ) (E, α E, d π E ) compatibly with α under the reduction (Z/p s Z) (Z/p s 1 Z). Set Γ := 1+pZ p. Letting J s be the Jacobian variety of X s, the inverse limit of the system induced by Albanese functoriality, (1.1) Ta p (J s ) Zp O Ta p (J s 1 ) Zp O, is equipped with an action of the Iwasawa algebras Λ O := O[[Z p ]] and Λ O := O[[Γ]]. Let h s be the O-algebra generated by the Hecke operators T l (l Np), U l := T l (l Np), and the diamond operators d (d (Z/p s Z) ) acting on the space S k (X s ) of cusp forms of weight k and level Γ s. Hida s ordinary projector e ord := lim U p n! n defines an idempotent of h s, projecting to the maximal subspace of h s where U p acts invertibly. We make each h s into a Λ O -algebra by letting the group-like element attached to z Z p act as z k 2 z. Taking the projective limit with respect to the restriction maps induced by the natural inclusion S k (X s 1 ) S k (X s ) we obtain a Λ O -algebra (1.2) h ord := lim e ord h s s which can be seen to be independent of the weight k 2 used in its construction. After a highly influential work [Hid86] of Hida, one can associate with f o a certain local domain I quotient of h ord, finite flat over Λ O, with the following properties. For each n, let a n I be the image of T n under the projection h ord I, and consider the formal q-expansion f = n>0 a n q n I[[q]]. We say that a continuous O-algebra homomorphism : I Q p is an arithmetic prime if there is an integer k 2, called the weight of, such that the composition Γ I Q p agrees with γ γ k 2 on an open subgroup of Γ of index p s 1 1. Denote by X arith (I) the set of arithmetic primes of I, which will often be seen as sitting inside Spf(I)(Q p ). If X arith (I), F will denote its residue field. Then:

3 SPECIALIZATIONS OF BIG HEEGNER POINTS 3 for every X arith (I), there exists an ordinary p-stabilized newform 1 f S k (X s ) such that (f) F [[q]] gives the q-expansion of f ; if s = 1 and k k (mod 2(p 1)), there exists a normalized newform f S k (X 0 (N)) such that (1.3) f (q) = f (q) pk 1 (a p ) f (q p ); there exists a unique o X arith (I) such that f o = f o. In particular, after p-stabilization (1.3), the form f o fits in the p-adic family f. Similarly for the associated Galois representation V fo : the continuous h ord -linear action of the absolute Galois group G Q on the module (1.4) T := T ord h ord I, where T ord := lim e ord (Ta p (J s ) Zp O), s gives rise to a big Galois representation ρ f : G Q Aut(T) such that (ρ f ) = ρ f for every X arith (I), where ρ f is the contragredient of the (cohomological) p-adic Galois representation ρ f : G Q Aut(V f ) attached to f by Deligne; in particular, one recovers ρ f o from ρ f by specialization at o. Assume from now on that the residual representation ρ fo is irreducible; then T can be shown to be free of rank 2 over I. (See [MT90, Théorème 7] for example.) Let K be an imaginary quadratic field with ring of integers O K containing an ideal N O K with (1.5) O K /N = Z/NZ, and denote by H the Hilbert class field of K. Under this Heegner hypothesis relative to N (but with no extra assumptions on the prime p), the work [How07b] of Howard produces a compatible sequence Up s X s of cohomology classes with values in a certain twist of the ordinary part of (1.1), giving rise to a canonical big cohomology class X, the big Heegner point (of conductor 1), in the cohomology of a self-dual twist T of T. Moreover, if every prime factor of N splits in K, it follows from his results that the class Z := Cor H/K (X) lies in Nekovář s extended Selmer group H 1 f (K, T ). In particular, for every X arith (I) with s = 1 and k k (mod 2(p 1)) as above, the specialization Z belongs to the Bloch-Kato Selmer group H 1 f (K, V f (k /2)) of the self-dual representation T I F = Vf (k /2). The classes Z may thus be regarded as a natural higher weight analogue of the Kummer images of Heegner points, on modular Abelian varieties (associated with weight 2 eigenforms). But for any of the above f, one has an alternate (and completely different!) method of producing such a higher weight analogue. Briefly, if k = 2r > 2, associated to any elliptic curve A with CM by O K, there is a null-homologous cycle heeg A,r, a so-called Heegner cycle, on the (2r 1)-dimensional Kuga-Sato variety 1 As defined in [NP00, (1.3.7)].

4 4 F. CASTELLA W r giving rise to an H-rational class in the Chow group CH r+1 (W r ) 0 with Q- coefficients. Since the representation V f (r ) appears in the étale cohomology of W r : H 2r 1 ét (W r, Q p )(r ) π f by taking the images of the cycles heeg A,r V f (r ), under the p-adic étale Abel-Jacobi map Φét H : CH r+1 (W r ) 0 (H) H 1 (H, H 2r 1 ét (W r, Q p )(r )) and composing with the map induced by π f Φét f,k ( heeg r on H 1 s, we may consider the classes ) := Cor H/K (π f Φét H( heeg A,r )). By the work [Nek00] of Nekovář, these classes are known to lie in the same Selmer group as Z, and the question of their comparison thus naturally arises. Main Theorem (Thm. 5.11). Assume that p splits in K and that Z is not I-torsion. Then for any X arith (I) of weight k = 2r > 2 with k k (mod 2(p 1)) and trivial character, we have ) 4 Φ Z, Z K = (1 pr 1 ét f,k ( heeg r ), Φét f,k ( heeg r ) K (a p ) u 2 (4D) r 1, where, K is the cyclotomic p-adic height pairing on H 1 f (K, V f (r )), u := O K /2, and D < 0 is the discriminant of K. Thus assuming the non-degeneracy of the p-adic height pairing, it follows that the étale Abel-Jacobi images of Heegner cycles are p-adically interpolated by Z. We also note that Z is conjecturally always not I-torsion ([How07b, Conj ]), and that by [How07a, Cor. 5] this conjecture can be verified in any given case by exhibiting the non-vanishing of an appropriate L-value (a derivative, in fact). This paper is organized as follows. Section 1 is aimed at proving an expression for the formal group logarithms of ordinary CM points on X s using Coleman s theory of p-adic integration. Our methods here are drawn from [BDP, Sect. 3], which we extend in weight 2 to the case of level divisible by an arbitrary power of p, but with ramification restricted to a potentially crystalline setting. Not quite surprisingly, this restriction turns out to make our computations essentially the same as theirs, and will suffice for our purposes. In Sect. 2 we recall the generalised Heegner cycles and the formula for their p- adic Abel-Jacobi images from loc.cit., and discuss the relation between these and the more classical Heegner cycles. In Sect. 3 we deduce from the work [Och03] of Ochiai a big logarithm map that will allow as to move between different weights in the Hida family. Finally, in Sect. 4 we prove our main results. The key observation is that, when p splits in K, the combination of CM points on X s taken in Howard s construction appears naturally in the evaluation of the critical twist of a p-adic modular form at a canonical trivialized elliptic curve. The expression from Sect. 1 thus yields, for infinitely many of weight 2, a formula for the p-adic logarithm of the localization of Z in terms of certain values of a p-adic modular form of weight 0 associated with f (Thm. 5.8). When extended by p-adic continuity to an arithmetic prime of higher even weight, this expression is seen to agree with the formula from Sect. 2,

5 SPECIALIZATIONS OF BIG HEEGNER POINTS 5 and by the interpolation properties of the big logarithm map it corresponds to the p-adic logarithm of the localization of Z. Our main results follow easily from this. Finally, we note that an extension of the results in this paper, and in particular of the Main Theorem above, has a number of arithmetic applications arising from the connection with the theory of p-adic L-functions. (See [Cas12].) 2. Preliminaries 2.1. p-adic modular forms. To avoid some issues related to the representability of certain moduli problems, in this section we change notations from the Introduction, letting X s be the compactified modular curve of level Γ s := Γ 1 (Np s ), viewed as a scheme over Spec(Q p ). Let π : E s X s be the universal generalized elliptic curve over X s, and let ω Xs := π Ω 1 E s/x s (log Z s ) be the invertible sheaf on X s given by the pushforward of the relative differentials on E s /X s with log-poles along the inverse image of the cuspidal subscheme Z s X s. Algebraically, H 0 (X s, ω 2 X s ) gives the space of modular forms of weight 2 and level Γ s (defined over Q p ). Consider the complex (2.1) Ω X s/q p (log Z s ) : 0 O Xs d Ω 1 X s/q p (log Z s ) 0 of sheaves on X s. The algebraic de Rham cohomology of X s H 1 dr(x s /Q p ) := H 1 (X s, Ω X s/q p (log Z s )) is a finite-dimensional Q p -vector space equipped with a Hodge filtration 0 H 0 (X s, Ω 1 X s/q p (log Z s )) H 1 dr(x s /Q p ), and by the Kodaira-Spencer isomorphism ω 2 X s = Ω 1 Xs/Q p (log Z s ), every cusp form f S 2 (X s ) (in particular) defines a cohomology class ω f HdR 1 (X s/q p ). Let X be the complete modular curve of level Γ 1 (N), also viewed over Spec(Q p ), and consider the subspaces of the associated rigid analytic space X an : X ord X <1/(p+1) X <p/(p+1) X an. To define these, let X /Zp be the canonical integral model of X over Spec(Z p ), and let X Fp := X Zp F p denote its special fiber. The supersingular points SS X Fp (F p ) is the finite set of points corresponding to the moduli of supersingular elliptic curves (with Γ 1 (N)-level structure) in characteristic p. Let E p 1 be the Eisenstein series of weight p 1 and level 1, seen as a global section of the sheaf ω (p 1) X. (Recall that we are assuming p 5.) The reduction of E p 1 to X Fp is the Hasse invariant, which defines a section of the reduction of ω (p 1) X with SS as its locus of (simple) zeroes. If x X(Q p ), let x X Fp (F p ) denote its reduction. Each point x SS is smooth in X Fp, and the ordinary locus of X X ord := X an x SS is defined to be the complement of their residue discs D x X an. The function E p 1 (x) p defines a local parameter on D x, and with the normalization p p = p 1, X <1/(p+1) (resp. X <p/(p+1) ) is defined to be complement in X an of the subdiscs of D x where E p 1 (x) p p 1/(p+1) (resp. E p 1 (x) p p p/(p+1) ), for all x SS. D x

6 6 F. CASTELLA Using the canonical subgroup H E (of order p) attached to every elliptic curve E corresponding to a closed point in X <p/(p+1), the Deligne-Tate map φ 0 : X <1/(p+1) X <p/(p+1) is defined by sending E E/H E (with the induced action on the level structure) under the moduli interpretation. This map is a finite morphism which by definition lifts to characteristic zero the absolute Frobenius on X Fp. (See [Kat73, Thm. 3.1].) For every s > 0, the Deline-Tate map φ 0 can be iterated s 1 times on the open rigid subspace X <p 2 s /(p+1) of X an where E p 1 (x) p > p p2 s /(p+1). Letting α s : X s X be the map forgetting the Γ 1 (p s )-part of the level structure, define W 1 (p s ) X an s to be the open rigid subspace of X s whose closed points correspond to triples (E, α E, π E ) whose image under α s lands inside X <p 2 s (p+1) and are such that π E generates the canonical subgroup of E of order p s (as in [Buz03, Def. 3.4]). Define W 2 (p s ) Xs an is the same manner, replacing p 2 s /(p+1) by p 1 s /(p+1) in the definition of W 1 (p s ). Then we obtain a lifting of Frobenius φ = φ s on X s making the diagram W 2 (p s ) φ W 1 (p s ) α s X <p 1 s (p+1) φ 0 α s X <p 2 s (p+1). commutative by sending a point x = (E, α E, ı E ) W 2 (p s ), where ı E : µ p s E[p s ] is an embedding giving the Γ 1 (p s )-level structure on E, to x = (φ 0 E, φ 0 α E, ı E ), where ı E is determined by requiring that α s(x ) lands in X <p 2 s /(p+1) and for each ζ µ p s {1}, ı E (ζ) = φ 0Q if ı E (ζ) = pq. (Cf. [Col97b, Sect. B.2].) Let I s := {v Q : 0 v < p 2 s /(p + 1)}, and for v I s define the affinoid subdomain X s (v) of Xs an inside W 1 (p s ) whose closed points x satisfy E p 1 (x) p p v. Then X s (0) is the connected component of the ordinary locus of X s containing the cusp. Let k Z, and denote by ω X an the rigid analytic sheaf on X an s s deduced from ω Xs. The space of p-adic modular forms of weight k and level Γ s (defined over Q p ) is the p-adic Banach space Mk ord (X s ) := H 0 (X s (0), ω k X ), s an and the space of overconvergent p-adic modular forms of weight k and level Γ s is the p-adic Fréchet space M rig k (X s) := lim v H 0 (X s (v), ω k X an s ), where the limit is with respect to the natural restriction maps as v I s increasingly approaches p 2 s /(p + 1). By restriction, a classical modular form in H 0 (X s, ω k X s ) defines an (obviously) overconvergent p-adic modular form of the same weight an level. Moreover, the action of the diamond operators on X s gives rise to an action of (Z/p s Z) on the spaces of p-adic modular forms which agrees with the action on H 0 (X s, ω k X s ) under restriction. We say that a ring R is a p-adic ring if the natural map R lim R/p n R is an isomorphism. For varying s > 0, the data of a compatible sequence of embeddings

7 SPECIALIZATIONS OF BIG HEEGNER POINTS 7 µ p s E as R-group schemes, amounts to the data of an embedding µ p E[p ] of p-divisible groups, and also to the given of a trivialization of E over R, i.e. an isomorphism ı E : Ê Ĝm of the associated formal groups. The space M(N) of Katz p-adic modular functions of tame level N (over Z p ) is the space of functions f on trivialized elliptic curves with Γ 1 (N)-level structure over arbitrary p-adic rings, assigning to the isomorphism class of a triple (E, α E, ı E ) over R a value f(e, α E, ı E ) R whose formation is compatible under base change. If R is a fixed p-adic ring, by only considering p-adic rings which are R-algebras, we obtain the notion of Katz p-adic modular functions defined over R, forming the space M(N) Zp R which will also be denoted by M(N) by an abuse of notation. The action of z Z p on a trivialization gives rise to an action of Z p on M(N): z f(e, α E, ı E ) := f(e, α E, z 1 ı E ), and given a character χ Hom cont (Z p, R ), we say that f M(N) has weightnebentypus χ if z f = χ(z)f for all z Z p. If k is an integer, denoting by z k the k-th power character on Z p, the subspace Mk ord (Np s, ε) of Mk ord (X s ) consisting of p-adic modular forms with nebentypus ε : (Z/p s Z) R can be recovered as (2.2) M ord k (Np s, ε) = {f M(N) : z f = z k ε(z)f, for all z Z p }. Since it will play an important role later, we next recall from [Gou88, Sect. III.6.2] the definition in terms of moduli of the twist of p-adic modular forms by characters of not necessarily finite order. Let R be a p-adic ring, and let (E, α E, ı E ) be a trivialized elliptic curve with Γ 1 (N)-level structure over R. For each s > 0, consider the quotient E 0 := E/ı 1 E (µ p s), and let ϕ 0 : E E 0 denote the projection. Since p N, ϕ 0 induces a Γ 1 (N)-level structure α E0 on E 0, and since ker(ϕ 0 ) = µ p s, the dual ˇϕ 0 : E 0 E is étale, inducing an isomorphism of the associated formal groups. Thus (with a slight abuse of notation) ı E0 := ı E ˇϕ 0 : Ê0 Ĝm is a trivialization of E 0, and since we have an embedding j : Z/p s Z = ker( ˇϕ 0 ) E 0 [p s ], we deduce an isomorphism E 0 [p s ] = µ p s Z/p s Z which we use to bijectively attach a p s -th root of unity ζ C to every étale subgroup C E 0 [p s ] of order p s, in such a way that 1 is attached to ker( ˇϕ 0 ). Now for f M(N) and a Z p, define f 1 a+p s Z p to be the rule on trivialized elliptic curves given by (2.3) f 1 a+ps Z p (E, α E, ı E ) = 1 p s ζ a C f(e 0/C, α C, ı C ) where the sum is over the étale subgroups C E 0 [p s ] of order p s, and where α C (resp. ı C ) denotes the Γ 1 (N)-level structure (resp. trivialization) on the quotient E 0 /C naturally induced by α E0 (resp. ı E0 ). Lemma 2.1. The assignment a + p s Z p ( ) f f 1 a+p s Z p gives rise to an End R M(N)-valued measure µ Gou on Z p. Proof. Let n a nq n be the q-expansion of f, i.e. the value that it takes at the triple (Tate(q), α can, ı can ) = (G m /q Z, ζ N, µ p G m /q Z ) over the p-adic completion C

8 8 F. CASTELLA of R((q)). By the q-expansion principle, the claim follows immediately from the equality f 1 a+ps Z p (q) = a n q n, n a mod p s which is shown by adapting the arguments in [Gou88, p. 102]. Definition 2.2 (Gouvêa). Let f M(N) and χ : Z p R be any continuous multiplicative function. The twist of f by χ is ( ) f χ := χ(x)dµ Gou (x) (f) M(N). Z p This operation is compatible with the usual character twist of Hecke eigenforms: Lemma 2.3. Let χ : Z p R be a continuous character extended by zero on pz p. If f M(N) has q-expansion n a nq n, then f χ has q-expansion n χ(n)a nq n, and if f has weight-nebentypus κ Hom cts (Z p, R ), then f χ has weight-nebentypus χ 2 κ. Proof. See [Gou88, Cor. III.6.8.i] and [Gou88, Cor. III.6.9]). In particular, twisting by the identity function of Z p we obtain an operator d : M(N) M(N) whose effect on q-expansions is q d dq. For every k Z, we see from (2.2) and Lemma 2.3, that this restricts to a map d : Mk ord (X s ) Mk+2(X ord s ) which increases the weight by 2 and preserves the nebentypus. Moreover, for k = 0, the arguments in [Col96, Prop. 4.3] can be adapted to show that d restricts to a linear map M rig 0 (X s) M rig 2 (X s), viewing M rig (X s) Mk ord (X s ) by restriction Comparison isomorphisms. Let ζ s be a primitive p s -th root of unity, and let F be a finite extension of Q p (ζ s ) over which X s acquires stable reduction, i.e. such that the base extension X s Qp F admits a stable model over the ring of integers O F of F. For the ease of notation, from now on we will denote X s Qp F (as well as the associated rigid analytic space) simply by X s. Let X s be the minimal regular model of X s over O F, and denote by F 0 the maximal unramified subfield of F. The work [HK94] of Hyodo-Kato endows the F -vector space HdR 1 (X s/f ) with a canonical F 0 -structure (2.4) H 1 log cris(x s ) H 1 dr(x s /F ) equipped with a semi-linear Frobenius operator ϕ. After the proof [Tsu99] of the Semistable conjecture of Fontaine Jannsen, these structures are known to agree with those attached by Fontaine s theory to the p-adic G F -representation (2.5) V s := H 1 ét(x s, Q p ). More precisely, since X s has semistable reduction, V s is semistable in the sense of Fontaine, and there is a canonical isomorphism D st (V s ) H 1 log cris (X s), inducing an isomorphism (2.6) D dr (V s ) H 1 dr(x s /F ) as filtered ϕ-modules after extension of scalars to F. k

9 SPECIALIZATIONS OF BIG HEEGNER POINTS 9 Consider the étale Abel-Jacobi map CH 1 (X s ) 0 (F ) H 1 (F, V s (1)) constructed in [Nek00], which in this case agrees with the usual Kummer map δ F : J s (F ) H 1 (F, Q p Ta p (J s )), where J s = Pic 0 (X s ) is the connected Picard variety of X s. (See [loc.cit., Ex. (2.3)].) Let g S 2 (X s ) be a newform with primitive nebentypus, let V g the p-adic Galois representation attached to g, which is equipped with a Galois-equivariant projection V s V g, and let Vg be the representation contragredient to V g, so that V g (1) and Vg are in Kummer duality. Also let L g be a finite extension of Q p over which the Hecke eigenvalues of g are defined. By [BK90, Ex. 3.11], the image of the induced composite map (2.7) δ g,f : J s (F ) δ F H 1 (F, V s (1)) H 1 (F, V g (1)) lies in the Bloch-Kato finite subspace H 1 f (F, V g(1)), and by our assumption on g, the Bloch-Kato exponential map gives an isomorphism (2.8) exp F,Vg(1) : D dr (V g (1)) Fil 0 D dr (V g (1)) H1 f (F, V g (1)) whose inverse will be denoted by log F,Vg(1). Our aim in this section is to compute the images of certain degree 0 divisors on X s under the p-adic Abel-Jacobi map δ (p) g,f, defined as the composition (2.9) J s (F ) δ g,f Hf 1 (F, V g (1)) log F,Vg (1) D dr(v g (1)) Fil 0 (Fil 0 D dr (Vg )), D dr (V g (1)) where the last identification arises from the de Rham pairing (2.10), : D dr (V g (1)) D dr (V g ) D dr (Q p (1)) Qp L g = Lg with respect to which Fil 0 D dr (V g (1)) and Fil 0 D dr (V g ) are exact annihilators of each other. A basic ingredient for this computation will be the following alternate description of the logarithm map log F,Vg(1). Recall the interpretation of H 1 (F, V g (1)) as the space Ext 1 Rep(G F )(Q p, V g (1)) of extensions of V g (1) by Q p in the category of p-adic G F -representations. Since F contains Q p (ζ s ), V g is a crystalline G F -representation in the sense of Fontaine, and under that interpretation the Bloch-Kato finite subspace corresponds to those extensions which are crystalline (see [Nek93, Prop. 1.26], for example): (2.11) H 1 f (F, V g (1)) = Ext 1 Rep cris (G F )(Q p, V g (1)). Now consider a crystalline extension (2.12) 0 V g (1) W Q p 0. Since D cris (V g (1)) ϕ=1 = 0 by our assumptions, the resulting extension of ϕ-modules (2.13) 0 D cris (V g (1)) D cris (W ) F 0 0 admits a unique section s frob W : F 0 D cris (W ) with s frob W (1) D cris(w ) ϕ=1. Extending scalars from F 0 to F in (2.13) and taking Fil 0 -parts, we take an arbitrary section s fil W : F Fil0 D dr (W ) of the resulting exact sequence of F -vector spaces (2.14) 0 Fil 0 D dr (V g (1)) Fil 0 D dr (W ) F 0

10 10 F. CASTELLA and form the difference t W := s fil W (1) s frob W (1), which can be seen in D dr (V g (1)), and whose image modulo Fil 0 D dr (V g (1)) is welldefined. Lemma 2.4. Under the identification (2.11), the above assignment 0 V g (1) W Q p 0 t W mod Fil 0 D dr (V g (1)) defines an isomorphism which agrees with the Bloch-Kato logarithm map log F,Vg(1) : H 1 f (F, V g (1)) D dr(v g (1)) Fil 0 D dr (V g (1)). Proof. See [Nek93, Lem. 2.7], for example. Let J s (F ) be the class of a degree 0 divisor on X s with support contained in the finite set of points S X s (F ). The extension class W = W (2.12) corresponding to δ g,f ( ) can then be constructed from the étale cohomology of the open curve Y s := X s S, as explained in [BDP, Sect. 3.1]. We describe the associated s fil W and s frob W. By [Tsu99] (or also [Fal02]), denoting g-isotypical components by the superscript g, there is a canonical isomorphism of F 0 Qp L g -modules (2.15) D cris (V g ) = H 1 log cris(x s ) g compatible with ϕ-actions and inducing an F Qp L g -module isomorphism (2.16) D dr (V g ) = H 1 dr(x s /F ) g after extension of scalars. Writing = Q S n Q.Q for some n Q Z, we assume from now on that the reductions of the points Q S are smooth and pair-wise distinct. Assume from now on that the reduction of S in the special fiber is stable under the absolute Frobenius. Like H 1 dr (X s/f ), the F -vector space H 1 dr (Y s/f ) is equipped with a canonical F 0 -structure (2.17) H 1 log cris(y s ) H 1 dr(y s /F ), a Frobenius operator still denoted by ϕ, and a Hecke action compatible with that in (2.4). Thus for W = W the exact sequence (2.13) is obtained as the pullback (2.18) D cris (V g (1)) D cris (W ) ρ F 0 Qp L g H 1 log cris (X s) g (1) H 1 log cris (Y s) g (1) res Q (F 0 Qp L g ) S 0 of the bottom extension of ϕ-modules with respect to the F 0 Qp L g -linear map sending 1 (n Q ) Q S, where the subscript 0 indicates taking the degree 0 subspace.

11 SPECIALIZATIONS OF BIG HEEGNER POINTS 11 On the other hand, after extending scalars from F 0 to F and taking Fil 0 -parts, (2.14) is given by the pullback 2 (2.19) Fil 0 D dr (V g (1)) Fil 1 H 1 dr (X s/f ) g Fil 0 D dr (W ) ρ F Qp L g Fil 1 H 1 dr (Y s/f ) g res Q (F Qp L g ) S 0 of the bottom exact sequence of free F Qp L g -modules with respect to the F Qp L g - linear map sending 1 (n Q ) Q S. Let g S 2 (X s ) be the form dual to g, defined as the newform associated with the twist g ε 1 g, and let ω g H 0 (X s, Ω 1 X s/f ) be its associated differential, so that Fil 0 D dr (Vg ) = (F Qp L g ).ω g. Thus δ (p) ( ) is determined by the value (2.20) δ (p) g,f ( )(ω g ) = t W, ω g of the pairing (2.10), which corresponds to the Poincaré pairing on H 1 dr (X s/f ) under the identification (2.16). Using rigid analysis, we now give an expression for the latter pairing that will make (2.20) amenable to computations. Let X s be the canonical balanced model of X s over Z p [ζ s ] constructed by Katz and Mazur (see [KM85, Ch. 13]). The special fiber X s Zp[ζ s]f p is a reduced disjoint union of Igusa curves over F p intersecting at the supersingular points. Exactly two of these components are isomorphic to the Igusa curve Ig(Γ s ) representing the moduli problem ([Γ 1 (N)], [Γ 1 (p s )]) over F p, and we let I be the one that contains the reduction of W 1 (p s ) Qp Q p (ζ s ), and I 0 be the other. (These two are the two good components in the terminology of [MW86].) By the universal property of the regular minimal model, there exists a morphism g,f (2.21) X s X s Zp[ζ s] O F which reduces to a sequence of blow-ups on the special fiber. Letting κ be the residue field of F, define W X s (resp. W 0 X s ) to be the inverse image under the reduction map via X s of the unique irreducible component of X s OF κ mapping bijectively onto I Fp κ (resp. I 0 Fp κ) in X s Zp[ζ s] κ via the reduction of (2.21). Similarly, define U X s by considering the irreducible components of X s Zp[ζ s] κ different from I Fp κ and I 0 Fp κ. Letting SS denote (the degree of) the supersingular divisor of Ig(Γ s ), it follows that U intersects W (resp. W 0 ) in a union of SS supersingular annuli. Since they reduce to smooth points, the residue class D Q of each Q S is conformal to the open unit disc D C p. Fix an isomorphism h Q : D Q D that sends Q to 0, and for a real number r Q < 1 in p Q, denote by V Q D Q the annulus consisting of the points x D Q with r Q < h Q (x) p < 1. In the same manner, we define annuli V z for each z in the cuspidal subscheme Z s X s. Attached to any (oriented) annulus V, there is a p-adic annular residue map Res V : Ω 1 V C p defined by expanding ω Ω 1 V as ω = n Z a nt n dt T for a fixed uniformizing parameter T on V (compatible with the orientation), and setting Res V (ω) = a 0. This descends to a linear functional on Ω 1 V /do V. (Cf. [Col89, Lem. 2.1].) 2 Notice the effect of the Tate twist on the filtrations.

12 12 F. CASTELLA For any basic wide-open W (as in [Buz03, p. 34]), define (2.22) H 1 rig(w) := H 1 (W, Ω (log Z)) = Ω 1 W/dO W, where Ω (log Z) denotes the complex of rigid analytic sheaves on W deduced from (2.1) by analytification and pullback, and consider the basic wide-opens W := W Q S (D Q V Q ) and W0 := W 0 Q S(D Q V Q ). As follows from the arguments in [BE10, Lem ], the spaces H 1 rig ( W ) and H 1 rig ( W 0 ) are each equipped with a natural action of the Hecke operators T l (l Np) compatible with the Hecke action on H 1 dr (Y s/f ) under restriction. Lemma 2.5. The natural restriction maps induce an isomorphism H 1 dr(y s /F ) g H 1 rig( W ) g H 1 rig( W 0 ) g. A class ω HdR 1 (Y s/f ) g belongs to the natural image of HdR 1 (X s/f ) g if an only if it can be represented by a pair of differentials (ω, ω 0 ) Ω 1 W Ω 1 W0 with vanishing p-adic annular residues. If η and ω are any two classes in HdR 1 (X s/f ) g, their Poincaré pairing can be computed as η, ω = (2.23) Res V (F ω V η V ) + Res V (F ω0 V η 0 V ), V W V W 0 where for each annulus V, F ωv denotes any solution to df ωv = ω V on V. Proof. By an excision argument, the first assertion is easily deduced from [Col97a, Thm. 2.1] as in [BE10, Lem ]; the second and third are shown by adapting the arguments in [Col96, 5] for each of the two components, as they are proven in [Col94a, Prop. 1.3] for s = 1. (See also [Col97a, 3].) 2.3. Coleman p-adic integration. Coleman s theory provides a coherent choice of local primitives that will allow us to compute (2.20) using the formula (2.23). Recall the lift of Frobenius φ : W 2 (p s ) W 1 (p s ) described in Sect. 2.1, where W i (p s ) are the strict neighborhoods of the connected component X s (0) of the ordinary locus of X s containing the cusp described there. Recall also the wide open space W described in the preceding section, which also contains X s (0) by construction. Proposition 2.6 (Coleman). Let g = n>1 b nq n S 2 (X s ) be a normalized newform with primitive nebentypus of p-power conductor, so that b p is such that U p g = b p g. Then there exists a locally analytic function F ωg on W which is unique up to a constant on W and such that df ωg = ω g on W, and F ωg bp p φ F ωg M rig 0 (X s). Proof. This follows from the general result of Coleman [Col94b, Thm. 10.1] on p-adic integration. Indeed, a computation on q-expansions shows that the action of the Frobenius lift φ on differentials agrees with that of pv, with V the map

13 SPECIALIZATIONS OF BIG HEEGNER POINTS 13 acting as q q p on q-expansions, in the sense that φ ω g = pω V g on W := φ 1 (W W 1 (p s )). Since the differential ω g [p] = ω g b p ω V g attached to g [p] = b n q n (n,p)=1 becomes exact upon restriction to W, this shows that the polynomial L(T ) = 1 bp p T is such that L(φ )ω g = 0. Finally, since g has primitive nebentypus, b p has complex absolute value p 1/2, and hence [Col94b, Thm. 10.1] can be applied with L(T ) as above. Attached to a primitive p s -th root of unity ζ, there is an automorphism w ζ of X s which interchanges the components W and W 0 (see [BE10, Lem ]). Corollary 2.7. Set φ = w ζ φ w ζ. With hypotheses as in Proposition 2.6, there exists a unique locally analytic function F ω g on W 0 which vanishes at 0, satisfies df ω g = ω g on W 0, and F ω g bp p (φ ) F ω g is rigid analytic on a wide-open neighborhood W 0 of w ζ X s (0) in W 0. Proof. Proposition 2.6 applied to the differential ω g := wζ ω g gives the existence of a locally analytic function F ω g with F ω g := wζ F ω g having the desired properties. The uniqueness of F ω g follows immediately from that of F ω g. We refer to the locally analytic function F ωg (resp. F ω g ) appearing in Prop. 2.6 as the Coleman primitive of g on W (resp. W 0 ). Let g = n>0 b nq n be as in Proposition 2.6. The q-expansion b nn (n,p)=1 q n corresponds to a p-adic modular form g vanishing at and satisfying dg = g [p], where d is the operator described at the end of Sect. 2.1, which here corresponds to the differential operator O W Ω 1 W for any subspace W X s. Set d 1 g [p] := g. Corollary 2.8. If F ωg is the Coleman primitive of g on W which vanishes at, then F ωg b p p φ F ωg = d 1 g [p]. Proof. Since d 1 g [p] is an overconvergent rigid analytic primitive of ω g [p], and the operator L(φ ) = 1 bp p φ acting on the space of locally analytic functions on W is invertible, we see that L(φ ) 1 (d 1 g [p] ) satisfies the defining properties of F ωg. Since d 1 g [p] vanishes at, the result follows. Now we can give a closed formula for the p-adic Abel-Jacobi images of certain degree 0 divisors on X s. Proposition 2.9. Assume s > 1. Let g S 2 (X s ) be a normalized newform with primitive nebentypus of p-power conductor, let P be an F -rational point of X s factoring through X s (0) X s, and let J s (F ) be the divisor class of (P ) ( ). Then (2.24) δ (p) g,f ( )(ω g ) = F ω g (P ), where F ωg is the Coleman primitive of ω g on W which vanishes at.

14 14 F. CASTELLA Proof. By (2.20), we must compute t W, ω g = s fil W, ω g s frob W, ω g, where s fil W Fil 1 D dr (W ) is such that ρ(s fil W ) = 1 in (2.19), and s frob W D cris (W ) ϕ=1 is such that ρ(s frob W ) = 1 in (2.18). By Lemma 2.5, we see that these can be represented, respectively, by η fil a section of Ω1 X over Y s/f s with simple poles at P and and with Res P (η fil) = 1, while Res Q(η fil ) = 0 for all Q S {P }; Res (η fil) = 1, while Res z(η fil) = 0 for all z Z s { }, η frob = (ηfrob, η0 frob ) Ω 1 W Ω 1 W0 with (φ η frob, (φ ) η frob ) = (p ηfrob + dg, p η0 frob + dg 0 ) with G and G 0 rigid analytic on φ 1 W and (φ ) 1 W0, respectively; Res V (η frob ) = 0 for all supersingular annuli V; and Res VQ (η frob) = Res Q(η fil) (Q S), Res V z (η frob) = Res z(η fil) (z Z s). The arguments in [BDP, Prop. 3.21] can now be straightforwardly adapted to deduce the result. Indeed, using the defining properties of the Coleman primitives F ωg and F ω g of ω g on W and W 0, respectively, one first shows that (2.25) Res V (F ωg η frob ) = 0 and Res V (F ω g η0 frob ) = 0 V W V W 0 as in [loc.cit., Lemma 3.20]. On the other hand, using the same primitives, one shows as in [loc.cit., Lemma 3.19] that (2.26) ) = F ωg (P ) and ) = 0. V W Res V (F ωg η fil V W 0 Res V (F ω g η fil Substituting (2.26) and (2.25) into the formula (2.23) for the Poincaré pairing (and using that s > 1, so that there is no overlap between the supersingular annuli in W and the supersingular annuli in W 0 ), the result follows. 3. Generalised Heegner cycles Let X 1 (N) be the compactified modular curve of level Γ 1 (N) defined over Q, and let E be the universal generalized elliptic curve over X 1 (N). (Recall that N > 4.) For r > 1, denote by W r the (2r 1)-dimensional Kuga-Sato variety 3, defined as the canonical desingularization of the (2r 2)-nd fiber product of E with itself over X 1 (N). By construction, the variety W r is equipped with a proper morphism π r : W r X 1 (N) whose fibers over a noncuspidal closed point of X 1 (N) corresponding to an elliptic curve E with Γ 1 (N)-level structure is identified with 2r 2 copies of E. (For a more detailed description, see [BDP, Sect. 2.1].) Let K be an imaginary quadratic field of odd discriminant D < 0. It will be assumed throughout that K satisfies the following hypothesis: Assumption 3.1. All the prime factors of N split in K. Denote by O K the ring of integers of K, and note that by this assumption we may choose an ideal N O K with O K /N = Z/NZ that we fix once and for all. 3 Perhaps most commonly denoted by W2r 2 ; cf. [Zha97] and [Nek95], for example.

15 SPECIALIZATIONS OF BIG HEEGNER POINTS 15 Let A be a fixed elliptic curve with CM by O K. The pair (A, A[N]) defines a point P A on X 0 (N) rational over H, the Hilbert class field of K. Choose one of the square-roots D O K, let Γ D A A be the graph of D, and define Υ heeg A,r := Γ D (r 1) Γ D viewed inside W r by the natural inclusion (A A) r 1 W r as the fiber of π r over a point on X 1 (N) lifting P A. Let ɛ W be the projector from [BDP, (2.1.2)], and set (3.1) heeg A,r := ɛ W Υ heeg A,r, which is an (r 1)-dimensional null-homologous cycle on W r defining an H-rational class in the Chow group CH r (W r ) 0 (taken with Q-coefficients, as always here). These cycles (3.1) are usually referred to as Heegner cycles (of conductor one, weight 2r), and they share with classical Heegner points (as in [Gro84]) many of their arithmetic properties (see [Nek92, Nek95, Zha97]). We next recall a variation of the previous construction introduced in the recent work [BDP] of Bertolini, Darmon, and Prasanna. Let A be the CM elliptic curve fixed above, and consider the variety 4 X r := W r A 2r 2. For each class [a] Pic(O K ), represented by an ideal a O K prime to N, let A a := A/A[a] and denote by ϕ a the degree Na-isogeny ϕ a : A A a. The pair a (A, A[N]) := (A a, A a [N]) defines a point P Aa in X 0 (N) rational over H. Let Γ t ϕ a A a A be the transpose of the graph of ϕ a, and set Υ bdp ϕ a,r := Γ t ϕ a (2r 2) Γ t ϕ a (A a A) 2r 2 = A 2r 2 a A 2r 2 (ıa,id A) X r, where ı a is the natural inclusion A 2r 2 a W r as the fiber of π r over a point on X 1 (N) lifting P Aa. Letting ɛ A be the projector from [BDP, (1.4.4)], the cycles (3.2) bdp ϕ a,r := ɛ A ɛ W Υ bdp ϕ a,r define classes in CH 2r 1 (X r ) 0 (H) and are referred to as generalised Heegner cycles. We will assume for the rest of this paper that K also satisfies the following: Assumption 3.2. The prime p splits in K. Let g S 2r (X 0 (N)) be a normalized newform, and let V g be the p-adic Galois representation associated to g by Deligne. By the Künneth formula, there is a map H 4r 3 ét (X r, Q p (2r 1)) H 2r 1 ét (W r, Q p (1)) Sym 2r 2 Hét(A, 1 Q p (1)), which composed with the natural Galois-equivariant projection induces a map H 2r 1 ét (W r, Q p (1)) Sym 2r 2 Hét(A, 1 Q p (1)) πg π N r 1 V g (r) π g,n r 1 : H 1 (F, H 4r 3 ét (X r, Q p (2r 1))) H 1 (F, V g (r)) over any number field F. In the following we fix a number field F containing H. 4 Notice that our indices differ from those in [BDP].

16 16 F. CASTELLA Now consider the étale Abel-Jacobi map Φét F : CH 2r 1 (X r ) 0 (F ) H 1 (F, H 4r 3 ét (X r, Q p )(2r 1)) constructed in [Nek00]. Let F p be the completion of ı p (F ), and denote by loc p the induced localization map from G F to Gal(Q p /F p ). Then we may define the p-adic Abel-Jacobi map AJ Fp by the commutativity of the diagram (3.3) CH 2r 1 (X r ) 0 (F ) π g,n r 1 Φét F H 1 (F, V g (r)) loc p H 1 (F p, V g (r)) AJ Fp H 1 f (F p, V g (r)) log Fp,Vg (r) Fil 1 D dr (V g (r 1)), where the existence of the dotted arrow follows from [Nek00, Thm.(3.1)(i)], and the vertical map is given by the logarithm map of Bloch-Kato, as it appeared in (2.9) for r = 1. Using the comparison isomorphism of Faltings [Fal89], the map AJ Fp may be evaluated at the class ω g e r 1 ζ, with e ζ an F p -basis of D dr (Q p (1)) = F p. The main result of [BDP] yields the following formula for the p-adic Abel-Jacobi images of the generalised Heegner cycles (3.2) which we will need. Theorem 3.3 (Bertolini Darmon Prasanna). Let g = n b nq n S 2r (X 0 (N)) be a normalized newform of weight 2r 2 and level N prime to p. Then (1 b p p r + p 1 ) Na 1 r AJ Fp ( bdp ϕ a,r)(ω g e r 1 [a] Pic(O K ) = ( 1) r 1 (r 1)! where g [p] = (n,p)=1 b nq n is the p-depletion of g. Proof. See the proof of [BDP, Thm. 5.13]. [a] Pic(O K ) ζ ) d r g [p] (a (A, A[N])), We end this section by relating the images of Heegner cycles and of generalised Heegner cycles under the p-adic height pairing. (Cf. [BDP, Sect. 2.4].) Consider Π r := W r A r 1 seen as a subvariety of W r X r = W r W r (A 2 ) r 1 via the map (id Wr, id Wr, (id A, D) r 1 ). Denoting by π W and π X the projections onto the first and second factors of W r X r, the rational equivalence class of the cycle Π r gives rise to a map on Chow groups induced by Π r ( ) = π W, (Π r π X ). Lemma 3.4. We have Π r : CH 2r 1 (X r ) CH r+1 (W r ) heeg A,r, heeg A,r W r = (4D) r 1 bdp id A,r, bdp id A,r X r, where, Wr and, Xr are the p-adic height pairings of [Nek93] on CH r+1 (W r ) 0 and CH 2r 1 (X r ) 0, respectively.

17 SPECIALIZATIONS OF BIG HEEGNER POINTS 17 Proof. The image Φét F ( heeg A,r ) remains unchanged if we replace Γ D by Z A := Γ D (A {0}) D({0} A) (see [Nek95, II(3.6)]). Since Z A Z A = 2D, we easily see from the construction of Π r that (3.4) Φét F ( heeg A,r ) = ( 2D)r 1 Φét F (Π r ( bdp id A,r )). On the other hand, if, A denotes the Poincaré pairing on HdR 1 (A/F ), we have ( D) ω, ( D) ω A = D ω, ω A, for all ω, ω HdR 1 (A/F ). By the definition of the p-adic height pairings, W r and, Xr (factoring through Φét F ), we thus see that (3.5) bdp id A,r, bdp id A,r X r = D r 1 Π r ( bdp id A,r ), Π r( bdp id A,r ) W r. Combining (3.4) and (3.5), the result follows. 4. Big logarithm map Let f = n>0 a nq n I[[q]] be a Hida family passing through (the ordinary p-stabilization of) a p-ordinary newform f o S k (X 0 (N)) as described in the Introduction. We begin this section by recalling the definition of a certain twist of f such that all of its specializations at arithmetic primes of even weight correspond to p-adic modular forms with trivial weight-nebentypus. Decompose the p-adic cyclotomic character ε cyc as the product ε cyc = ω ɛ : G Q Z p = µ p 1 Γ. Since k is even, the character ω k 2 admits a square root ω k 2 2 : G Q µ p 1, and in fact two different square roots, corresponding to the two different lifts of k 2 Z/(p 1)Z to Z/2(p 1)Z. Fix for now a choice of ω k 2 2, and define the critical character to be (4.1) Θ := ω k 2 2 [ɛ 1/2 ] : G Q Λ O, where ɛ 1/2 : G Q Γ denotes the unique square root of ɛ taking values in Γ. Remark 4.1. As noted in [How07b, Rem ], the above choice of Θ is for most purposes largely indistinguishable from the other choice, namely ω p 1 2 Θ, where ω p 1 2 : Gal(Q( p )/Q) {±1} (p = ( 1) p 1 2 p). Nonetheless, for a given f o as above, our main result (Theorem 5.11) will specifically apply to only one of the two possible choices for the critical character. The critical twist of T is then defined to be the module (4.2) T := T I I equipped with the diagonal G Q -action, where I = I(Θ 1 ) is I as a module over Θ itself with G Q acting via the character G 1 Q Λ O I. Lemma 4.2. Let ρ T : G Q Aut(T ) be the Galois representation carried by T. Then for every X arith (I) of even weight k = 2r 2 we have (ρ T ) = ρ f ε r cyc,

18 18 F. CASTELLA where f is a character twist of f of the same weight and with trivial nebentypus. In other words, defining V := T I F and letting V f be the representation space of ρ f, we have (4.3) V = V f (r ), and in particular V is isomorphic to its Kummer dual. Proof. This follows from a straightforward computation explained in [NP00, (3.5.2)] for example (where T is denoted by T ). Let θ : Z p Λ O be such that Θ = θ ε cyc. It follows from the preceding lemma that the formal q-expansion f = f θ 1 := n>0 θ 1 (n)a n q n I[[q]] (where we put θ 1 (n) = 0 whenever p n) is such that, for every X arith (I) of even weight, V is the Galois representation attached to the specialization f θ 1 of f, which by Lemma 2.3 is a p-adic modular form of weight 0 and trivial nebentypus. We next recall some of the local properties of the big Galois representation T. Let I w D w G Q be the inertia and decompositon groups at the place w p induced by our fixed embedding ı p : Q Q p. In the following we will identify D w with the absolute Galois group G Qp. Then by a result of Mazur and Wiles (see [Wil88, Thm ]) there exists a filtration of I[D w ]-modules (4.4) 0 F + w T T F w T 0 with F ± w T free of rank one over I and with the Galois action on F w T unramified, given by the character α : D w /I w I sending an arithmetic Frobenius σ p to a p. Twisting (4.4) by Θ 1 we define F ± w T in the natural manner. Let T := Hom I (T, I) be the contragredient 5 of T, and consider the I-module (4.5) D := (F + w T Zp Ẑ nr p ) G Qp, where F w + T := Hom I (F T, I) T, and Ẑnr p is the completion of the ring of integers of the maximal unramified extension of Q p in Q p. Fix once and for all a compatible system ζ = {ζ s } of primitive p s -th roots of unity, and denote by e ζ the basis of D dr (Q p (1)) corresponding to 1 Q p under the resulting identification D dr (Q p (1)) = Q p. Lemma 4.3. The module D is free of rank one over I, and for every X arith (I) of even weight k = 2r 2 there is a canonical isomorphism (4.6) D D dr (Q p (r )) = D dr(v f (r )) Fil 0 D dr (V f (r )). Proof. Since the action on F w + T is unramified, the first claim follows from [Och03, Lemma 3.3] in light of the definition (4.5) of D. The second can be deduced from [Och03, Lemma 3.2] as in the proof of [Och03, Lemma 3.6]. With the same notations as in Lemma 4.3, we denote by, dr the pairing (4.7), dr : D D dr (Q p (r )) Fil 1 D dr (V f (r 1)) F 5 So that T I F = Vf for every X arith (I).

19 SPECIALIZATIONS OF BIG HEEGNER POINTS 19 deduced from the usual de Rham pairing D dr (V f (r )) Fil 0 D dr (V f (r )) Fil0 D dr (V f (1 r )) F via the identification (4.6) and the isomorphism V f = Vf (k 1). Theorem 4.4 (Ochiai). Assume that the residual representation ρ fo is irreducible, fix an I-basis η of D, and set λ := a p 1. There exists an I-linear map Log (η) T : H 1 (Q p, F + w T ) I[λ 1 ] such that if Y H 1 (Q p, F + w T ) and X arith (I) has weight k = 2r 2, then (Log (η) (Y)) = ( 1)r 1 T (r 1)! (1 (4.8) pr 1 (a p) ) 1 ( 1 (ap) p r ) log Vf (r )(Y ), η dr if ϑ = 1; ( ) s 1 (ap) G(ϑ 1 ) p r 1 logs,vf (r )(Y ), η dr if ϑ 1, where log Vf (r ) (resp. log s,vf (r )) is the Bloch-Kato logarithm map for V f (r ) over Q p (resp. Q p,s := Q p (µ p s)), η Fil 1 D dr (V f (r 1)) is such that η e r ζ, η dr = 1, ϑ : Z p F is the finite order character z θ (z)z 1 r, s > 0 is such that the conductor of ϑ is p s, and G(ϑ 1 ) is the Gauss sum x mod p s ϑ 1 (x)ζs x. Proof. Let Λ(C ) := Z p [[C ]], where C is the Galois group of the cyclotomic Z p -extension of Q p, and let Λ cyc be the module Λ(C ) equipped with the natural action of G Qp on group-like elements. Also, let γ o be a topological generator of C and define I := (λ, γ o 1), seen as an ideal of height 2 inside I Zp Λ(C ) = I[[C ]]. Consider the I Zp Λ(C )-modules D := D Zp Λ(C ), F + w T := F + w T Zp Λ cyc ω k 2 2, the latter being equipped with the diagonal action of G Qp. By [Och03, Prop. 5.3] there exists an injective I Zp Λ(C )-linear map Exp F + w T : ID H 1 (Q p, F + w T ), with cokernel killed by λ, which interpolates the Bloch Kato exponential map over the arithmetic primes of I and of Λ(C ). As in (4.1), let ɛ 1/2 : C Γ Z p be the unique square-root of the wild component of the p-adic cyclotomic character ε cyc, and let Tw : I[[C ]] I[[C ]] be the I-algebra isomorphism given by Tw ([γ]) = ɛ 1/2 w (γ)[γ] for all γ C. Then letting F + w T be the I[[C ]]-module F + w T with the C -action twisted by ɛ 1/2, there is a natural projection Cor : F + w T F + w T induced by the augmentation map I[[C ]] I.

20 20 F. CASTELLA Setting D := ID Zp I[[C ]]/(ɛ 1/2 (γ o )[γ o ] 1), the composition ID (Tw ) 1 ID Exp F + w T H 1 (Q p, F + w T ) factors through an injective I-linear map ɛ 1/2 H 1 (Q p, F w + T ) Cor H 1 (Q p, F w + T ), (4.9) Exp F + w T : D H 1 (Q p, F + w T ) making, for every X arith (I) as in the statement, the diagram D Exp + F w T H 1 (Q p, F w + T ) Sp Sp D dr (F + w V f ) H 1 (Q p, F + w V f ) commutative, where the bottom horizontal arrow is given by ) ( ) 1 1 pr (1 (a p) 1 (ap) p r expv ( 1) r 1 f (r 1)! ( ) p r 1 s (a expv p) f if ϑ = 1; if ϑ 1, with exp V f D dr (V f ) Fil 0 D dr (V f ) the Bloch Kato exponential map for V f, which factors as D dr (F + w V f ) exp V f H 1 (Q p, F + w V f ) H 1 (Q p, V f ). Now if Y is an arbitrary class in H 1 (Q p, F w + T ), then λ Y lands in the image of the map Exp F + w T and so Log T (Y) := λ 1 Exp 1 F w + T (λ Y) is a well-defined element in I[λ 1 ] I D. Thus defining Log η T (Y) I[λ 1 ] by the relation the result follows. Log T (Y) = Log η T (Y) η 1, 5. The big Heegner point In this chapter we prove the main results of this paper, relating the étale Abel- Jacobi images of Heegner cycles to the specializations at higher even weights of the big Heegner point Z (whose definition is recalled below), from where a deformation of the p-adic Gross-Zagier formula of Nekovář over a Hida family follows at once. There are two key points to the proof: the properties of the big logarithm map deduced from the work of Ochiai as explained in the preceding section, and the local study of (almost all) the weight 2 specializations of Z taken up in the following.

21 SPECIALIZATIONS OF BIG HEEGNER POINTS Weight two specializations. Recall form Sect. 3 that K is a fixed imaginary quadratic field in which all prime factors of N split, and that N O K is a fixed cyclic N-ideal, i.e. such that O K /N = Z/NZ. We also assume that p splits in K, and let p be the prime of K above p induced by our fixed embedding ı p, and by p the other. Finally, A is a fixed elliptic curve with CM by O K defined over the Hilbert class field H of K. Let R 0 = Ẑnr p be the completion of the ring of integers of the maximal unramified extension of Q p, which we view as an overfield of H via ı p. Since p splits in K, A admits a trivialization ı A : Â Ĝm over R 0 with ı 1 A (µ p s) = A[ps ] for every s > 0. Letting α A be the cyclic N-isogeny on A with kernel A[N], the triple (A, α A, ı A ) thus defines a trivialized elliptic curve with Γ 0 (N)-level structure defined over R 0. Set A 0 := A/A[p s ] and let (A 0, α A0, ı A0 ) be the trivialized elliptic curve deduced from (A, α A, ı A ) via the projection A A 0. Let C A 0 [p s ] be any étale subgroup of order p s, and set A s := A 0 /C. Finally, let (A s, α As, ı As ) be the trivialized elliptic curve with Γ 0 (N)-level structure deduced from (A 0, α A0, ı A0 ) via the projection A 0 A s, and consider the triple (5.1) h s = (A s, α As, ı As (ζ s )), which defines an algebraic point on the modular curve X s. Write p = ( 1) p 1 2 p, and let ϑ be the unique continuous character (5.2) ϑ : G Q( p ) Z p /{±1} such that ϑ 2 = ε cyc. Notice the inclusion G Hp s G Q( p ) for any s > 0, where H p s denotes the ring class field of K of conductor p s. Lemma 5.1. The curve A s has CM by the order O p s of K of conductor p s, and the point h s is rational over L p s := H p s(µ p s). In fact we have (5.3) h σ s = ϑ(σ) h s for all σ Gal(L p s/h p s). Proof. The first assertion is clear, and immediately from the construction we also see that α As is the cyclic N-isogeny on A s with kernel A s [N O p s]. It follows that the point (5.1) gives rise to precisely the point h s X s (C) in [How07b, Eq. (4)]. The result thus follows from [loc.cit., Cor ]. If is an arithmetic prime of I, we let ψ denote its wild character, defined as the composition of : I Q p with the structure map Γ = 1 + pz p I. The nebentypus of f is then given by ε f = ψ ω k k, where ω : (Z/pZ) µ p 1 Z p is the Teichmüller character. Recall the critical characters Θ and θ from Sect. 4, and for every X arith (I) of weight 2, consider the F -valued Hecke character of K given by (5.4) χ (x) = Θ (art Q (N K/Q (x))) for all x A K. Notice that since χ has finite order, it may alternately be seen as character on G K via the Artin reciprocity map art K : A K Gab K.