Heegner points in Coleman families

We construct two-parameter analytic families of Galois cohomology classes interpolating the etale Abel--Jacobi images of generalised Heegner cycles, with both the modular form and Grossencharacter varying in p-adic families.

Our proof relies heavily on the techniques developed by two of us in the paper [LZ16] to interpolate Euler systems for GL 2 × GL 2 in Coleman families. As in op.cit., the main tool is the theory of overconvergentétale cohomology introduced by Andreatta-Iovita-Stevens [AIS15].
In the final section of the paper, we impose the additional assumption that p splits in K. We sketch a proof that the Bertolini-Darmon-Prasanna p-adic L-functions L BDP (f ) associated to specialisations f of F interpolate to a 2-parameter p-adic L-function L BDP (F), which is a rigid-analytic function onṼ ; and we prove the following explicit reciprocity law: Theorem B. The image of the class z F under the Perrin-Riou big logarithm map is the p-adic L-function L BDP (F).
1.2. Relation to earlier work. Theorem A extends earlier work of a number of authors. In particular, if F is an ordinary Coleman family (i.e. a Hida family), then Howard [How07] has constructed a family of cohomology classes z F interpolating the Heegner points at weight 2 specialisations of F. It is, however, not clear from Howard's construction whether the specialisations of the family z F at higher-weight classical points (f, χ) coincide with the generalised Heegner cycles, which is the content of Theorem A in this case.
This compatibility is known if χ has infinity-type ( k 2 , k 2 ) and p is split in K, by work of Castella [Cas19]. Castella has informed us that his method could also be made to work for χ of more general infinity-type; but the restriction to p split is fundamental, since his strategy involves deducing Theorem A from a version of Theorem B for Hida families proved in [CH18], rather than the other way around as in our approach. A direct proof of Theorem A for ordinary families, without assuming p split, can be extracted from recent work of Disegni [Dis19]; Disegni's methods are rather closer in spirit to those of the present paper.
The case of non-ordinary Coleman families does not seem to have received so much attention hitherto. The only result in this direction known to the authors is in unpublished work of Kobayashi, who has announced a construction of a one-parameter family of cohomology classes interpolating the Heegner classes for (f, χ), for f a fixed, possibly non-ordinary eigenform and varying χ (assuming p is split in K and f has trivial character). This result can be recovered from Theorem A by specialising our family z F to the fibre ofṼ over the point of V corresponding to f α .
1.3. Potential generalisations. Our result raises several natural questions which we hope will be addressed in future work. Firstly, although we have imposed a strong Heegner hypothesis which allows us to work with classical modular curves, the construction should extend straightforwardly to Heegner points arising from quaternionic Shimura curves attached to (indefinite) quaternion division algebras over Q. It should also be possible to extend the construction to Shimura curves over totally real fields; Disegni's results for ordinary families already apply in this generality, and we hope that it may be possible to extend our results to treat the non-ordinary case.
One can also consider whether the cohomology classes z F glue together for different F's. The spaceṼ is a small local piece of an eigenvariety for the quasi-split unitary group GU(1, 1) associated to K, and it is natural to ask whether the family z F can be "globalised" over the whole eigenvariety. A result of this kind for Kato's Euler system has been announced in a preprint of Hansen [Han15], and it would be interesting to generalise this to the Heegner setting.
The original motivation for this project was to study the behaviour of Heegner classes in the neighbourhood of a critical-slope Eisenstein series f . Here the eigenspace in classicalétale cohomology associated to f is 1-dimensional, and the projection of the Heegner class to this eigenspace is trivially zero; but the eigenspace in overconvergentétale cohomology is larger, and we hope that it may be possible to obtain interesting cohomology classes by projecting our families z F to these "shadow" Eisenstein eigenspaces.

Notation and conventions
2.1. Heegner pairs. We recall the setting in which Heegner points and Heegner cycles are considered, following [BDP13, §4.1]. We shall fix an integer N , and a Dirichlet character ε : (Z/N Z) × → Q × (not necessarily primitive); these will be the tame level and tame character of the modular forms we shall consider. We shall suppose that N 4 throughout this paper. (The case of N = 1, 2, 3 can be dealt with by introducing auxiliary level structure in the usual way, but we leave the details to the interested reader.) Let K be an imaginary quadratic field satisfying the classical Heegner hypothesis, that all primes q | N are split 1 in K. This hypothesis implies that there exist ideals N O K such that O K /N ∼ = Z/N Z. We shall choose such an ideal N. For any integer c coprime to N , there is an isomorphism where O c is the O K -order of conductor c; so we can regard ε as a character of (O c /N ∩ O c ) × and hence of the profinite completion O × c . Definition 2.1.1. A Heegner pair of finite type (c, N, ε) is a pair (f, χ), where • f is a normalised cuspidal newform of level Γ 1 (N ), character ε, and some weight k + 2 2; • χ is an algebraic Grössencharacter of infinity-type 2 (a, b), with a, b 0 and a + b = k, whose conductor is divisible 3 by c and whose restriction to O × c is ε −1 (where we regard ε as a character of O × c as above). Remark 2.1.2. Let N denote the Grössencharacter of ∞-type (1, 1) such that for all primes q of K, N maps a uniformizer at q to # (O K /q). Then (f, χ) is a Heegner pair precisely when the character χ · N lies in the set Σ (1) cc (N) associated to f , as defined in §4.1 of [BDP13]. We find it more convenient to work with χ rather than with χ · Nm, which has the effect of shifting the centre of the functional equation from s = 0 to s = 1.
If (f, χ) is a Heegner pair, then L(f, χ −1 , s) has a functional equation centred at s = 1, and the sign in the functional equation is −1, so L(f, χ −1 , 1) = 0. The construction of [BDP13] attaches to each Heegner pair (f, χ), and each prime p, a p-adicétale cohomology class with coefficients in the self-dual p-adic Galois representation V p (f ) * ⊗ χ. Our goal in this paper is to interpolate these p-adically as f and χ vary, taking c to be a power of p.

Coleman families.
Let p be prime not dividing N , and fix an embedding Q → Q p . For simplicity, we suppose p = 2. Recall that a p-stabilisation of a newform f ∈ S k+2 (Γ 1 (N ), ε) is a normalised eigenform of level Γ 1 (N ) ∩ Γ 0 (p) with the same Hecke eigenvalues as f away from p; such a form is uniquely determined by its U p -eigenvalue, which is a root of the polynomial X 2 − a p (f )X + p k+1 ε(p), and we write f α for the p-stabilisation associated to the root α.
Definition 2.2.1. A p-stabilised Heegner pair (f α , χ) of tame type (N, ε) consists of a Heegner pair (f, χ) of finite type (p s , N, ε), for some s 0, and a choice of p-stabilisation f α of f . These are naturally parametrised by a rigid-analytic space, as we now explain. Let L be a finite extension of Q p containing the values of ε. We write W for the 1-dimensional rigid-analytic group variety over L parametrising continuous p-adic characters of Z × p ; we consider Z as a subset of W(L) by mapping k to the character x → x k .
There is a rigid-analytic space E(N, ε) → W, the N -new, cuspidal, character ε part of the Coleman-Mazur-Buzzard eigencurve [CM98,Buz07], and p-stabilised eigenforms f α as above are naturally points of E(N, ε). (Here, as in [LZ16], our conventions are such that the fibre over k ∈ Z corresponds to overconvergent eigenforms of weight k + 2.) There is also a rigid-analytic space X (N, ε) parametrising continuous p-adic characters of A × K,f /K × whose restriction to ( O (p) K ) × agrees with the inverse of ε. Restricting characters to Z × p ⊆ O × K,p defines a morphism X (N, ε) → W. Definition 2.2.2. With the above notations, we define E K (N, ε) to be the fibre product This will be the parameter space for our p-adic families; it is clear that any p-stabilised Heegner pair of tame type (N, ε) gives a point on E K (N, ε).
1 One could also allow some primes q | N to ramify in K subject to assumptions on the epsilon-factors, as in [BDP13], but for simplicity we shall stick to the above setting.
3 Together with the condition on χ| O × c this implies that the conductor of χ is exactly cNε, where Nε is the unique factor of N whose norm is the conductor of ε.

Generalised Heegner cycles
3.1. Groups and embeddings. If τ ∈ P 1 K − P 1 Q , there is a unique embedding of Q-algebras ι : K → Mat 2×2 (Q) such that ι(K) fixes τ and acts on the corresponding line in A 2 K by the natural scalar multiplication. Clearly, the algebra embedding corresponding to g · τ is gιg −1 . We can also regard ι as an embedding of algebraic groups H → GL 2 , where H is the torus Res K/Q (G m ).
Clearly, the vector v = τ 1 is an eigenvector for ι(K × ) acting via the standard representation of GL 2 (K) on K 2 : we have ι(u) · v = uv for all u ∈ K × . On the other hand, if (K 2 ) ∨ denotes the dual of the standard representation of GL 2 (K), then v * = τ * 1 is an eigenvector for ι(K × ) acting on (K 2 ) ∨ , i.e.  Proof. Choose any τ 0 with positive imaginary part generating the abelian group quotientN −1 /Z. Then (C/Zτ 0 + Z, 1 N ) = (C/N −1 , 1 N ) = (C/N, 1) so (i) follows. Moreover, since (p, N) = 1, we know that τ 0 ⊗ 1 generates (N −1 ⊗ Z p )/Z p = (O K ⊗ Z p )/Z p so (ii) also follows. It remains to prove (iii). If p is inert or ramified in K, then at least one of τ 0 , τ 0 + 1 is a unit above p; if p is split in K then (since p = 2) at least one of τ 0 , τ 0 + 1, τ 0 + 2 is a unit above p. This gives the required τ .
Notation 3.2.2. We fix (for the remainder of this paper) a τ satisfying the conditions of the lemma, and let ι : H → GL 2 be the corresponding embedding. For m 0, we let τ m = p −m τ , and we let ι m = p −m 0 0 1 ι p m 0 0 1 be the embedding corresponding to τ m . We We let F m be the abelian extension of K corresponding to the group U N,p m = {x ∈ O p m : x = 1 mod N}, so that the pair (A m , t Am ) and the isogeny φ m are defined over F m .

Note that the endomorphism ring of
3.3. Algebraic representations. If E is any commutative ring, we write Sym k E 2 for the left representation of GL 2 (E) afforded by the space of homogenous polynomials of degree k over E in two variables X, Y , The dual of Sym k E 2 as a GL 2 (E)-representation is given by TSym k ((E 2 ) ∨ ), where (E 2 ) ∨ is the dual of the standard representation, and TSym k denotes symmetric tensors (see (see [Kin15,§2.2] or [KLZ17, §2.2] for further details).
We now assume that E is a field, and that there exists an embedding σ : K → E (and we fix such a choice). Then σ and its conjugateσ are 1-dimensional representations of H over E. Definition 3.3.1. We let G = GL 2 ×H, and for integers a, b 0, we define V a,b to be the following representation of G over E: We write δ for the diagonal embedding (ι, id) : H → G, where ι is the embedding H → GL 2 fixed in Notation 3.2.2, and similarly δ m = (ι m , id). We write δ * m for the restriction of representations from G to H via δ m . The following computation is straightforward: ) has a unique summand isomorphic to the trivial representation of H.
If ι is given by τ ∈ P 1 K as above, then we let e m = On this vector e m , the group ι m (H) acts as σ −1 . Hence, for every a, b 0, we can consider the vector where · denotes the symmetrised tensor product in the algebra TSym for each k ∈ Z 0 , where W k is a compactification of the fibre product E k of the universal elliptic curve over Y 1 (N ), W and A are certain idempotents defined in op.cit., and F m is as in Notation 3.2.2. Via pullback, we regard ∆ φm simply as a cycle on E k × A k . In the language of relative Chow motives, we have for the projection of ∆ φm to this direct summand. We can reinterpret this more slickly as follows. Let us write S m for the canonical model over K of the Shimura variety for H of level U N,p m . Since our Shimura data for both H and GL 2 are of PEL type, there is a functor from algebraic representations of G to relative Chow motives on Y 1 (N ) × S m , compatible with tensor products and duals, constructed by Ancona [Anc15,LSZ17]; and it follows from the definition of this functor that TSym k (h 1 (E)(1)) ⊗ h (a,b) (A) is the relative motive associated to V ∨ a,b . A result due to Torzewski [Tor18] shows that this functor is natural with respect to morphisms of PEL data, so we have a diagram where top arrow δ * m is restriction of algebraic representations, and the bottom arrow is pullback of relative Chow motives. For any a, b 0, let V a,b be the image of V a,b under Ancona's functor. Since S m has codimension 1 in Y 1 (N ) × S m , the pushforward (Gysin) map δ m * gives a morphism for any a, b, where the last isomorphism comes from the fact that S m is isomorphic as a K-variety to the Gal(F m /K)-orbit of τ m . The left-hand side is just (V ∨ a,b ) δm(H) , and the pushforward of our basis vector e [a,b] m is exactly the Heegner class. If L denotes a p-adic field with an embedding σ : K → L, then the above motivic cohomology groups map naturally to theirétale analogues with coefficients in L; and the p-adic realisation of V a,b is exactly the lissé etale L-sheaf on Sh G (U ) associated to the representation V a,b of G(Q p ). (We shall abuse notation a little by using the same symbol V a,b both for the relative Chow motive and itsétale realisation.) We write mot,m . The realisation map is compatible with pushforward maps, so we deduce that z Remark 3.4.1. There is a corresponding description of the realisations of the Heegner class in other cohomology theories admitting functorial realisation maps from motivic cohomology, such as absolute Hodge cohomology, or p-adic syntomic cohomology.
3.5. Projection to eigenspaces. Since Y 1 (N ) is affine, its base-change to Q has cohomological dimension 1, so the Hochschild-Serre spectral sequence gives a natural map , and σ,σ are interpreted as characters Gal(K ab /F ) → Q × p via the Artin map. As the groups H 1 (K, −) are not in general finite-dimensional, it is convenient to introduce the following abuse of notation. Let us fix a large but finite set of primes Σ containing p, all primes ramified in K, and all primes dividing N . Then all the modular curves and Kuga-Sato varieties that we consider have smooth models over the ring Z[Σ −1 ], so the elements we consider will in fact land in the finite-dimensional subspace H 1 (O K [1/Σ], −) (the cohomology unramified outside Σ). We shall abusively write H 1 (K, −) when we mean H 1 (O K [1/Σ], −) henceforth, and similarly for finite extensions of K.
Let f be any cuspidal Hecke eigenform of level Γ 1 (N ) and weight k + 2 2. Via our fixed embedding Q → Q p , we can consider the Hecke eigenvalues a (f ) as elements of Q p . We write V p (f ) for Deligne's Galois representation associated to f , which the unique isomorphism class of two-dimensional Q p -linear representations of Gal(Q/Q) on which the trace of geometric Frobenius at a prime N p is a (f ). The dual representation V p (f ) * has a canonical realisation as a quotient of theétale cohomology of Y 1 (N ), which gives a Gal(Q/Q)-equivariant projection map On the other hand, if χ is a Grössencharacter of K of infinity-type (a, b) which is trivial on U N,p m for some m, then the character Gal(K ab /K) → Q × p associated to χ restricts to σ aσb on Gal(K ab /F m ), so we have a Gal(K ab /K)-equivariant projection map pr χ : Ind K Fm σ aσb χ.
Tensoring these together we obtain a Gal(Q/K)-equivariant map If (f, χ) is a Heegner pair of finite type (p m , N, ε) and infinity-type (a, b), then H 1 et (Y 1 (N ) Q , V ∨ k (1)) ⊗ Q p has a 2-dimensional quotient that realises the dual Galois representation V p (f ) * . Similarly, Ind K Fm (σ aσb ) has a 1-dimensional quotient on which Gal(K ab /K) acts via the character associated to χ (so geometric Frobenius at a prime q Np acts as χ( q )). Thus we have a Gal(K/K)-equivariant projection Definition 3.5.1. If f is an eigenform of level N , weight k + 2 and character ε, then for m 0 and 0 j k we define If (f, χ) is a Heegner pair of finite type (p m , N, ε) and infinity-type (k − j, j), we define For applications to p-adic deformation, it is convenient to extend this by defining cohomology classes on the modular curves Y 1 (N (p n )) of level Γ 1 (N ) ∩ Γ 0 (p n ), for any n 0. If m n, then the point on Y 1 (N (p n )) corresponding to τ m is defined over F m ; using this modular curve in place of Y 1 (N ) in the above constructions, we obtain a class for (f α , χ) a p-stablised Heegner pair, in the same way as above. If f α is a p-stabilisation of a prime-to-p level eigenform f , then V p (f ) * and V p (f α ) * are isomorphic as abstract Galois representations, but realised differently as quotients of cohomology of the tower of modular curves. An explicit isomorphism between the two is given by the map (Pr α ) * :

Interpolating coefficient systems
4.1. Spaces of distributions. Let L be a finite extension of Q p , and k 0 an integer. Let T = O 2 L , considered as a left GL 2 (O L )-module via the defining representation of GL 2 ; and recall the explicit model of the GL 2 (O L )-module Sym k T described above. As we have seen, the dual of Sym k T is the module TSym k (T ∨ ) of symmetric tensors of degree k over T ∨ .  c d · f = (bZ + d) k f aZ + c bZ + d .
We consider A k,n as a Banach space in the supremum norm (as functions on p n O Cp ), with unit ball the O L -lattice A • k,n of functions whose supremum norm is 1. (4) We let D • k,n = Hom O L A • k,n , O L , and D k,n = D • k,n ⊗ L = Hom cts (A k,n , L). We equip these with the dual action of K 0 (p n ) defined by for all γ ∈ K 0 (p n ), φ ∈ D k,n , and f ∈ A k,n . Dualising the inclusions of (3) gives us K 0 (p n )equivariant moment maps mom k : D • k,n → TSym k (T ∨ ).
Note that the action (4.1.a) is well-defined, since c ∈ p n Z p and d ∈ Z × p , and hence Z → c+aZ d+bZ preserves p n O Cp .
We can extend the action of K 0 (p n ) on A k,0 to the monoid Σ 0 (p n ) ⊂ GL 2 (Q p ) generated by K 0 (p n ) together with the element p 0 0 1 , which acts on A k,n as f (Z) → f (pZ). Hence we obtain a left action on D k,n (or D • k,n ) of the monoid Σ 0 (p n ) generated by K 0 (p n ) and p −1 0 0 1 . This is compatible with the specialisations mom k .

4.2.
Interpolation over weight space. Let W denote the rigid space parameterising characters of Z × p , as above. We shall interpolate the distribution spaces D k,n over W, following [AIS15, Han15,LZ16].
More precisely, for n 1, let W n be the locus in W consisting of "n-analytic" characters, which are the characters κ such that v p (κ(1 + p) − 1) > 1 p n−1 (p−1) , where v p is the valuation such that v p (p) = 1. (Geometrically, W is a union of p − 1 discs of radius 1, and W n is the union of slightly smaller open discs inside each component of W.) If κ is n-analytic, then its restriction to each coset a + p n Z p is given by a single convergent power series.
We let U be an open disc contained in W n and defined over L. Associated to U is an algebra ] of functions on U bounded by 1, which is a local ring with maximal ideal m U and finite residue field; and a universal character κ U : Z × p → Λ × U . We define A • U,n to be the space of power series s 0 a s ( Z p n ) s with a s ∈ Λ U tending to 0 in the m U -adic topology. This space has a right action of Σ 0 (p n ), via the same formulae as before. (This relies crucially on the fact that κ U is n-analytic, which follows from the assumption that U ⊆ W n .) If we let D • U,n be the set of Λ U -linear maps A • U,n → Λ U , then D • U,n is a profinite topological Λ U -module with a left action of Σ 0 (p n ), equipped with a filtration by open Σ 0 (p n )-invariant Λ U -submodules, and with Σ 0 (p n )-equivariant moment maps Proof. Unravelling the definitions, if u ∈ K ⊗ Q p maps to a b c d ∈ GL 2 (Q p ) under ι m , then aσ(τ * m ) + c = σ(u)τ * m and bσ(τ * m ) + d = σ(u). Thus . Hence a b c d · e k,0,m = σ(u) −k e k,0,m as required. The restriction of this linear functional e k,0,m to P k = Sym k T is given by evaluation of polynomials at (X, Y ) = (σ(τ * m ), 1), which is exactly the definition of e For an open disc U ⊆ W n as above, we have a class e U,0,m defined similarly, for any m n. This is an , for a fixed j 0 and varying integers k j, via the following simple trick: we consider the tensor product of the vector e k−j,0,m ∈ (D • k−j,n ) with e [0,j] m ∈ TSym j (T ∨ ). This gives, clearly, a vector in the space D • k−j,n ⊗ TSym j T ∨ , on which ι m ((O p m ⊗ Z p ) × ) acts via σ −(k−j)σ−j ; and its image under the composite where the last map is the symmetrised tensor product, is e . This evidently interpolates over discs U , as above. More subtly, for any j 0 there is a map of Σ 0 (p n )-modules the "overconvergent projector". (This is the map denoted by δ * j in [LZ16, §5.2], but we use a different notation here to avoid conflict with the morphism denoted δ m elsewhere in this paper.) The map Π j does not preserve the natural O L -lattices in general, but its denominator is bounded by an explicit function of n and j. By construction, this vector transforms under (O p m ⊗ Z p ) × via the character σ −(κ U −j)σ−j , and its image under mom k for k ∈ U is given by Remark 4.4.2. The class e U,j,m is independent of n: more precisely, the classes with the same name for different n's are compatible under the natural maps D U,n → D U,n−1 . One can make the value of e U,j,m on f ∈ A U,n explicit using the formulae of op.cit.: it is a linear combination of the first j derivatives of f at p m τ * . However, we shall not need this explicit formula here.
The following lemma will be crucial in §5.3 below: Lemma 4.4.3. Let h 0. There is a constant C, depending on n and h but independent of m, such that where ∇ ∈ Λ U [1/p] acts in weight k as multiplication by k.
Proof. We note first that e m −ē m = p m · τ * −τ * 0 as elements of On the other hand, we know there is a constant C (depending only on n and h) such that Using the identity (cf. Lemma 5.1.5 of [LZ16]), we deduce the result.

Families of Heegner classes
5.1. Families over weight space. As above, let U be an open disc in weight space defined over L, and let κ U be the associated universal weight; and suppose that U ⊆ W n , for some n 1, so that κ U is n-analytic in the sense of §4.2.
Since D • U,n is a profinite left K 0 (p n )-module, we can interpret it as anétale sheaf on the modular curve Y 1 (N (p n )). For each m n and j 0, we can regard e U,j,m as a section e U,j,m ∈ H 0 et S m , δ * m D • U,n ⊗ σ κ U −jσj in the notation of §3.4, whose image under the moment map mom k , for any k ∈ U with k j, is theétale realisation of e [k−j,j] m . We can thus define U,m,n = (δ m ) * (e U,j,m ) ∈ H 2 et Y 1 (N (p n )) Fm , D U,n (1) ⊗ σ (κ U −j)σj , and we automatically obtain the following interpolation property: Proposition 5.1.1. For every m n, and every k ∈ U ∩ Z 0 , we have et,m,n is as defined in (3.5.a).
These classes have the following straightforward but crucial norm-compatibility property. LetŶ be the modular curve of level {g ∈ Γ 1 (N ) : g = 0 mod * p p n * }; and consider the diagram of modular curveŝ where Φ is given by the action of p 0 0 1 ∈ GL 2 (Q p ) (which corresponds to τ → τ /p on the upper half-plane). On the cohomology of the Q p -sheaves V k and V ∨ k , this diagram gives rise to two Hecke operators: the operator (pr) * • Φ * • (pr) * , which is the usual Hecke operator U p ; and the operator U p = (pr) * • (Φ −1 ) * • (pr) * , which is the dual of U p under Poincaré duality. See [KLZ17,§2.4] for further details. The action of U p makes sense on the cohomology of the sheaves D k,n , compatibly with the moment maps, since pullback by Φ −1 corresponds to acting on the sheaf by p −1 0 0 1 ∈ Σ 0 (p). Proposition 5.1.2. For all m n 1, we have U,m,n .
Proof. One computes that the CM point at levelŶ corresponding to τ m is defined over F m+1 , and its orbit under the action of Gal(F m+1 /F m ) is exactly the preimage underpr of the Γ 1 (N (p n ))-orbit of τ m . Applying the same constructions as before to this CM point onŶ gives a clasŝ U,m,n =pr * z U,m,n . Hence we have U,m,n = norm U,m+1,n . So we can conclude that (pr) U,m+1,n . Combining these last two formulae gives norm U,m,n as required.
5.2. Projection to a Coleman family. Let f α be a p-stabilised newform of some weight k 0 + 2 2, tame level N and character ε, as above. We impose the technical condition that f α be a noble eigenform in the sense of [Han15]. Then there exists an affinoid disc V k 0 , and a unique Coleman family F of eigenforms over V , specialising in weight k 0 to f α . Shrinking V if necessary, we may arrange that for every k ∈ V ∩ Z 0 , the specialisation of F at k is a p-stabilisation g δ of some level N newform g of weight k + 2.
As explained in [LZ16, §4.6], after possibly further shrinking V , we may find an open disc U ⊃ V contained in W 1 with the following property: the Galois representation has a direct summand M V (F) * which is free of rank 2 over O(V ), and interpolates the (dual) Galois representations of the classical specialisations g δ of F. See [LZ16, §4.6] for further details.
Our assumption that all classical-weight specialisations of F are classical forms implies that the image of z These have the following interpolation property: let k ∈ V ∩ Z j , and let g δ be the specialisation of F at k. Then, for any Grössencharacter χ of infinity-type (k − j, j) and finite type (p m , N, ε), the image of z F ,m under specialisation at k and projection to the

Interpolation in j.
Let F ∞ = n 1 F n , and let Γ ac = Gal(F ∞ /F 1 ) ∼ = (O p ⊗ Z p ) × /Z × p . This is an abelian group isomorphic to the additive group Z p , and κ ac = σ/σ is a character Γ ac → Q × p of infinite order. Let W ac denote the rigid-analytic variety parametrising characters of Γ ac , which is equipped with a universal character j : Γ ac → O(W ac ) × . We regard Z as a subgroup of W ac (Q p ) via the powers of κ ac , so the specialisation of j at n ∈ Z is (κ ac ) n .
Proposition 5.3.1. There exists a cohomology class Proof. Let λ 0 be such that p λ is the supremum norm of α −1 F ∈ O(V ). We choose an integer h λ , and consider the family of cohomology classes defined by is the image of the quantity h j=0 (−1) j ∇−j h−j e U,j,m considered in the lemma. Since F is cuspidal, we have H 0 (F ∞ , M V (F) * ) = 0; so we can now invoke the general construction of [LZ16,Proposition 2.3.3] to give a class which interpolates these classes, where D λ (Γ ac ) is the subspace of O(W ac ) consisting of distributions of order λ. Moreover, if we take two different values h, h , then the classes ∇ h c(h ) and ∇ h c(h) have the same specialisations at locally algebraic characters of degree up to min(h, h ) λ , so they must in fact agree. We can map c(h) into the slightly larger module H 1 . This is the sections of a torsion-free sheaf over V × W ac ; and the image of c(h) is divisible by ∇ h (because sufficiently many of its specialisations are so). Hence the quotient c(h)/ ∇ h is well-defined and independent of h, and therefore enjoys an interpolating property at characters of all weights j 0. 5.4. Re-parametrisation of weights. Since σ κ V is not well-defined as a character of Gal(Q/K) but only of Gal(Q/F 1 ), it is convenient to "re-parametrise" V × W ac , as follows.
Consider the group Γ 1 K = (Z p + pO K,p ) × . Then we have a short exact sequence of abelian profinite groups 1 → Z × p → Γ 1 K → Γ ac → 1, (which is, in fact, split, although there is no canonical splitting). There is therefore a natural map W 1 K → W, whose fibres are the orbits of W ac . Over V , this morphism admits a canonical section (since 1-analytic characters of Z × p extend to Γ 1 K ). Hence the preimage of V in W 1 K is isomorphic as a rigid-analytic space to V × W ac .
On the other hand, we have defined above a space X = X (N, ε), and restriction of characters defines a finite map X (N, ε) → W 1 K . Some careful book-keeping shows that ifṼ denotes the preimage of V × W ac in X (N, ε), then the pullback of z F ,∞ is an element of H 1 (F 1 , MṼ (F) * (kṼ )), where kṼ is the universal character Gal(K ab /K) → O(Ṽ ) × , and MṼ (F) * = O(Ṽ ) × O(V ) M V (F) * . We define z F to be the image of z F ,∞ under projection to H 1 (K, MṼ (F) * (kṼ )). This is the "universal" version of the projection to the χ-component in the definition of theétale classes, so one has the following interpolation formula: Theorem 5.4.1 (Theorem A). Let x be a point ofṼ corresponding to a p-stabilised Heegner pair (g δ , χ), with χ of infinity-type (k−j, j) and conductor p m at p. Then the specialisation of the class z F ∈ H 1 (K, MṼ (F) * (kṼ )) at x is given by can be "globalized" over the whole rigid-analytic variety E K (N, ε).

Explicit reciprocity laws in families
We now assume that p is split in K and that K has odd discriminant; and we write (p) = pp, where p is the prime corresponding to our embedding K ⊂ Q → Q p .
By a crystalline point ofṼ , we shall mean a point ofṼ corresponding to a p-stabilised Heegner pair (g δ , χ), where g δ is a p-stabilisation (with U p -eigenvalue δ) of a newform g of prime-to-p level, and χ is an algebraic Grössencharacter that is unramified above p. We write δ for the other root of the Hecke polynomial of f , so that δδ = pχ(p).
6.1. Triangulations. Let R be the Robba ring. By a result of Ruochuan Liu [Liu15, Theorem 0.3.4], we know that (after possibly shrinkingṼ slightly) the (ϕ, Γ)-module over R⊗O(Ṽ ) given by D p := D † rig (K p , MṼ (F) * (kṼ )) is trianguline: it has a natural (ϕ, Γ)-invariant rank 1 submodule F + D p . After some unravelling we find that if (g δ , χ) is a crystalline point of weight (k − j, j), then the specialisation of F + D p at (g δ , χ) is crystalline with Hodge-Tate weight j + 1 and crystalline Frobenius eigenvalue χ(p) δ = δ pχ(p) . Proposition 6.1.1. The map is injective, and loc p (z F ) lies in the image of this map.
Proof. Compare [LZ16, Theorem 7.1.2]. ShrinkingṼ slightly further if necessary, we can arrange that for every x ∈Ṽ we have H 0 (K p , F − D p,x ) = 0, where D p,x is the fibre at x. This certainly implies that H 0 (K p , F − D p ) vanishes, and also that H 1 (K p , F − D p ) is torsion-free over O(Ṽ ). The vanishing of H 0 (K p , F − D p ) implies that H 1 (K p , F + D p ) → H 1 (K p , D p ) is injective, and its image is the kernel of H 1 (K p , D p ) → H 1 (K p , F − D p ). So it suffices to show that loc p (z F ) maps to zero in the latter module; and the torsion-freeness means that it suffices to check this after specialisation at a Zariski-dense set of points of V .
The specialisations of loc p z F at crystalline points x of weight (k −j, j), with 0 j k, are in the image of theétale cycle class map; hence they lie in the Bloch-Kato H 1 f subspace (see e.g. [BDP13,§3.4]). However, if we have the stronger inequality 0 j < k, then the projection of H 1 f (K p , D p,x ) to H 1 (K p , F − D p,x ) is zero, since F − D p,x has strictly negative Hodge-Tate weight. So loc p z F must vanish at all crystalline points satisfying this inequality, and these are clearly Zariski-dense.
Proposition 6.1.2. There exists a locally-free O(Ṽ )-module D cris (F + D p ) of rank 1, and a homomorphism of O(Ṽ )-modules L : H 1 (K p , D p ) → D cris (F + D p ), with the following interpolation property: for any crystalline point P = (g δ , χ), the fibre of D cris (F + D p ) at P is canonically identified with the ϕ = χ(p) δ eigenspace in D cris (K p , V p (g δ ) * (χ)), and the fibre at P of the map L is given by where log and exp * are the Bloch-Kato logarithm and dual-exponential maps for the crystalline Galois representation V p (f ) * (χ)| G Kp .
(We assume here that P is such that the Euler factor does not vanish, which is true for almost all crystalline points P . The interpolating property at non-crystalline algebraic points can be made explicit, but we do not need this here.) If we apply this to z F , then two of the four Euler factors cancel out 4 , and we obtain an element of D cris (F + D p ) interpolating the classes To interpret this as a p-adic L-function, we need to trivialise the sheaf D cris (F + D p ) = D cris Q p , F + M V (F) * ⊗ O(V ) D cris (K p , OṼ (kṼ )) .
The first term has a canonical trivialisation (after possibly extending the coefficient field L by the N -th roots of unity), given by pairing with the family of differential forms ω F of [LZ16, Theorem 6.4.1]. The second term can only be canonically trivialised after base-extending to the field Q nr p ; to descend this to a finite extension we need to renormalise by the CM period Ω k−2j p . 4 Note that the construction of z F is symmetric in p andp (but involves a choice of p-stabilisation δ), whereas the definition of the BDP L-function depends crucially on the choice of p but is insensitive to the choice of δ. The Euler factors in the map L precisely balance out this difference.
6.2. The BDP L-function for a Coleman family. Let Ω ∞ ∈ C × and Ω p ∈ C × p be complex and p-adic CM periods, as in §5 of [BDP13]. These depend on K and p, and are only well-defined up to various scaling factors.
Theorem 6.2.1 (Bertolini-Darmon-Prasanna). Let f be a newform of level N , character ε and weight k + 2 2, and let X(k, N, ε) be the fibre of X(N, ε) above k ∈ W. There exists a rigid-analytic function L BDP p (f ) on X(k, N, ε) whose value at a crystalline character χ of infinity-type (k − j, j) is given by