Heegner cycles and higher weight specializations of big Heegner points

Abstract

Let ${\mathbf{{f}}}$ 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 $\Gamma _0(N)\cap \Gamma _1(p^s)$, Howard has constructed a canonical class $\mathfrak{Z }$ in the cohomology of a self-dual twist of the big Galois representation associated to ${\mathbf{{f}}}$. If a $p$-ordinary eigenform $f$ on $\Gamma _0(N)$ of weight $k>2$ is the specialization of ${\mathbf{{f}}}$ at $\nu$, one thus obtains from $\mathfrak{Z }_{\nu }$ a higher weight generalization of the Kummer images of Heegner points. In this paper we relate the classes $\mathfrak{Z }_{\nu }$ to the étale Abel-Jacobi images of Heegner cycles when $p$ splits in $K$.

1 Introduction

Fix a prime $p>3$ and an integer $N>4$ such that $p\not \mid N\phi (N)$. Let

$$\begin{aligned} f_o=\sum _{n>0}a_nq^n\in S_k(X_0(N)) \end{aligned}$$

be a $p$-ordinary newform of even weight $k=2r\ge 2$ 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 $\iota _\infty :\overline{{\mathbf{{Q}}}}\hookrightarrow {{\mathbf{{C}}}}$ and $\iota _p:\overline{{\mathbf{{Q}}}}\hookrightarrow {\overline{{\mathbf{{Q}}}}}_p$ that will remain fixed throughout this paper. Also let $\mathcal{O }$ denote the ring of integers of a (sufficiently large) finite extension $L/{\mathbf{{Q}}}_p$ containing all the $a_n$.

For $s>0$, let $X_s$ be the compactified modular curve of level

$$\begin{aligned} \Gamma _s:=\Gamma _0(N)\cap \Gamma _1(p^s), \end{aligned}$$

and consider the tower

$$\begin{aligned} \cdots \rightarrow X_s\xrightarrow {\alpha }X_{s-1}\rightarrow \cdots \end{aligned}$$

with respect to the degeneracy maps described on the non-cuspidal moduli by

$$\begin{aligned} (E,\alpha _E,\pi _E)\mapsto (E,\alpha _E,p\cdot \pi _E), \end{aligned}$$

where $\alpha _E$ denotes a cyclic $N$-isogeny on the elliptic curve $E$, and $\pi _E$ a point of $E$ of exact order $p^s$. The group $({\mathbf{{Z}}}/p^s{\mathbf{{Z}}})^\times$ acts on $X_s$ via the diamond operators

$$\begin{aligned} \langle d\rangle :(E,\alpha _E,\pi _E)\mapsto (E,\alpha _E,d\cdot \pi _E) \end{aligned}$$

compatibly with $\alpha$ under the reduction $({\mathbf{{Z}}}/p^s{\mathbf{{Z}}})^\times \rightarrow ({\mathbf{{Z}}}/p^{s-1}{\mathbf{{Z}}})^\times$. Set $\Gamma :=1+p{\mathbf{{Z}}}_p$. Letting $J_s$ be the Jacobian variety of $X_s$, the inverse limit of the system induced by Albanese functoriality,

$$\begin{aligned} \cdots \rightarrow \mathrm{Ta}_p(J_s)\otimes _{{\mathbf{{Z}}}_p}\mathcal{O }\rightarrow \mathrm{Ta}_p(J_{s-1})\otimes _{{\mathbf{{Z}}}_p}\mathcal{O }\rightarrow \cdots , \end{aligned}$$

(1.1)

is equipped with an action of the Iwasawa algebras $\widetilde{\Lambda }_{\mathcal{O }}:=\mathcal{O }[[{\mathbf{{Z}}}_p^\times ]]$ and

$$\begin{aligned} \Lambda _\mathcal{O }:=\mathcal{O }[[\Gamma ]]. \end{aligned}$$

Let $\mathfrak{h }_s$ be the $\mathcal{O }$-algebra generated by the Hecke operators $T_\ell$ ($\ell \not \mid Np$), $U_\ell :=T_\ell$ ($\ell \vert Np$), and the diamond operators $\langle d\rangle$ ($d\in ({\mathbf{{Z}}}/p^s{\mathbf{{Z}}})^\times$) acting on the space $S_k(X_s)$ of cusp forms of weight $k$ and level $\Gamma _s$. Hida’s ordinary projector

$$\begin{aligned} e^\mathrm{ord}:=\lim _{n\rightarrow \infty }U_p^{n!} \end{aligned}$$

defines an idempotent of $\mathfrak{h }_s$, projecting to the maximal subspace of $\mathfrak{h }_s$ where $U_p$ acts invertibly. We make each $\mathfrak{h }_s$ into a $\widetilde{\Lambda }_\mathcal{O }$-algebra by letting the group-like element attached to $z\in {\mathbf{{Z}}}_p^\times$ act as $z^{k-2}\langle z\rangle$.

Taking the projective limit with respect to the restriction maps induced by the natural inclusion $S_k(X_{s-1})\hookrightarrow S_k(X_s)$, we obtain a $\widetilde{\Lambda }_\mathcal{O }$-algebra

$$\begin{aligned} \mathfrak{h }^\mathrm{ord}:={\mathop {\leftarrow } \limits ^{\text{ lim}}_{s}}e^\mathrm{ord}\mathfrak{h }_s \end{aligned}$$

(1.2)

which can be seen to be independent of the weight $k\ge 2$ used in its construction.

After a highly influential work [16] of Hida, one can associate with $f_o$ a certain local domain $\mathbb{I }$ quotient of $\mathfrak{h }^\mathrm{ord}$, finite flat over $\Lambda _\mathcal{O }$, with the following properties. For each $n$, let ${\mathbf{{a}}}_n\in \mathbb{I }$ be the image of $T_n$ under the projection $\mathfrak{h }^\mathrm{ord}\rightarrow \mathbb{I }$, and consider the formal $q$-expansion

$$\begin{aligned} {\mathbf{{f}}}=\sum _{n>0}{\mathbf{{a}}}_nq^n\in \mathbb{I }[[q]]. \end{aligned}$$

We say that a continuous $\mathcal{O }$-algebra homomorphism $\nu :\mathbb{I }\rightarrow \overline{{\mathbf{{Q}}}}_p$ is an arithmetic prime if there is an integer $k_\nu \ge 2$, called the weight of $\nu$, such that the composition $\Gamma \rightarrow \mathbb{I }^\times \rightarrow \overline{{\mathbf{{Q}}}}_p^\times$ agrees with $\gamma \mapsto \gamma ^{k_\nu -2}$ on an open subgroup of $\Gamma$ of index $p^{s_\nu -1}\ge 1$. Denote by $\mathcal{X }_\mathrm{arith}(\mathbb{I })$ the set of arithmetic primes of $\mathbb{I }$, which will often be seen as sitting inside $\mathrm{Spf}(\mathbb{I })(\overline{{\mathbf{{Q}}}}_p)$. If $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$, $F_\nu$ will denote its residue field. Then:

• for every $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$, there exists an ordinary $p$-stabilized newform¹

  $$\begin{aligned} {\mathbf{{f}}}_\nu \in S_{k_\nu }(X_{s_\nu }) \end{aligned}$$

  such that $\nu ({\mathbf{{f}}})\in F_\nu [[q]]$ gives the $q$-expansion of ${\mathbf{{f}}}_\nu$;

• if $s_\nu =1$ and $k_\nu \equiv k\pmod {2(p-1)}$, there exists a normalized newform ${\mathbf{{f}}}_\nu ^\sharp \in S_{k_\nu }(X_0(N))$ such that

  $$\begin{aligned} {\mathbf{{f}}}_\nu (q)={\mathbf{{f}}}_\nu ^\sharp (q)-\frac{p^{k_\nu -1}}{\nu ({\mathbf{{a}}}_p)} {\mathbf{{f}}}_\nu ^\sharp (q^p); \end{aligned}$$

  (1.3)

• there exists a unique $\nu _o\in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ such that $f_o={\mathbf{{f}}}_{\nu _o}^\sharp$.

In particular, after “$p$-stabilization” (1.3), the form $f_o$ fits in the $p$-adic family ${\mathbf{{f}}}$.

Similarly for the associated Galois representation $V_{f_o}$: the continuous $\mathfrak{h }^\mathrm{ord}$-linear action of the absolute Galois group $G_{\mathbf{{Q}}}$ on the module

$$\begin{aligned} \mathbb{T }:=\mathbb{T }^\mathrm{ord}\otimes _{\mathfrak{h }^\mathrm{ord}}\mathbb{I },\quad \quad \mathrm{where} \ \mathbb{T }^\mathrm{ord}:={\mathop {\leftarrow } \limits ^{\text{ lim}}_{s}{{e}^\mathrm{ord}}}(\mathrm{Ta}_p(J_s)\otimes _{{\mathbf{{Z}}}_p}\mathcal{O }), \end{aligned}$$

(1.4)

gives rise to a “big” Galois representation $\rho _{{\mathbf{{f}}}}:G_{{\mathbf{{Q}}}}\rightarrow \mathrm{Aut}(\mathbb{T })$ such that

$$\begin{aligned} \nu (\rho _{{\mathbf{{f}}}})\cong \rho _{{\mathbf{{f}}}_\nu }^* \quad \quad \mathrm{for \ every}\, \nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I }), \end{aligned}$$

where $\rho _{{\mathbf{{f}}}_\nu }^*$ is the contragredient of the (cohomological) $p$-adic Galois representation $\rho _{{\mathbf{{f}}}_\nu }:G_{\mathbf{{Q}}}\rightarrow \mathrm{Aut}(V_{{\mathbf{{f}}}_\nu })$ attached to ${\mathbf{{f}}}_\nu$ by Deligne; in particular, one recovers $\rho _{f_o}^*$ from $\rho _{{\mathbf{{f}}}}$ by specialization at $\nu _o$.

Assume from now on that the residual representation $\bar{\rho }_{f_o}$ is irreducible; then $\mathbb T$ can be shown to be free of rank 2 over $\mathbb{I }$. (See [23, Théorème 7].) Let $K$ be an imaginary quadratic field with ring of integers $\mathcal{O }_K$ containing an ideal $\mathfrak{N }\subset \mathcal{O }_K$ with

$$\begin{aligned} \mathcal{O }_K/\mathfrak{N }\cong {\mathbf{{Z}}}/N{\mathbf{{Z}}}, \end{aligned}$$

(1.5)

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 [19] of Howard produces a compatible sequence $U_p^{-s}\cdot \mathfrak{X }_{s}$ of cohomology classes with values in a certain twist of the ordinary part of (1.1), giving rise to a canonical “big” cohomology class $\mathfrak{X }$, the big Heegner point (of conductor 1), in the cohomology of a self-dual twist $\mathbb{T }^\dagger$ of $\mathbb{T }$. Moreover, if every prime factor of $N$ splits in $K$, it follows from his results that the class

$$\begin{aligned} \mathfrak{Z }:=\mathrm{Cor}_{H/K}(\mathfrak{X }) \end{aligned}$$

lies in Nekovář’s extended Selmer group $\widetilde{H}_f^1(K,\mathbb{T }^\dagger )$. In particular, for every $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ with $s_\nu =1$ and $k_\nu \equiv k\pmod {2(p-1)}$ as above, the specialization $\mathfrak{Z }_\nu$ belongs to the Bloch-Kato Selmer group $H_{f}^1(K,V_{{\mathbf{{f}}}_\nu ^\sharp }(k_\nu /2))$ of the self-dual representation $\mathbb{T }^\dagger \otimes _{\mathbb{I }}F_\nu \cong V_{{\mathbf{{f}}}_\nu ^\sharp }(k_\nu /2)$. The classes $\mathfrak{Z }_\nu$ 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 ${\mathbf{{f}}}_\nu ^\sharp$, one has an alternate (and completely different!) method of producing such a higher weight analogue. Briefly, if $k_\nu =2r_\nu > 2$, associated to any elliptic curve $A$ with CM by $\mathcal{O }_K$, there is a null-homologous cycle $\Delta ^\mathrm{heeg}_{A,r_\nu }$, a so-called Heegner cycle, on the $(2r_\nu -1)$-dimensional Kuga–Sato variety $W_{r_\nu }$, giving rise to an $H$-rational class in the Chow group $\mathrm{CH}^{r_\nu +1}(W_{r_\nu })_0$ with ${\mathbf{{Q}}}$-coefficients. Since the representation $V_{{\mathbf{{f}}}_\nu ^\sharp }(r_\nu )$ appears in the étale cohomology of $W_{r_\nu }$:

$$\begin{aligned} H_{{{\acute{\mathrm{e}}}}\mathrm{t}}^{2r_\nu -1}(\overline{W}_{r_\nu }, {\mathbf{{Q}}}_{p})(r_\nu )\xrightarrow {\pi _{{\mathbf{{f}}}_\nu ^\sharp }} V_{{\mathbf{{f}}}_\nu ^\sharp }(r_\nu ), \end{aligned}$$

by taking the images of the cycles $\Delta ^\mathrm{heeg}_{A,r_\nu }$ under the $p$-adic étale Abel-Jacobi map

$$\begin{aligned} \Phi _{H}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}:\mathrm{CH}^{r_\nu +1}(W_{r_\nu })_0(H) \rightarrow H^1(H,H_{{{{\acute{\mathrm{e}}}}\mathrm{t}}}^{2r_\nu -1}(\overline{W}_{r_\nu },{\mathbf{{Q}}}_p)(r_\nu )) \end{aligned}$$

and composing with the map induced by $\pi _{{\mathbf{{f}}}_\nu ^\sharp }$ on $H^1$’s, we may consider the classes

$$\begin{aligned} \Phi _\mathbf{f _\nu ^\sharp ,K}^{{\acute{\mathrm{e}}}\mathrm{t}}(\Delta ^\mathrm{heeg}_{r_\nu }) :=\mathrm{Cor}_{H/K}(\pi _{{\mathbf{{f}}}_\nu ^\sharp }\Phi _{H}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Delta ^\mathrm{heeg}_{A,r_\nu })). \end{aligned}$$

By the work [28] of Nekovář, these classes are known to lie in the same Selmer group as $\mathfrak{Z }_\nu$, and the question of their comparison thus naturally arises.

Main Theorem

(Thm. 5.11) Assume that $p$ splits in $K$ and that $\mathfrak{Z }$ is not $\mathbb{I }$-torsion. Then for any $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ of weight $k_\nu =2r_\nu > 2$ with $k_\nu \equiv k\pmod {2(p-1)}$ and trivial character, we have

$$\begin{aligned} \langle \mathfrak{Z }_{\nu },\mathfrak{Z }_\nu \rangle _K =\left(1-\frac{p^{r_\nu -1}}{\nu ({\mathbf{{a}}}_p)}\right)^4 \frac{\langle \Phi _{{\mathbf{{f}}}_\nu ^\sharp ,K}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Delta ^\mathrm{heeg}_{r_\nu }), \Phi _{{\mathbf{{f}}}_\nu ^\sharp ,K}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Delta ^\mathrm{heeg}_{r_\nu })\rangle _K}{{u^2(4D)^{r_\nu -1}}}, \end{aligned}$$

where $\langle ,\rangle _K$ is the cyclotomic $p$-adic height pairing on $H_f^1(K,V_{{\mathbf{{f}}}_\nu ^\sharp }(r_\nu ))$, $u:=\vert \mathcal{O }_K^\times \vert /2$, 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 $\mathfrak{Z }$. We also note that $\mathfrak{Z }$ is conjecturally always not $\mathbb{I }$-torsion ([19, Conj. 3.4.1]), and that by [18, 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 2 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 [1], Sect. 4], 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. 3 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. 4 we deduce from the work [30] of Ochiai a “big” logarithm map that will allow as to move between different weights in the Hida family.

Finally, in Sect. 5 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. 2 thus yields, for infinitely many $\nu$ of weight 2, a formula for the $p$-adic logarithm of the localization of $\mathfrak{Z }_\nu$ in terms of certain values of a $p$-adic modular form of weight 0 associated with ${\mathbf{{f}}}_\nu$ (Thm. 5.8). When extended by $p$-adic continuity to an arithmetic prime $\nu$ of higher even weight, this expression is seen to agree with the formula from Sect. 3, and by the interpolation properties of the big logarithm map it corresponds to the $p$-adic logarithm of the localization of $\mathfrak{Z }_\nu$. 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 [5].)

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 $\Gamma _s:=\Gamma _1(Np^s)$, viewed as a scheme over $\mathrm{Spec}(\mathbf Q _p)$. Let $\pi :\mathcal{E }_s\rightarrow X_s$ be the universal generalized elliptic curve over $X_s$, and let

$$\begin{aligned} \underline{\omega }_{X_s}:=\pi _{*}\Omega ^1_{\mathcal{E }_s/X_s}(\log Z_s) \end{aligned}$$

be the pushforward of the invertible sheaf of relative differentials on $\mathcal{E }_s/X_s$ with possible log-poles along the inverse image of the cuspidal subscheme $Z_s\subset X_s$.

Algebraically, $H^0(X_s,\underline{\omega }_{X_s}^{\otimes {2}})$ gives the space of modular forms of weight 2 and level $\Gamma _s$ (defined over ${\mathbf{{Q}}}_p$). Consider the complex

$$\begin{aligned} \Omega _{X_s/{\mathbf{{Q}}}_p}^\bullet (\log {Z}_s):0\longrightarrow \mathcal{O }_{X_s}\xrightarrow {\;\;d\;\;} \Omega _{X_s/{\mathbf{{Q}}}_p}^1(\log {Z}_s)\longrightarrow 0 \end{aligned}$$

(2.1)

of sheaves on $X_s$. The algebraic de Rham cohomology of $X_s$

$$\begin{aligned} H_\mathrm{dR}^1(X_s/{\mathbf{{Q}}}_p):=\mathbb{H }^1(X_s, \Omega _{X_s/{\mathbf{{Q}}}_p}^\bullet (\log {Z_s})) \end{aligned}$$

is a finite-dimensional ${\mathbf{{Q}}}_p$-vector space equipped with a Hodge filtration

$$\begin{aligned} 0\;\subset \; H^0(X_s,\Omega _{X_s/{\mathbf{{Q}}}_p}^1(\log {Z_s}))\;\subset \; H_\mathrm{dR}^1(X_s/{\mathbf{{Q}}}_p), \end{aligned}$$

and by the Kodaira–Spencer isomorphism $\underline{\omega }_{X_s}^{\otimes 2}\cong \Omega ^1_{X_s/{\mathbf{{Q}}}_p}(\log {Z_s})$, every cusp form $f\in S_2(X_s)$ (in particular) defines a cohomology class $\omega _f\in H_\mathrm{dR}^1(X_s/{\mathbf{{Q}}}_p)$.

Let $X$ be the complete modular curve of level $\Gamma _1(N)$, also viewed over $\mathrm{Spec}({\mathbf{{Q}}}_p)$, and consider the subspaces of the associated rigid analytic space $X^\mathrm{an}$:

$$\begin{aligned} X^\mathrm{ord}\;\subset \;X_{<1/(p+1)}\;\subset \; X_{<p/(p+1)}\;\subset \;X^\mathrm{an}. \end{aligned}$$

To define these, let $\mathcal{X }_{/{\mathbf{{Z}}}_p}$ be the canonical integral model of $X$ over $\mathrm{Spec}({\mathbf{{Z}}}_p)$, and let $X_{{\mathbf{F}}_p}:=\mathcal{X }\times _{{\mathbf{{Z}}}_p}\mathbf{F}_p$ denote its special fiber. The supersingular points $SS\subset X_{{\mathbf{F}}_p}({\overline{\mathbf{F}}}_{p})$ is the finite set of points corresponding to the moduli of supersingular elliptic curves (with $\Gamma _1(N)$-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 $\underline{\omega }_{X}^{\otimes (p-1)}$. (Recall that we are assuming $p\ge 5$.) The reduction of $E_{p-1}$ to $X_{\mathbf{F}_p}$ is the Hasse invariant, which defines a section of the reduction of $\underline{\omega }_{X}^{\otimes (p-1)}$ with $SS$ as its locus of (simple) zeroes. If $x\in X(\overline{\mathbf{Q }}_p)$, let $\bar{x}\in X_\mathbf{F _p}(\overline{\mathbf{F }}_p)$ denote its reduction. Each point $\bar{x}\in SS$ is smooth in $X_{\mathbf{F}_p}$, and the ordinary locus of $X$.

$$\begin{aligned} X^\mathrm{ord}:=X^\mathrm{an}\backslash \bigcup _{\bar{x}\in SS}D_{\bar{x}} \end{aligned}$$

is defined to be the complement of their residue discs $D_{\bar{x}}\subset X^\mathrm{an}$. The function $\vert E_{p-1}(x)\vert _p$ defines a local parameter on $D_{\bar{x}}$, and with the normalization $\vert {p}\vert _p=p^{-1}$, $X_{<1/(p+1)}$ (resp. $X_{<p/(p+1)}$) is defined to be complement in $X^\mathrm{an}$ of the subdiscs of $D_{\bar{x}}$ where $\vert E_{p-1}(x)\vert _p\le p^{-1/(p+1)}$ (resp. $\vert E_{p-1}(x)\vert _p\le p^{-p/(p+1)}$), for all $\bar{x}\in SS$.

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

$$\begin{aligned} \phi _0:X_{<1/(p+1)}\rightarrow X_{<p/(p+1)} \end{aligned}$$

is defined by sending $E\mapsto E/H_E$ (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 $X_{{\mathbf{F}}_p}$. (See [21, Thm. 3.1].)

For every $s>0$, the Deligne–Tate map $\phi _0$ can be iterated $s-1$ times on the open rigid subspace $X_{<p^{2-s}/(p+1)}$ of $X^\mathrm{an}$ where $\vert E_{p-1}(x)\vert _p>p^{-p^{2-s}/(p+1)}$. Letting $\alpha _s:X_s\rightarrow X$ be the map forgetting the “$\Gamma _1(p^s)$-part” of the level structure, define

$$\begin{aligned} \mathcal{W }_1(p^s)\subset X_s^\mathrm{an} \end{aligned}$$

to be the open rigid subspace of $X_s$ whose closed points correspond to triples $(E,\alpha _E,\pi _E)$ whose image under $\alpha _s$ lands inside $X_{<p^{2-s}(p+1)}$ and are such that $\pi _E$ generates the canonical subgroup of $E$ of order  $p^s$ (as in [4, Def. 3.4]).

Define $\mathcal W _2(p^s)\subset X_s^\mathrm{an}$ is the same manner, replacing $p^{2-s}/(p+1)$ by $p^{1-s}/(p+1)$ in the definition of $\mathcal W _1(p^s)$. Then we obtain a lifting of Frobenius $\phi =\phi _s$ on $X_s$ making the diagram

commutative by sending a point $x\!=\!(E,\alpha _E,\imath _E)\!\in \!\mathcal{W }_2(p^s)$, where $\imath _E\!:\!\varvec{\mu }_{p^s}\!\hookrightarrow \! E[p^s]$ is an embedding giving the $\Gamma _1(p^s)$-level structure on $E$, to $x^{\prime }=(\phi _0E,\phi _0\alpha _E,\imath _E^{\prime })$, where $\imath _E^{\prime }$ is determined by requiring that $\alpha _s(x^{\prime })$ lands in $X_{<p^{2-s}/(p+1)}$ and for each $\zeta \in \varvec{\mu }_{p^s}-\{1\}$, $\imath _E^{\prime }(\zeta )=\phi _0Q$ if $\imath _E(\zeta )=pQ$. (Cf. [11, Sect. B.2].)

Let $k\in {\mathbf{{Z}}}$, and denote by $\underline{\omega }_{X_s^\mathrm{an}}$ the rigid analytic sheaf on $X_s^\mathrm{an}$ deduced from $\underline{\omega }_{X_s}$. Let $I_s:=\{v\in \mathbf Q :0\le v< p^{2-s}/(p+1)\}$, and for $p^{-v}\in I_s$ define the affinoid subdomain $X_s(v)$ of $X_s^\mathrm{an}$ inside $\mathcal W _1(p^s)$ whose closed points $x$ satisfy $\vert E_{p- 1}(x)\vert _p\ge p^{-v}$. Then $X_s(0)$ is the connected component of the ordinary locus of $X_s$ containing the cusp $\infty$. The space of $p$ -adic modular forms of weight $k$ and level $\Gamma _s$ (defined over ${\mathbf{{Q}}}_p$) is the $p$-adic Banach space

$$\begin{aligned} M_k^\mathrm{ord}(X_s):=H^0(X_s(0),\underline{\omega }_{X_s^\mathrm{an}}^{\otimes k}), \end{aligned}$$

and the space of overconvergent  $p$ -adic modular forms of weight $k$ and level $\Gamma _s$ is the $p$-adic Fréchet space

$$\begin{aligned} M_k^\mathrm{rig}(X_s):={\mathop {\leftarrow } \limits ^{\text{ lim}}_{v}} H^0(X_s(v),\underline{\omega }_{X_s^\mathrm{an}}^{\otimes k}), \end{aligned}$$

where the limit is with respect to the natural restriction maps as $v\in I_s$ increasingly approaches $p^{2-s}/(p+1)$. By restriction, a classical modular form in $H^0(X_s,\underline{\omega }_{X_s}^{\otimes k})$ 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 $({\mathbf{{Z}}}/p^s{\mathbf{{Z}}})^\times$ on the spaces of $p$-adic modular forms which agrees with the action on $H^0(X_s,\underline{\omega }_{X_s}^{\otimes k})$ under restriction.

We say that a ring $R$ is a $p$ -adic ring if the natural map $R\rightarrow {\mathop {\leftarrow } \limits ^{\text{ lim}}} R/p^nR$ is an isomorphism. For varying $s>0$, the data of a compatible sequence of embeddings $\varvec{\mu }_{p^s}\hookrightarrow E$ as $R$-group schemes, amounts to the data of an embedding $\varvec{\mu }_{p^\infty }\hookrightarrow E[p^\infty ]$ of $p$-divisible groups, and also to the given of a trivialization of $E$ over $R$, i.e. an isomorphism

$$\begin{aligned} \imath _E:\hat{E}\rightarrow \hat{\mathbf{G}}_m \end{aligned}$$

of the associated formal groups. The space $\mathbf{M}(N)$ of Katz  $p$-adic modular functions of tame level $N$ (over ${\mathbf{{Z}}}_p$) is the space of functions $f$ on trivialized elliptic curves with $\Gamma _1(N)$-level structure over arbitrary $p$-adic rings, assigning to the isomorphism class of a triple $(E,\alpha _E,\imath _E)$ over $R$ a value $f(E,\alpha _E,\imath _E)\in R$ 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 $\mathbf M (N)\widehat{\otimes }_\mathbf{Z _p}R$ which will also be denoted by $\mathbf M (N)$ with an abuse of notation.

The action of $z\in {\mathbf{{Z}}}_p^\times$ on a trivialization gives rise to an action of ${\mathbf{{Z}}}_p^\times$ on $\mathbf{M}(N)$:

$$\begin{aligned} \langle z\rangle f(E,\alpha _E,\imath _E):=f(E,\alpha _E,z\cdot \imath _E), \end{aligned}$$

and given a character $\chi \in \mathrm{Hom}_\mathrm{cont}({\mathbf{{Z}}}_p^\times ,R^\times )$, we say that $f\in \mathbf{M}(N)$ has weight-nebentypus $\chi$ if $\langle z\rangle f=\chi (z)f$ for all $z\in {\mathbf{{Z}}}_p^\times$. If $k$ is an integer, denoting by $z^k$ the $k$-th power character on ${\mathbf{{Z}}}_p^\times$, the subspace $M_k^\mathrm{ord}(Np^s,\varepsilon )$ of $M_k^\mathrm{ord}(X_s)$ consisting of $p$-adic modular forms with nebentypus $\varepsilon :({\mathbf{{Z}}}/p^s{\mathbf{{Z}}})^\times \rightarrow R^\times$ can be recovered as

$$\begin{aligned} M_k^\mathrm{ord}(Np^s,\varepsilon )\cong \{f\in \mathbf{M}(N):\langle z\rangle f=z^k\varepsilon (z)f,\quad \mathrm{for \ all}\,\, z\in {\mathbf{{Z}}}_p^\times \}. \end{aligned}$$

(2.2)

Since it will play an important role later, we next recall from [14, 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 $(E,\alpha _E,\imath _E)$ be a trivialized elliptic curve with $\Gamma _1(N)$-level structure over $R$. For each $s>0$, consider the quotient $E_0:=E/\imath _E^{-1}(\varvec{\mu }_{p^s})$, and let $\varphi _0:E\rightarrow E_0$ denote the projection. Since $p\not \mid N$, $\varphi _0$ induces a $\Gamma _1(N)$-level structure $\alpha _{E_0}$ on $E_0$, and since $\mathrm{ker}(\varphi _0)\cong \varvec{\mu }_{p^s}$, the dual $\check{\varphi }_0:E_0\rightarrow E$ is étale, inducing an isomorphism of the associated formal groups. Thus (with a slight abuse of notation) $\imath _{E_0}:=\imath _E\circ \check{\varphi }_0: \hat{E}_0\xrightarrow {\sim }\hat{\mathbf{G}}_m$ is a trivialization of $E_0$, and since we have an embedding $\jmath :{\mathbf{{Z}}}/p^s{\mathbf{{Z}}}\cong \mathrm{ker}(\check{\varphi }_0)\hookrightarrow E_0[p^s]$, we deduce an isomorphism

$$\begin{aligned} E_0[p^s]\cong \varvec{\mu }_{p^s}\oplus {\mathbf{{Z}}}/p^s{\mathbf{{Z}}}\end{aligned}$$

which we use to bijectively attach a $p^s$-th root of unity $\zeta _{C}$ to every étale subgroup $C\subset E_0[p^s]$ of order $p^s$, in such a way that 1 is attached to $\mathrm{ker}(\check{\varphi }_0)$.

Now for $f\in \mathbf{M}(N)$ and $a\in {\mathbf{{Z}}}_p$, define $f\otimes \mathbb 1 _{a+p^s{\mathbf{{Z}}}_p}$ to be the rule on trivialized elliptic curves given by

$$\begin{aligned} f\otimes \mathbb 1 _{a+p^s{\mathbf{{Z}}}_p}(E,\alpha _E,\imath _E) =\frac{1}{p^s}\sum _{C}\zeta _C^{-a}\cdot f(E_0/C,\alpha _C,\imath _C) \end{aligned}$$

(2.3)

where the sum is over the étale subgroups $C\subset E_0[p^s]$ of order $p^s$, and where $\alpha _C$ (resp. $\imath _C$) denotes the $\Gamma _1(N)$-level structure (resp. trivialization) on the quotient $E_0/C$ naturally induced by $\alpha _{E_0}$ (resp. $\imath _{E_0}$).

Lemma 2.1

The assignment $a+p^s{\mathbf{{Z}}}_p\rightsquigarrow \left(f\mapsto f\otimes \mathbb 1 _{a+p^s{\mathbf{{Z}}}_p}\right)$ gives rise to an $\mathrm{End}_R\mathbf{M}(N)$-valued measure $\mu _\mathrm{Gou}$ on ${\mathbf{{Z}}}_p$.

Proof

Let $\sum _na_nq^n$ be the $q$-expansion of $f$, i.e. the value that it takes at the triple $(\mathrm{Tate}(q),\alpha _\mathrm{can},\imath _\mathrm{can})=(\mathbf{G}_m/q^{\mathbf{{Z}}},\zeta _N,\varvec{\mu }_{p^\infty } \hookrightarrow \mathbf{G}_m/q^{\mathbf{{Z}}})$ over the $p$-adic completion of ${R((q))}$. By the $q$-expansion principle, the claim follows immediately from the equality

$$\begin{aligned} f\otimes \mathbb 1 _{a+p^s{\mathbf{{Z}}}_p}(q)=\sum _{n\equiv a\;\mathrm{mod}\;{p^s}}a_nq^n, \end{aligned}$$

which is shown by adapting the arguments in [14, p. 102]. $\square$

Definition 2.2

(Gouvêa) Let $f\in \mathbf{M}(N)$ and $\chi :{\mathbf{{Z}}}_p\rightarrow R$ be any continuous multiplicative function. The twist of $f$ by $\chi$ is

$$\begin{aligned} f\otimes \chi :=\left(\int _{{\mathbf{{Z}}}_p}\chi (x)d\mu _\mathrm{Gou}(x)\right) (f)\in \mathbf{M}(N). \end{aligned}$$

This operation is compatible with the usual character twist of Hecke eigenforms:

Lemma 2.3

Let $\chi :{\mathbf{{Z}}}_p^\times \rightarrow R^\times$ be a continuous character extended by zero on $p{\mathbf{{Z}}}_p$. If $f\in \mathbf{M}(N)$ has $q$-expansion $\sum _na_nq^n$, then $f\otimes \chi$ has $q$-expansion $\sum _n\chi (n)a_nq^n$, and if $f$ has weight-nebentypus $\kappa \in \mathrm{Hom}_\mathrm{cts}({\mathbf{{Z}}}_p^\times ,R^\times )$, then $f\otimes \chi$ has weight-nebentypus $\chi ^2\kappa$.

Proof

See [14], Cor. III.6.8.$i$] and [14, Cor. III.6.9]). $\square$

In particular, twisting by the identity function of ${\mathbf{{Z}}}_p$ we obtain an operator $d:\mathbf{M}(N)\rightarrow \mathbf{M}(N)$ whose effect on $q$-expansions is $q\frac{d}{dq}$. For every $k\in {\mathbf{{Z}}}$, we see from (2.2) and Lemma 2.3, that this restricts to a map

$$\begin{aligned} d:M_k^\mathrm{ord}(X_s)\rightarrow M_{k+2}^\mathrm{ord}(X_s) \end{aligned}$$

which increases the weight by 2 and preserves the nebentypus. Moreover, for $k=0$, the arguments in [9, Prop. 4.3] can be adapted to show that $d$ restricts to a linear map $M_0^\mathrm{rig}(X_s)\rightarrow M_{2}^\mathrm{rig}(X_s)$, viewing $M_k^\mathrm{rig}(X_s)\hookrightarrow M_k^\mathrm{ord}(X_s)$ by restriction.

2.2 Comparison isomorphisms

Let $\zeta _s$ be a primitive $p^s$-th root of unity, and let ${F}$ be a finite extension of ${\mathbf{{Q}}}_p(\zeta _s)$ over which $X_s$ acquires stable reduction, i.e. such that the base extension $X_s\times _{{\mathbf{{Q}}}_p}F$ admits a stable model over the ring of integers $\mathcal{O }_{F}$ of ${F}$. For the ease of notation, from now on we will denote $X_s\times _{{\mathbf{{Q}}}_p}{F}$ (as well as the associated rigid analytic space) simply by $X_s$.

Let ${\fancyscript{X}}_s$ be the minimal regular model of $X_{s}$ over $\mathcal{O }_{F}$, and denote by $F_0$ the maximal unramified subfield of ${F}$. The work [17] of Hyodo–Kato endows the ${F}$-vector space $H_\mathrm{dR}^1(X_s/{F})$ with a canonical $F_0$-structure

$$\begin{aligned} H_\mathrm{log-cris}^1({\fancyscript{X}}_s) \hookrightarrow H_\mathrm{dR}^1(X_s/{F}) \end{aligned}$$

(2.4)

equipped with a semi-linear Frobenius operator $\varphi$.

After the proof [33] 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

$$\begin{aligned} V_s:=H_{{{{\acute{\mathrm{e}}}}\mathrm{t}}}^1(\overline{X}_s,{\mathbf{{Q}}}_p). \end{aligned}$$

(2.5)

More precisely, since $X_s$ has semistable reduction, $V_s$ is semistable in the sense of Fontaine, and there is a canonical isomorphism $D_\mathrm{st}(V_s)\xrightarrow {} H_\mathrm{log-cris}^1({\fancyscript{X}}_s)$, inducing an isomorphism

$$\begin{aligned} D_\mathrm{dR}(V_s)\xrightarrow {\sim } H_\mathrm{dR}^1(X_s/{F}) \end{aligned}$$

(2.6)

as filtered $\varphi$-modules after extension of scalars to $F$.

Consider the étale Abel-Jacobi map $\mathrm{CH}^1(X_s)_0({F})\rightarrow H^1({F},V_s(1))$ constructed in [28], which in this case agrees with the usual Kummer map

$$\begin{aligned} {\delta }_{{F}}:J_s({F})\rightarrow H^1({F},{\mathbf{{Q}}}_p\otimes \mathrm{Ta}_p(J_s)), \end{aligned}$$

where $J_s=\mathrm{Pic}^0(X_s)$ is the connected Picard variety of $X_s$. (See [loc.cit., Ex. (2.3)]).

Let $g\in S_2(X_s)$ be a newform with primitive nebentypus of $p$-power conductor, let $V_g$ the $p$-adic Galois representation attached to $g$, which is equipped with a Galois-equivariant projection $V_s\rightarrow V_g$, and let $V_g^*$ be the representation contragredient to $V_g$, so that $V_g(1)$ and $V_g^*$ are in Kummer duality. Also let $L_g$ be a finite extension of ${\mathbf{{Q}}}_p$ over which the Hecke eigenvalues of $g$ are defined. By [3, Ex. 3.11], the image of the induced composite map:

$$\begin{aligned} \delta _{g,F}:J_s({F})\xrightarrow {\delta _{{F}}}H^1({F},V_s(1))\rightarrow H^1({F},V_g(1)) \end{aligned}$$

(2.7)

lies in the Bloch-Kato “finite” subspace $H_f^1({F},V_g(1))$, and by our assumption on the nebentypus of $g$, the Bloch-Kato exponential map gives an isomorphism

$$\begin{aligned} \mathrm{exp}_{{F},V_g(1)}:\frac{D_\mathrm{dR}(V_g(1))}{\mathrm{Fil}^0D_\mathrm{dR}(V_g(1))}\rightarrow H^1_f({F},V_g(1)) \end{aligned}$$

(2.8)

whose inverse will be denoted by $\mathrm{log}_{{F},V_g(1)}$.

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 $\delta _{g,F}^{(p)}$, defined as the composition

$$\begin{aligned} \delta _{g,{F}}^{(p)}:J_s({F})\xrightarrow {{\delta }_{g,{F}}} H_f^1({F},V_g(1))\xrightarrow {\mathrm{log}_{{F},V_g(1)}}\frac{D_\mathrm{dR}(V_g(1))}{\mathrm{Fil}^0D_\mathrm{dR}(V_g(1))}\xrightarrow {\sim }(\mathrm{Fil}^0D_\mathrm{dR}(V_g^*))^\vee ,\nonumber \\ \end{aligned}$$

(2.9)

where the last identification arises from the de Rham pairing

$$\begin{aligned} \langle \;,\;\rangle :D_\mathrm{dR}(V_g(1))\times D_\mathrm{dR}(V_g^*)\rightarrow D_\mathrm{dR}({\mathbf{{Q}}}_p(1))\otimes _{{\mathbf{{Q}}}_p}L_g\cong L_g \end{aligned}$$

(2.10)

with respect to which $\mathrm{Fil}^0D_\mathrm{dR}(V_g(1))$ and $\mathrm{Fil}^0D_\mathrm{dR}(V_g^*)$ are exact annihilators of each other. A basic ingredient for this computation will be the following alternate description of the logarithm map $\mathrm{log}_{{F},V_g(1)}$.

Recall the interpretation of $H^1({F},V_g(1))$ as the space $\mathrm{Ext}_{\underline{\mathrm{Rep}}(G_{{F}})}^1({\mathbf{{Q}}}_p,V_g(1))$ of extensions of $V_g(1)$ by ${\mathbf{{Q}}}_p$ in the category of $p$-adic $G_{F}$-representations. Since ${F}$ contains ${\mathbf{{Q}}}_p(\zeta _s)$, $V_g$ 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 [26, Prop. 1.26], for example):

$$\begin{aligned} H_f^1({F},V_g(1))\cong \mathrm{Ext}_{\underline{\mathrm{Rep}}_\mathrm{cris}(G_{{F}})}^1({\mathbf{{Q}}}_p,V_g(1)). \end{aligned}$$

(2.11)

Now consider a crystalline extension

$$\begin{aligned} 0\rightarrow V_g(1)\rightarrow W\rightarrow {\mathbf{{Q}}}_p\rightarrow 0. \end{aligned}$$

(2.12)

Since $D_\mathrm{cris}(V_g(1))^{\varphi =1}=0$ by our assumptions, the resulting extension of $\varphi$-modules

$$\begin{aligned} 0\rightarrow D_\mathrm{cris}(V_g(1))\rightarrow D_\mathrm{cris}(W)\rightarrow F_0\rightarrow 0 \end{aligned}$$

(2.13)

admits a unique section $s_W^\mathrm{frob}:F_0\rightarrow D_\mathrm{cris}(W)$ with $s_W^\mathrm{frob}(1)\in D_\mathrm{cris}(W)^{\varphi =1}$. Extending scalars from $F_0$ to ${F}$ in (2.13) and taking $\mathrm{Fil}^0$-parts, we take an arbitrary section $s_W^\mathrm{fil}:{F}\rightarrow \mathrm{Fil}^0D_\mathrm{dR}(W)$ of the resulting exact sequence of ${F}$-vector spaces

$$\begin{aligned} 0\rightarrow \mathrm{Fil}^0D_\mathrm{dR}(V_g(1))\rightarrow \mathrm{Fil}^0D_\mathrm{dR}(W)\rightarrow {F}\rightarrow 0 \end{aligned}$$

(2.14)

and form the difference

$$\begin{aligned} t_W:=s_W^\mathrm{fil}(1)-s_W^\mathrm{frob}(1), \end{aligned}$$

which can be seen in $D_\mathrm{dR}(V_g(1))$, and whose image modulo $\mathrm{Fil}^0D_\mathrm{dR}(V_g(1))$ is well-defined.

Lemma 2.4

Under the identification (2.11), the above assignment

$$\begin{aligned} 0\rightarrow V_g(1)\rightarrow W\rightarrow {\mathbf{{Q}}}_p\rightarrow 0\quad \rightsquigarrow \quad t_W\;\mathrm{mod}\;\mathrm{Fil}^0D_\mathrm{dR}(V_g(1)) \end{aligned}$$

defines an isomorphism which agrees with the Bloch-Kato logarithm map

$$\begin{aligned} \mathrm{log}_{{F},V_g(1)}:H_f^1({F},V_g(1))\xrightarrow {\sim }\frac{D_\mathrm{dR}(V_g(1))}{\mathrm{Fil}^0D_\mathrm{dR}(V_g(1))}. \end{aligned}$$

Proof

See [26, Lem. 2.7], for example. $\square$

Let $\Delta \in J_s({F})$ be the class of a degree 0 divisor on $X_s$ with support contained in the finite set of points $S\subset X_s({F})$. The extension class $W=W_\Delta$ (2.12) corresponding to ${\delta }_{g,{F}}(\Delta )$ can then be constructed from the étale cohomology of the open curve $Y_s:=X_s\!\backslash \! S$, as explained in [1, Sect. 4.1]. We describe the associated $s_{W_\Delta }^\mathrm{fil}$ and $s_{W_\Delta }^\mathrm{frob}$.

By [33] (or also [13]), denoting $g$-isotypical components by the superscript $g$, there is a canonical isomorphism of $F_0\otimes _{{\mathbf{{Q}}}_p}L_g$-modules

$$\begin{aligned} D_\mathrm{cris}(V_g)\cong H_\mathrm{log-cris}^1({\fancyscript{X}}_s)^g \end{aligned}$$

(2.15)

compatible with $\varphi$-actions and inducing an ${F}\otimes _{{\mathbf{{Q}}}_p}L_g$-module isomorphism

$$\begin{aligned} D_\mathrm{dR}(V_g)\cong H_\mathrm{dR}^1(X_s/{F})^g \end{aligned}$$

(2.16)

after extension of scalars.

Writing $\Delta =\sum _{Q\in S}n_Q.Q$ for some $n_Q\in {\mathbf{{Z}}}$, we assume from now on that the reductions of the points $Q\in S$ 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 $H^1_\mathrm{dR}(X_s/{F})$, the ${F}$-vector space $H^1_\mathrm{dR}(Y_s/{F})$ is equipped with a canonical $F_0$-structure

$$\begin{aligned} H_\mathrm{log-cris}^1({\fancyscript{Y}}_s)\hookrightarrow H_\mathrm{dR}^1(Y_s/{F}), \end{aligned}$$

(2.17)

a Frobenius operator still denoted by $\varphi$, and a Hecke action compatible with that in (2.4). Thus for $W=W_\Delta$ the exact sequence (2.13) is obtained as the pullback

(2.18)

of the bottom extension of $\varphi$-modules with respect to the $F_0\otimes _{{\mathbf{{Q}}}_p}L_g$-linear map sending $1\mapsto (n_Q)_{Q\in S}$, where the subscript 0 indicates taking the degree 0 subspace.

On the other hand, after extending scalars from $F_0$ to ${F}$ and taking $\mathrm{Fil}^0$-parts, (2.14) is given by the pullback²

(2.19)

of the bottom exact sequence of free ${F}\otimes _{{\mathbf{{Q}}}_p}L_g$-modules with respect to the ${F}\otimes _{{\mathbf{{Q}}}_p} L_g$-linear map sending $1\mapsto (n_Q)_{Q\in S}$.

Let $g^*\in S_2(X_s)$ be the form dual to $g$, defined as the newform associated with the twist $g\otimes \varepsilon _g^{-1}$, and let $\omega _{g^*}\in H^0(X_s,\Omega _{X_s/{F}}^1)$ be its associated differential, so that $\mathrm{Fil}^0D_\mathrm{dR}(V_{g}^*)=\mathrm{Fil}^1D_\mathrm{dR}(V_{g^*})=({F}\otimes _{{\mathbf{{Q}}}_p}L_g).\omega _{g^*}$. Thus $\delta _{g,{F}}^{(p)}(\Delta )$ is determined by the value

$$\begin{aligned} {\delta }_{g,{F}}^{(p)}(\Delta )(\omega _{g^*})=\langle t_{W_{\Delta }},\omega _{g^*}\rangle \end{aligned}$$

(2.20)

of the pairing (2.10), which corresponds to the Poincaré pairing on $H_\mathrm{dR}^1(X_s/{F})$ 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 $\mathcal{X }_{s}$ be the canonical balanced model of $X_s$ over ${\mathbf{{Z}}}_p[\zeta _s]$ constructed by Katz and Mazur (see [22, Ch. 13]). The special fiber $\mathcal{X }_s\times _{{\mathbf{{Z}}}_p[\zeta _s]}\mathbf{F}_p$ is a reduced disjoint union of Igusa curves over $\mathbf{F}_p$ intersecting at the supersingular points. Exactly two of these components are isomorphic to the Igusa curve $\mathrm{Ig}(\Gamma _s)$ representing the moduli problem $([\Gamma _1(N)],[\mathrm{bal}.\Gamma _1(p^s)^\mathrm{can}])$ over $\mathbf{F}_p$, and we let $I_\infty$ be the one that contains the reduction of $\mathcal{W }_1(p^s)\times _{{\mathbf{{Q}}}_p}{{\mathbf{{Q}}}_p(\zeta _s)}$, and $I_0$ be the other. (These two are the two “good” components in the terminology of [24]).

By the universal property of the regular minimal model, there exists a morphism

$$\begin{aligned} {\fancyscript{X}}_s\rightarrow \mathcal{X }_s\times _{{\mathbf{{Z}}}_p[\zeta _s]}\mathcal{O }_{F}\end{aligned}$$

(2.21)

which reduces to a sequence of blow-ups on the special fiber. Letting $\kappa$ be the residue field of ${F}$, define $\mathcal{W }_{\infty }\subset X_s$ (resp. $\mathcal{W }_{0}\subset X_s$) to be the inverse image under the reduction map via ${\fancyscript{X}}_s$ of the unique irreducible component of ${\fancyscript{X}}_s\times _{\mathcal{O }_{F}}\kappa$ mapping bijectively onto $I_\infty \times _{\mathbf{F}_p}\kappa$ (resp. $I_0\times _{\mathbf{F}_p}\kappa$) in $\mathcal{X }_s\times _{{\mathbf{{Z}}}_p[\zeta _s]}\kappa$ via the reduction of (2.21). Similarly, define $\mathcal{U }\subset X_s$ by considering the irreducible components of $\mathcal{X }_s\times _{{\mathbf{{Z}}}_p[\zeta _s]}\kappa$ different from $I_\infty \times _{\mathbf{F}_p}\kappa$ and $I_0\times _{\mathbf{F}_p}\kappa$. Letting $SS$ denote (the degree of) the supersingular divisor of $\mathrm{Ig}(\Gamma _s)$, it follows that $\mathcal{U }$ intersects $\mathcal{W }_{\infty }$ (resp. $\mathcal{W }_{0}$) in a union of $SS$ supersingular annuli.

Since they reduce to smooth points, the residue class $D_Q$ of each $Q\in S$ is conformal to the open unit disc $D\subset {\mathbf{{C}}}_p$. Fix an isomorphism $h_Q:D_Q\xrightarrow {\sim }D$ that sends $Q$ to 0, and for a real number $r_Q<1$ in $p^{\mathbf{{Q}}}$, denote by $\mathcal{V }_Q\subset D_Q$ the annulus consisting of the points $x\in D_Q$ with $r_Q<\vert h_Q(x)\vert _p<1$. In the same manner, we define annuli $\mathcal{V }_z$ for each $z$ in the cuspidal subscheme $Z_s\subset X_s$.

Attached to any (oriented) annulus $\mathcal{V }$, there is a $p$ -adic annular residue map

$$\begin{aligned} \mathrm{Res}_{\mathcal{V }}:\Omega _\mathcal{V }^1\rightarrow {\mathbf{{C}}}_p \end{aligned}$$

defined by expanding $\omega \in \Omega ^1_\mathcal{V }$ as $\omega =\sum _{n\in {\mathbf{{Z}}}}a_nT^n\frac{dT}{T}$ for a fixed uniformizing parameter $T$ on $\mathcal{V }$ (compatible with the orientation), and setting $\mathrm{Res}_\mathcal{V }(\omega )=a_0$. This descends to a linear functional on $\Omega _\mathcal{V }^1/d\mathcal{O }_\mathcal{V }$. (Cf. [6, Lem. 2.1]).

For any basic wide-open  $\mathcal{W }$ (as in [4, p. 34]), define

$$\begin{aligned} H_\mathrm{rig}^1(\mathcal{W }):=\mathbb{H }^1(\mathcal{W },\Omega _{}^\bullet (\log {Z})) \cong {\Omega _{\mathcal{W }}^1}/{d\mathcal{O }_{\mathcal{W }}}, \end{aligned}$$

(2.22)

where $\Omega _{}^\bullet (\log {Z})$ denotes the complex of rigid analytic sheaves on $\mathcal{W }$ deduced from (2.1) by analytification and pullback, and consider the basic wide-opens

$$\begin{aligned} \widetilde{\mathcal{W }}_\infty :=\mathcal{W }_\infty \backslash \bigcup _{Q\in S}(D_Q\backslash \mathcal{V }_Q)\quad \mathrm{and}\quad \widetilde{\mathcal{W }}_0:=\mathcal{W }_0\backslash \bigcup _{Q\in S}(D_Q\backslash \mathcal{V }_Q). \end{aligned}$$

As follows from the arguments in [2, Lem. 4.4.1], the spaces $H_\mathrm{rig}^1(\widetilde{\mathcal{W }}_\infty )$ and $H_\mathrm{rig}^1(\widetilde{\mathcal{W }}_0)$ are each equipped with a natural action of the Hecke operators $T_\ell$ ($\ell \not \mid Np$) compatible with the Hecke action on $H^1_\mathrm{dR}(Y_s/{F})$ under restriction.

Lemma 2.5

• The natural restriction maps induce an isomorphism

  $$\begin{aligned} H_\mathrm{dR}^1(Y_s/{F})^g\;\xrightarrow {}\;H_\mathrm{rig}^1(\widetilde{\mathcal{W }}_{\infty })^g\oplus H_\mathrm{rig}^1(\widetilde{\mathcal{W }}_0)^g. \end{aligned}$$

• A class $\omega \in H^1_\mathrm{dR}(Y_s/{F})^g$ belongs to the natural image of $H^1_\mathrm{dR}(X_s/{F})^g$ if an only if it can be represented by a pair of differentials $(\omega _\infty ,\omega _0)\in \Omega ^1_{\widetilde{\mathcal{W }}_\infty } \times \Omega ^1_{\widetilde{\mathcal{W }}_0}$ with vanishing $p$-adic annular residues.

• If $\eta$ and $\omega$ are any two classes in $H_\mathrm{dR}^1(X_s/{F})^g$, their Poincaré pairing can be computed as

  $$\begin{aligned} \langle \eta ,\omega \rangle = \sum _{\mathcal{V }\subset \widetilde{\mathcal{W }}_\infty }\mathrm{Res}_{\mathcal{V }}(F_{\omega _\infty \vert _\mathcal{V }}\cdot \eta _{\infty }\vert _\mathcal{V })+ \sum _{\mathcal{V }\subset \widetilde{\mathcal{W }}_0}\mathrm{Res}_{\mathcal{V }}(F_{\omega _0\vert _\mathcal{V }}\cdot \eta _0\vert _\mathcal{V }), \end{aligned}$$

  (2.23)

  where for each annulus $\mathcal{V }$, $F_{\omega _\mathcal{V }}$ denotes any solution to $dF_{\omega _\mathcal{V }}=\omega _\mathcal{V }$ on $\mathcal{V }$.

Proof

By an excision argument, the first assertion is easily deduced from [10, Thm. 2.1] as in [2, Lem. 4.4.2]; the second and third are shown by adapting the arguments in [9, §5] for each of the two components, as they are proven in [7, Prop. 1.3] for $s=1$. (See also [10, §${3}$].) $\square$

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 $\phi :\mathcal{W }_2(p^s)\rightarrow \mathcal{W }_1(p^s)$ described in Sect. 2.1, where $\mathcal{W }_i(p^s)$ are the strict neighborhoods of the connected component $X_s(0)$ of the ordinary locus of $X_s$ containing the cusp $\infty$ described there. Recall also the wide open space $\mathcal W _\infty$ described in the preceding section, which also contains $X_s(0)$ by construction.

Proposition 2.6

(Coleman) Let $g=\sum _{n>0}b_nq^n\in S_2(X_s)$ be a normalized newform with primitive nebentypus of $p$-power conductor, so that $b_p$ is such that $U_pg=b_pg$. Then there exists a locally analytic function $F_{\omega _g}$ on $\mathcal{W }_{\infty }$ which is unique up to a constant on $\mathcal{W }_\infty$ and such that

• $dF_{\omega _g}=\omega _{g}$ on ${\mathcal{W }_\infty }$, and

• $F_{\omega _g}-\frac{b_p}{p}\phi ^*F_{\omega _g} \in M_0^\mathrm{rig}(X_s).$

Proof

This follows from the general result of Coleman [8, Thm. 10.1]. Indeed, a computation on $q$-expansions shows that the action of the Frobenius lift $\phi$ on differentials agrees with that of $pV$, with $V$ the map sending $q\mapsto q^p$, in the sense that $\phi ^*\omega _g=p\omega _{Vg}$ on $\mathcal{W }_\infty ^{\prime }:=\phi ^{-1}(\mathcal{W }_\infty \cap \mathcal{W }_1(p^s))$. Since the differential $\omega _{g^{[p]}}=\omega _{g}-b_p\omega _{Vg}$ attached to

$$\begin{aligned} g^{[p]}=\sum _{(n,p)=1}b_nq^n \end{aligned}$$

becomes exact upon restriction to $\mathcal{W }_\infty ^{\prime }$, this shows that the polynomial $L(T)=1-\frac{b_p}{p}T$ is such that

$$\begin{aligned} L(\phi ^*)\omega _g=0. \end{aligned}$$

Finally, since $g$ has primitive nebentypus, $b_p$ has complex absolute value $p^{1/2}$, and hence [8, Thm. 10.1] can be applied with $L(T)$ as above. $\square$

Attached to a primitive $p^s$-th root of unity $\zeta$, there is an automorphism $w_\zeta$ of $X_s$ which interchanges the components $\mathcal{W }_\infty$ and $\mathcal{W }_0$ (see [2, Lem. 4.4.3]).

Corollary 2.7

Define $\phi ^{\prime }:=w_\zeta \circ \phi \circ w_\zeta$. With hypotheses as in Proposition 2.6, there exists a unique locally analytic function $F_{\omega _g}^{\prime }$ on $\mathcal{W }_{0}$ which vanishes at 0, satisfies $dF_{\omega _g}^{\prime }=\omega _{g}$ on $\mathcal{W }_0$, and $F_{\omega _g}^{\prime }-\frac{b_p}{p}(\phi ^{\prime })^*F_{\omega _g}^{\prime }$ is rigid analytic on a wide-open neighborhood $\mathcal{W }_0^{\prime }$ of $w_\zeta X_s(0)$ in $\mathcal{W }_0$.

Proof

Proposition 2.6 applied to the differential $\omega _g^{\prime }:=w_\zeta ^*\omega _g$ gives the existence of a locally analytic function $F_{\omega _g^{\prime }}$ with $F_{\omega _g}^{\prime }:=w_\zeta ^*F_{\omega _g^{\prime }}$ having the desired properties. The uniqueness of $F_{\omega _g}^{\prime }$ follows immediately from that of $F_{\omega _g^{\prime }}$. $\square$

We refer to the locally analytic function $F_{\omega _g}$ (resp. $F_{\omega _g}^{\prime }$) appearing in Proposition 2.6 as the Coleman primitive of $g$ on $\mathcal{W }_\infty$ (resp. $\mathcal{W }_0$). Let $g=\sum _{n>0}b_nq^n$ be as in Proposition 2.6. The $q$-expansion $\sum _{(n,p)=1}\frac{b_n}{n}q^n$ corresponds to a $p$-adic modular form $g^{\prime }$ vanishing at $\infty$ and satisfying $dg^{\prime }=g^{[p]}$, where $d$ is the operator described at the end of Section 2.1, which here corresponds to the differential operator $\mathcal O _\mathcal{W }\rightarrow \Omega _\mathcal{W }^1$ for any subspace $\mathcal W \subset X_s$. Set $d^{-1}g^{[p]}:=g^{\prime }$.

Corollary 2.8

If $F_{\omega _g}$ is the Coleman primitive of $g$ on $\mathcal{W }_\infty$ which vanishes at $\infty$, then

$$\begin{aligned} F_{\omega _g}-\frac{b_p}{p}\phi ^*F_{\omega _g}=d^{-1}g^{[p]}. \end{aligned}$$

Proof

Since $d^{-1}g^{[p]}$ is an overconvergent rigid analytic primitive of $\omega _{g^{[p]}}$, and the operator $L(\phi ^*)=1-\frac{b_p}{p}\phi ^*$ acting on the space of locally analytic functions on $\mathcal{W }^{\prime }_\infty$ is invertible, we see that $L(\phi ^*)^{-1}(d^{-1}g^{[p]})$ satisfies the defining properties of $F_{\omega _g}$. Since $d^{-1}g^{[p]}$ vanishes at $\infty$, the result follows. $\square$

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 $g\in S_2(X_s)$ be a normalized newform with primitive nebentypus of $p$-power conductor, let $P$ be an ${F}$-rational point of $X_s$ factoring through $X_s(0)\subset X_s$, and let $\Delta \in J_s({F})$ be the divisor class of $(P)-(\infty )$. Then

$$\begin{aligned} \delta ^{(p)}_{g,{F}}(\Delta )(\omega _{g^*})=F_{\omega _{g^*}}(P), \end{aligned}$$

(2.24)

where $F_{\omega _{g^*}}$ is the Coleman primitive of $\omega _{g^*}$ on $\mathcal{W }_\infty$ which vanishes at $\infty$.

Proof

By (2.20), we must compute $\langle t_{W_\Delta },\omega _{g^*}\rangle = \langle s_{W_\Delta }^\mathrm{fil},\omega _{g^*}\rangle - \langle s_{W_\Delta }^\mathrm{frob},\omega _{g^*}\rangle$, where

• $s_{W_\Delta }^\mathrm{fil}\in \mathrm{Fil}^1D_\mathrm{dR}(W_\Delta )$ is such that $\rho (s_{W_\Delta }^\mathrm{fil})=1$ in (2.19), and

• $s_{W_\Delta }^\mathrm{frob}\in D_\mathrm{cris}(W_\Delta )^{\varphi =1}$ is such that $\rho (s_{W_\Delta }^\mathrm{frob})=1$ in (2.18).

By Lemma 2.5, we see that these can be represented, respectively, by

• $\eta _{\Delta }^\mathrm{fil}$ a section of $\Omega _{X_s/{F}}^1$ over $Y_s$ with simple poles at $P$ and $\infty$ and with

  • $\mathrm{Res}_{P}(\eta _{\Delta }^\mathrm{fil})=1$, while $\mathrm{Res}_{\infty }(\eta _{\Delta }^\mathrm{fil})=0$ for all $Q\in S-\{P\}$;

  • $\mathrm{Res}_{\infty }(\eta _{\Delta }^\mathrm{fil})=-1$, while $\mathrm{Res}_{z}(\eta _{\Delta }^\mathrm{fil})=0$ for all $z\in Z_s-\{\infty \}$,

• $\eta _{\Delta }^\mathrm{frob}=(\eta _{\infty }^\mathrm{frob},\eta _{0}^\mathrm{frob}) \in \Omega _{\widetilde{\mathcal{W }}_\infty }^1\times \Omega _{\widetilde{\mathcal{W }}_0}^1$ with

  • $(\phi ^*\eta _{\infty }^\mathrm{frob},(\phi ^{\prime })^*\eta _{\Delta }^\mathrm{frob})=(p\cdot \eta _{\infty }^\mathrm{frob}+dG_\infty ,p\cdot \eta _{0}^\mathrm{frob}+dG_0)$ with $G_\infty$ and $G_0$ rigid analytic on $\phi ^{-1}\widetilde{\mathcal{W }}_\infty$ and $(\phi ^{\prime })^{-1}\widetilde{\mathcal{W }}_0$, respectively;

  • $\mathrm{Res}_\mathcal{V }(\eta _\Delta ^\mathrm{frob})=0$ for all supersingular annuli $\mathcal{V }$; and

  • $\mathrm{Res}_{\mathcal{V }_Q}(\eta _\Delta ^\mathrm{frob})=\mathrm{Res}_Q(\eta _\Delta ^\mathrm{fil})$ ($Q\in S$), $\mathrm{Res}_{\mathcal{V }_z}(\eta _\Delta ^\mathrm{frob})=\mathrm{Res}_z(\eta _\Delta ^\mathrm{fil})$ ($z\in Z_s$).

The arguments in [1, Prop. 3.21] can now be straightforwardly adapted to deduce the result. Indeed, using the defining properties of the Coleman primitives $F_{\omega _{g^*}}$ and $F_{\omega _{g^*}}^{\prime }$ of $\omega _{g^*}$ on $\mathcal{W }_\infty$ and $\mathcal{W }_0$, respectively, one first shows that

$$\begin{aligned} \sum _{\mathcal{V }\subset \widetilde{\mathcal{W }}_\infty }\mathrm{Res}_\mathcal{V }(F_{\omega _{g^*}}\cdot \eta _\infty ^\mathrm{frob})=0\quad \mathrm{and}\quad \sum _{\mathcal{V }\subset \widetilde{\mathcal{W }}_0}\mathrm{Res}_\mathcal{V }(F_{\omega _{g^*}}^{\prime }\cdot \eta _0^\mathrm{frob})=0 \end{aligned}$$

(2.25)

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

$$\begin{aligned} \sum _{\mathcal{V }\subset \widetilde{\mathcal{W }}_\infty }\mathrm{Res}_\mathcal{V }(F_{\omega _{g^*}}\cdot \eta _\Delta ^\mathrm{fil}) \!=\!F_{\omega _{g^*}}(P) \quad \mathrm{and}\quad \sum _{\mathcal{V }\subset \widetilde{\mathcal{W }}_0}\mathrm{Res}_\mathcal{V }(F_{\omega _{g^*}}^{\prime }\cdot \eta _\Delta ^\mathrm{fil})=0.\qquad \end{aligned}$$

(2.26)

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 $\widetilde{\mathcal{W }}_\infty$ and the supersingular annuli in $\widetilde{\mathcal{W }}_0$), the result follows. $\square$

3 Generalised Heegner cycles

Let $X_1(N)$ be the compactified modular curve of level $\Gamma _1(N)$ defined over ${\mathbf{{Q}}}$, and let $\mathcal{E }$ 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 variety³, defined as the canonical desingularization of the $(2r-2)$-nd fiber product of $\mathcal{E }$ with itself over $X_1(N)$. By construction, the variety $W_r$ is equipped with a proper morphism

$$\begin{aligned} \pi _r:W_r\rightarrow X_1(N) \end{aligned}$$

whose fibers over a noncuspidal closed point of $X_1(N)$ corresponding to an elliptic curve $E$ with $\Gamma _1(N)$-level structure is identified with $2r-2$ copies of $E$. (For a more detailed description, see [1, Sect. 3.1].)

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 $\mathcal{O }_K$ the ring of integers of $K$, and note that by this assumption we may choose an ideal $\mathfrak{N }\subset \mathcal{O }_K$ with $\mathcal{O }_K/\mathfrak{N }\cong {\mathbf{{Z}}}/N{\mathbf{{Z}}}$ that we fix once and for all.

Let $A$ be a fixed elliptic curve with CM by $\mathcal{O }_K$. The pair $(A,A[\mathfrak{N }])$ defines a point $P_A$ on $X_0(N)_{}$ rational over $H$, the Hilbert class field of $K$. Choose one of the square-roots $\sqrt{-D}\in \mathcal{O }_K$, let $\Gamma _{\sqrt{-D}}\subset A\times A$ be the graph of $\sqrt{-D}$, and define

$$\begin{aligned} \Upsilon _{A,r}^\mathrm{heeg}:=\Gamma _{\sqrt{-D}}\times \overset{(r-1)}{\cdots } \times \Gamma _{\sqrt{-D}} \end{aligned}$$

viewed inside $W_r$ by the natural inclusion $(A\times A)^{r-1}\rightarrow W_r$ as the fiber of $\pi _r$ over a point on $X_1(N)$ lifting $P_A$. Let $\epsilon _W$ be the projector from [1, (2.1.2)], and set

$$\begin{aligned} \Delta ^\mathrm{heeg}_{A,r}:=\epsilon _W\Upsilon _{A,r}^\mathrm{heeg}, \end{aligned}$$

(3.1)

which is an $(r-1)$-dimensional null-homologous cycle on $W_r$ defining an $H$-rational class in the Chow group $\mathrm{CH}^r(W_r)_0$ (taken with ${\mathbf{{Q}}}$-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 [15]) many of their arithmetic properties (see [25, 27, 35]).

We next recall a variation of the previous construction introduced in the recent work [1] of Bertolini–Darmon–Prasanna. Let $A$ be the CM elliptic curve fixed above, and consider the variety⁴

$$\begin{aligned} X_r:=W_r\times A^{2r-2}. \end{aligned}$$

For each class $[\mathfrak a ]\in \mathrm{Pic}(\mathcal{O }_K)$, represented by an ideal $\mathfrak a \subset \mathcal{O }_K$ prime to $N$, let $A_\mathfrak a :=A/A[\mathfrak a ]$ and denote by $\varphi _\mathfrak a$ the degree $\mathrm{N}\mathfrak a$-isogeny

$$\begin{aligned} \varphi _\mathfrak a :A\rightarrow A_\mathfrak a . \end{aligned}$$

The pair $\mathfrak a *(A,A[\mathfrak{N }]):=(A_\mathfrak a ,A_\mathfrak a [\mathfrak{N }])$ defines a point $P_{A_\mathfrak a }$ in $X_0(N)$ rational over $H$. Let $\Gamma _{\varphi _\mathfrak a }^t\subset A_\mathfrak a \times A$ be the transpose of the graph of $\varphi _\mathfrak a$, and set

$$\begin{aligned} \Upsilon ^\mathrm{bdp}_{\varphi _\mathfrak a ,r}:=\Gamma _{\varphi _\mathfrak a }^t \times \overset{(2r-2)}{\cdots }\times \Gamma _{\varphi _\mathfrak a }^t\; \subset \;(A_\mathfrak a \times A)^{2r-2}=A_\mathfrak{a }^{2r-2}\times A^{2r-2}\xrightarrow {(\imath _\mathfrak a ,\mathrm{id}_A)} X_r, \end{aligned}$$

where $\imath _\mathfrak a$ is the natural inclusion $A_\mathfrak{a }^{2r-2}\rightarrow W_r$ as the fiber of $\pi _r$ over a point on $X_1(N)$ lifting $P_{A_\mathfrak a }$. Letting $\epsilon _A$ be the projector from [1, (1.4.4)], the cycles

$$\begin{aligned} \Delta _{\varphi _\mathfrak a ,r}^\mathrm{bdp}:=\epsilon _A\epsilon _W\Upsilon _{\varphi _\mathfrak a ,r}^\mathrm{bdp} \end{aligned}$$

(3.2)

define classes in $\mathrm{CH}^{2r-1}(X_r)_0(H)$ 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 $g\in S_{2r}(X_0(N))$ 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

$$\begin{aligned} H_{{{{\acute{\mathrm{e}}}}\mathrm{t}}}^{4r-3}(\overline{X}_r,{\mathbf{{Q}}}_p(2r-1))\longrightarrow H_{{{{\acute{\mathrm{e}}}}\mathrm{t}}}^{2r-1}(\overline{W}_r,{\mathbf{{Q}}}_p(1)) \otimes \mathrm{Sym}^{2r-2}H^1_{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\overline{A},{\mathbf{{Q}}}_p(1)), \end{aligned}$$

which composed with the natural Galois-equivariant projection

$$\begin{aligned} H_{{{{\acute{\mathrm{e}}}}\mathrm{t}}}^{2r-1}(\overline{W}_r,{\mathbf{{Q}}}_p(1)) \otimes \mathrm{Sym}^{2r-2}H^1_{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\overline{A},{\mathbf{{Q}}}_p(1))\xrightarrow {\pi _g\otimes \pi _{\mathrm{N}^{r-1}}} V_g(r) \end{aligned}$$

induces a map

$$\begin{aligned} \pi _{g,\mathrm{N}^{r-1}}:H^1(F,H_{{{{\acute{\mathrm{e}}}}\mathrm{t}}}^{4r-3}(\overline{X}_r,{\mathbf{{Q}}}_p(2r-1))) \longrightarrow H^1(F,V_g(r))\nonumber \end{aligned}$$

over any number field $F$. In the following we fix a number field $F$ containing $H$.

Now consider the étale Abel-Jacobi map

$$\begin{aligned} \Phi _F^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}:\mathrm{CH}^{2r-1}(X_r)_0(F)\rightarrow H^1(F,H_{{{{\acute{\mathrm{e}}}}\mathrm{t}}}^{4r-3}(\overline{X}_r,{\mathbf{{Q}}}_p)(2r-1)) \end{aligned}$$

constructed in [28]. Let $F_p$ be the completion of $\imath _p(F)$, and denote by $\mathrm{loc}_p$ the induced localization map from $G_F$ to $\mathrm{Gal}(\overline{{\mathbf{{Q}}}}_p/F_p)$. Then we may define the $p$ -adic Abel-Jacobi map $\mathrm{AJ}_{F_p}$ by the commutativity of the diagram

(3.3)

where the existence of the dotted arrow follows from [28, 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 [12], the map $\mathrm{AJ}_{F_p}$ may be evaluated at the class $\omega _g\otimes e_\zeta ^{\otimes r-1}$, with $e_\zeta$ an $F_p$-basis of $D_\mathrm{dR}({\mathbf{{Q}}}_p(1))\cong F_p$.

The main result of [1] 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 $g=\sum _nb_nq^n\in S_{2r}(X_0(N))$ be a normalized newform of weight $2r\ge 2$ and level $N$ prime to $p$. Then

$$\begin{aligned} (1-b_pp^{-r}+p^{-1})&\sum _{[\mathfrak a ]\in \mathrm{Pic}(\mathcal{O }_K)}\mathrm{N}\mathfrak a ^{1-r}\cdot \mathrm{AJ}_{F_p}(\Delta _{\varphi _\mathfrak a ,r}^\mathrm{bdp}) (\omega _g\otimes e_\zeta ^{\otimes r-1})\\&\quad \quad =(-1)^{r-1}(r-1)!\sum _{[\mathfrak a ]\in \mathrm{Pic}(\mathcal{O }_K)}d^{-r}g^{[p]}(\mathfrak a *(A,A[\mathfrak{N }])), \end{aligned}$$

where $g^{[p]}=\sum _{(n,p)=1}b_nq^n$ is the $p$-depletion of $g$.

Proof

See the proof of [1, Thm. 5.13]. $\square$

We end this section by relating the images of Heegner cycles and of generalised Heegner cycles under the $p$-adic height pairing. (Cf. [1, Sect. 3.4]).

Consider $\Pi _r:=W_r\times A^{r-1}$ seen as a subvariety of $W_r\times X_r=W_r\times W_r\times (A^2)^{r-1}$ via the map

$$\begin{aligned} (\mathrm{id}_{W_r},\mathrm{id}_{W_r},(\mathrm{id}_A,\sqrt{-D})^{r-1}). \end{aligned}$$

Denoting by $\pi _W$ and $\pi _X$ the projections onto the first and second factors of $W_r\times X_r$, the rational equivalence class of the cycle $\Pi _r$ gives rise to a map on Chow groups

$$\begin{aligned} \Pi _r:\mathrm{CH}^{2r-1}(X_r)\rightarrow \mathrm{CH}^{r+1}(W_{r}) \end{aligned}$$

induced by $\Pi _r(\Delta )=\pi _{W,*}(\Pi _r\cdot \pi _X^*\Delta )$.

Lemma 3.4

We have

$$\begin{aligned} \langle \Delta _{A,r}^\mathrm{heeg},\Delta _{A,r}^\mathrm{heeg}\rangle _{W_r}= (4D)^{r-1}\cdot \langle \Delta _{\mathrm{id}_A,r}^\mathrm{bdp},\Delta _{\mathrm{id}_A,r}^\mathrm{bdp}\rangle _{X_r}, \end{aligned}$$

where $\langle ,\rangle _{W_r}$ and $\langle ,\rangle _{X_r}$ are the $p$-adic height pairings of [26] on $\mathrm{CH}^{r+1}(W_r)_0$ and $\mathrm{CH}^{2r-1}(X_r)_0$, respectively.

Proof

The image $\Phi _{F}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Delta _{A,r}^\mathrm{{heeg}})$ remains unchanged if we replace $\Gamma _{\sqrt{-D}}$ by $Z_A:=\Gamma _{\sqrt{-D}}-(A\times \{0\})-D(\{0\}\times A)$ (see [27, §II(3.6)]). Since $Z_A\cdot Z_A=-2D$, we easily see from the construction of $\Pi _r$ that

$$\begin{aligned} \Phi _F^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Delta _{A,r}^\mathrm{heeg})=(-2D)^{r-1}\cdot \Phi _F^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Pi _r(\Delta _{\mathrm{id}_A,r}^\mathrm{bdp})). \end{aligned}$$

(3.4)

On the other hand, if $\langle ,\rangle _A$ denotes the Poincaré pairing on $H_\mathrm{dR}^1(A/F)$, we have

$$\begin{aligned} \langle (\sqrt{-D})^*\omega ,(\sqrt{-D})^*\omega ^{\prime }\rangle _A =D\cdot \langle \omega ,\omega ^{\prime }\rangle _A, \end{aligned}$$

for all $\omega , \omega ^{\prime }\in H_\mathrm{dR}^1(A/F)$. By the definition of the $p$-adic height pairings $\langle ,\rangle _{W_r}$ and $\langle ,\rangle _{X_{r}}$ (factoring through $\Phi _{F}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}$), we thus see that

$$\begin{aligned} \langle \Delta _{\mathrm{id}_A,r}^\mathrm{bdp},\Delta _{\mathrm{id}_A,r}^\mathrm{bdp}\rangle _{X_r}=D^{r-1}\cdot \langle \Pi _r(\Delta _{\mathrm{id}_A,r}^\mathrm{bdp}),\Pi _r(\Delta _{\mathrm{id}_A,r}^\mathrm{bdp})\rangle _{W_r}. \end{aligned}$$

(3.5)

Combining (3.4) and (3.5), the result follows. $\square$

4 Big logarithm map

Let ${\mathbf{{f}}}=\sum _{n>0}{\mathbf{{a}}}_nq^n\in \mathbb{I }[[q]]$ be a Hida family passing through (the ordinary $p$-stabilization of) a $p$-ordinary newform $f_o\in S_k(X_0(N))$ as described in the Introduction. We begin this section by recalling the definition of a certain twist of ${\mathbf{{f}}}$ 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 $\varepsilon _\mathrm{cyc}$ as the product

$$\begin{aligned} \varepsilon _\mathrm{cyc}=\omega \cdot \epsilon :G_{{\mathbf{{Q}}}}\rightarrow {\mathbf{{Z}}}_p^\times =\varvec{\mu }_{p-1}\times \Gamma . \end{aligned}$$

Since $k$ is even, the character $\omega ^{k-2}$ admits a square root $\omega ^{\frac{k-2}{2}}:G_{\mathbf{{Q}}}\rightarrow \varvec{\mu }_{p-1}$, and in fact two different square roots, corresponding to the two different lifts of $k-2\in {\mathbf{{Z}}}/(p-1){\mathbf{{Z}}}$ to ${\mathbf{{Z}}}/2(p-1){\mathbf{{Z}}}$. Fix for now a choice of $\omega ^{\frac{k-2}{2}}$, and define the critical character to be

$$\begin{aligned} \Theta :=\omega ^{\frac{k-2}{2}}\cdot [\epsilon ^{1/2}]:G_{\mathbf{{Q}}}\rightarrow \Lambda _\mathcal{O }^\times , \end{aligned}$$

(4.1)

where $\epsilon ^{1/2}:G_{\mathbf{{Q}}}\rightarrow \Gamma$ denotes the unique square root of $\epsilon$ taking values in $\Gamma$.

Remark 4.1

As noted in [19, Rem. 2.1.4], the above choice of $\Theta$ is for most purposes largely indistinguishable from the other choice, namely $\omega ^{\frac{p-1}{2}}\Theta$, where

$$\begin{aligned} \omega ^{\frac{p-1}{2}}:\mathrm{Gal}({\mathbf{{Q}}}(\sqrt{p^*})/{\mathbf{{Q}}})\xrightarrow {\sim }\{\pm {1}\}\quad \quad (p^*=(-1)^{\frac{p-1}{2}}p). \end{aligned}$$

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 $\mathbb{T }$ is then defined to be the module

$$\begin{aligned} \mathbb{T }^\dagger :=\mathbb{T }\otimes _\mathbb{I }\mathbb{I }^\dagger \end{aligned}$$

(4.2)

equipped with the diagonal $G_{\mathbf{{Q}}}$-action, where $\mathbb{I }^\dagger =\mathbb{I }(\Theta ^{-1})$ is $\mathbb{I }$ as a module over itself with $G_{\mathbf{{Q}}}$ acting via the character $G_{\mathbf{{Q}}}\xrightarrow {\Theta ^{-1}}\Lambda _\mathcal{O }^\times \rightarrow \mathbb{I }^\times .$

Lemma 4.2

Let $\rho _{\mathbb{T }^\dagger }:G_{\mathbf{{Q}}}\rightarrow \mathrm{Aut}(\mathbb{T }^\dagger )$ be the Galois representation carried by $\mathbb{T }^\dagger$. Then for every $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ of even weight $k_\nu =2r_\nu \ge 2$ we have

$$\begin{aligned} \nu (\rho _{\mathbb{T }^\dagger })\cong \rho _{{\mathbf{{f}}}_\nu ^{\prime }}\otimes \varepsilon _\mathrm{cyc}^{r_\nu }, \end{aligned}$$

where ${\mathbf{{f}}}_\nu ^{\prime }$ is a character twist of ${\mathbf{{f}}}_\nu$ of the same weight with trivial nebentypus. In other words, defining $\mathbb V _\nu ^\dagger :=\mathbb{T }^\dagger \otimes _{\mathbb{I }}F_\nu$ and letting $V_{{\mathbf{{f}}}_\nu ^{\prime }}$ be the representation space of $\rho _{{\mathbf{{f}}}_\nu ^{\prime }}$, we have

$$\begin{aligned} \mathbb V _\nu ^\dagger \cong V_{{\mathbf{{f}}}_\nu ^{\prime }}(r_\nu ), \end{aligned}$$

(4.3)

and in particular $\mathbb V _\nu ^\dagger$ is isomorphic to its Kummer dual.

Proof

This follows from a straightforward computation explained in [29, (3.5.2)] for example (where $\mathbb{T }^\dagger$ is denoted by $T$). $\square$

Let $\theta :{\mathbf{{Z}}}_p^\times \rightarrow \Lambda _\mathcal{O }^\times$ be such that $\Theta =\theta \circ \varepsilon _\mathrm{cyc}$. It follows from the preceding lemma that the formal $q$-expansion

$$\begin{aligned} {\mathbf{{f}}}^\dagger ={\mathbf{{f}}}\otimes \theta ^{-1} :=\sum _{n>0}\theta ^{-1}(n){\mathbf{{a}}}_nq^n\in \mathbb{I }[[q]] \end{aligned}$$

(where we put $\theta ^{-1}(n)=0$ whenever $p\vert n$) is such that, for every $\nu \in \mathcal X _\mathrm{arith}(\mathbb I )$ of even weight, $\mathbb V _\nu ^\dagger$ is the Galois representation attached to the specialization ${\mathbf{{f}}}_\nu \otimes \theta _\nu ^{-1}$ of ${\mathbf{{f}}}^\dagger$, 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 $\mathbb{T }$. Let $I_w\subset D_w\subset G_{\mathbf{{Q}}}$ be the inertia and decomposition groups at the place $w\vert p$ induced by our fixed embedding $\imath _p:\overline{{\mathbf{{Q}}}}\hookrightarrow \overline{{\mathbf{{Q}}}}_p$. In the following we will identify $D_w$ with the absolute Galois group $G_{{\mathbf{{Q}}}_p}$. Then by a result of Mazur and Wiles (see [34, Thm. 2.2.2]) there exists a filtration of $\mathbb{I }[D_w]$-modules

$$\begin{aligned} 0\rightarrow {\fancyscript{F}}_w^+\mathbb{T }\rightarrow \mathbb{T }\rightarrow {\fancyscript{F}}_w^-\mathbb{T }\rightarrow 0 \end{aligned}$$

(4.4)

with ${\fancyscript{F}}_w^{\pm }\mathbb{T }$ free of rank one over $\mathbb{I }$ and with the Galois action on ${\fancyscript{F}}_w^-\mathbb{T }$ unramified, given by the character $\alpha :D_w/I_w\rightarrow \mathbb{I }^\times$ sending an arithmetic Frobenius $\sigma _p$ to ${\mathbf{{a}}}_p$. Twisting (4.4) by $\Theta ^{-1}$ we define ${\fancyscript{F}}_w^\pm \mathbb{T }^\dagger$ in the natural manner.

Let $\mathbb{T }^*:=\mathrm{Hom}_{\mathbb{I }}(\mathbb{T },\mathbb{I })$ be the contragredient⁵ of $\mathbb{T }$, and consider the $\mathbb{I }$-module

$$\begin{aligned} \mathbb{D }:=({\fancyscript{F}}_w^+\mathbb{T }^*\widehat{\otimes }_{{\mathbf{{Z}}}_p} \widehat{{\mathbf{{Z}}}}_p^\mathrm{nr})^{G_{{\mathbf{{Q}}}_p}}, \end{aligned}$$

(4.5)

where ${\fancyscript{F}}_w^+\mathbb{T }^*:=\mathrm{Hom}_\mathbb{I }({\fancyscript{F}}^-\mathbb{T },\mathbb{I })\subset \mathbb{T }^*$, and $\widehat{{\mathbf{{Z}}}}_p^\mathrm{nr}$ is the completion of the ring of integers of the maximal unramified extension of ${\mathbf{{Q}}}_p$ in $\overline{{\mathbf{{Q}}}}_p$.

Fix once and for all a compatible system $\zeta =\{\zeta _s\}$ of primitive $p^s$-th roots of unity, and denote by $e_\zeta$ the basis of $D_\mathrm{dR}({\mathbf{{Q}}}_p(1))$ corresponding to $1\in {\mathbf{{Q}}}_p$ under the resulting identification $D_\mathrm{dR}({\mathbf{{Q}}}_p(1))\cong {\mathbf{{Q}}}_p$.

Lemma 4.3

The module $\mathbb{D }$ is free of rank one over $\mathbb{I }$, and for every $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ of even weight $k_\nu =2r_\nu \ge 2$ there is a canonical isomorphism

$$\begin{aligned} \mathbb{D }_\nu \otimes D_\mathrm{dR}({\mathbf{{Q}}}_p(r_\nu )) \cong \frac{D_\mathrm{dR}(V_{{\mathbf{{f}}}_\nu }(r_\nu ))}{\mathrm{Fil}^0D_\mathrm{dR}(V_{{\mathbf{{f}}}_\nu }(r_\nu ))}. \end{aligned}$$

(4.6)

Proof

Since the action on ${\fancyscript{F}}_w^+\mathbb{T }^*$ is unramified, the first claim follows from [30, Lemma 3.3] in light of the definition (4.5) of $\mathbb{D }$. The second can be deduced from [30, Lemma 3.2] as in the proof of [30, Lemma 3.6]. $\square$

With the same notations as in Lemma 4.3, we denote by $\langle ,\rangle _\mathrm{dR}$ the pairing

$$\begin{aligned} \langle ,\rangle _\mathrm{dR}:\mathbb{D }_\nu \otimes D_\mathrm{dR}({\mathbf{{Q}}}_p(r_\nu )) \times \mathrm{Fil}^1D_\mathrm{dR}(V_{{\mathbf{{f}}}_\nu ^*}(r_\nu -1))\rightarrow F_\nu \end{aligned}$$

(4.7)

deduced from the usual de Rham pairing

$$\begin{aligned} \frac{D_\mathrm{dR}(V_{{\mathbf{{f}}}_\nu }(r_\nu ))}{\mathrm{Fil}^0D_\mathrm{dR}(V_{{\mathbf{{f}}}_\nu }(r_\nu ))} \times \mathrm{Fil}^0D_\mathrm{dR}(V_{{\mathbf{{f}}}_\nu }^*(1-r_\nu ))\rightarrow F_\nu \end{aligned}$$

via the identification (4.6) and the isomorphism $V_{{\mathbf{{f}}}_\nu }^*\cong V_{{\mathbf{{f}}}_\nu ^*}(k_\nu -1)$.

Theorem 4.4

(Ochiai) Assume that the residual representation $\bar{\rho }_{f_o}$ is irreducible, fix an $\mathbb{I }$-basis $\mathbf{\eta }$ of $\mathbb{D }$, and set $\lambda :={\mathbf{{a}}}_p-1$. There exists an $\mathbb{I }$-linear map

$$\begin{aligned} \mathrm{Log}_{\mathbb{T }^\dagger }^{(\eta )}:H^1({\mathbf{{Q}}}_p,{\fancyscript{F}}_w^+\mathbb{T }^\dagger ) \rightarrow \mathbb{I }[\lambda ^{-1}] \end{aligned}$$

such that if $\mathfrak{Y }\in H^1({\mathbf{{Q}}}_p,{\fancyscript{F}}_w^+\mathbb{T }^\dagger )$ and $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ has weight $k_\nu =2r_\nu \ge 2$, then

$$\begin{aligned} \nu (\mathrm{Log}^{(\eta )}_{\mathbb{T }^\dagger }(\mathfrak{Y }))&= \frac{(-1)^{r_\nu -1}}{(r_\nu -1)!}\nonumber \\&\!\times \!\left\{ \begin{array}{l@{\quad }l} \left(1-\frac{p^{r_\nu -1}}{\nu ({\mathbf{{a}}}_p)}\right)^{-1} \left(1-\frac{\nu ({\mathbf{{a}}}_p)}{p^{r_\nu }}\right) \langle \mathrm{log}_{V_{{\mathbf{{f}}}_\nu }(r_\nu )} (\mathfrak{Y }_\nu ),\eta _\nu ^{\prime }\rangle _\mathrm{dR}&\mathrm{if}\,\vartheta _\nu =\mathbb 1 ;\\ \frac{1}{G(\vartheta _\nu ^{-1})} \left(\frac{\nu ({\mathbf{{a}}}_p)}{p^{r_\nu -1}}\right)^{s_\nu } \langle \mathrm{log}_{s,V_{{\mathbf{{f}}}_\nu }(r_\nu )}(\mathfrak{Y }_\nu ), \eta _\nu ^{\prime }\rangle _\mathrm{dR}&\mathrm{if} \,\vartheta _\nu \ne \mathbb 1 , \end{array}\right.\nonumber \\ \end{aligned}$$

(4.8)

where

• $\mathrm{log}_{V_{{\mathbf{{f}}}_\nu }(r_\nu )}$ (resp. $\mathrm{log}_{s,V_{{\mathbf{{f}}}_\nu }(r_\nu )}$) is the Bloch-Kato logarithm map for $V_{{\mathbf{{f}}}_\nu }(r_\nu )$ over ${\mathbf{{Q}}}_p$ (resp. ${\mathbf{{Q}}}_{p,s}:={\mathbf{{Q}}}_p(\varvec{\mu }_{p^s})$),

• $\eta _\nu ^{\prime }\in \mathrm{Fil}^1D_\mathrm{dR}(V_{{\mathbf{{f}}}_\nu ^*}(r_\nu -1))$ is such that $\langle \eta _\nu \otimes e_\zeta ^{\otimes r_\nu },\eta _\nu ^{\prime }\rangle _\mathrm{dR}=1$,

• $\vartheta _\nu :{\mathbf{{Z}}}_p^\times \rightarrow F_\nu ^\times$ is the finite order character $z\mapsto \theta _\nu (z)z^{1-r_\nu }$,

• $s>0$ is such that the conductor of $\vartheta _\nu$ is $p^{s}$, and

• $G(\vartheta _\nu ^{-1})$ is the Gauss sum $\sum _{x\;\mathrm{mod}\;p^{s}} \vartheta _\nu ^{-1}(x)\zeta _{s}^{x}$.

Proof

Let $\Lambda _\mathrm{cyc}={\mathbf{{Z}}}_p[[\Gamma _\mathrm{cyc}]]$ be the cyclotomic Iwasawa algebra, where

$$\begin{aligned} \Gamma _\mathrm{cyc}:=\mathrm{Gal}(\mathbf Q _{p,\infty }/\mathbf Q _p)\cong \mathbf Z _p, \end{aligned}$$

and consider the $\mathbb{I }\widehat{\otimes }_{{\mathbf{{Z}}}_p}\Lambda _\mathrm{cyc}$-modules $\mathcal{D }:=\mathbb{D }\widehat{\otimes }_{{\mathbf{{Z}}}_p}\Lambda _\mathrm{cyc}$ and ${\fancyscript{F}}_w^+\mathcal{T }^*:={\fancyscript{F}}_w^+\mathbb{T }^*\widehat{\otimes }_{{\mathbf{{Z}}}_p} \Lambda _\mathrm{cyc}\otimes \omega ^{\frac{k-2}{2}}$, the latter being equipped with the diagonal action of $G_{{\mathbf{{Q}}}_p}$. Also let $\gamma _o$ be a topological generator of $\Gamma _\mathrm{cyc}$ and $\mathcal I :=(\lambda ,\gamma _o)\subset \mathbb I \widehat{\otimes }_\mathbf{Z _p}\Lambda _\mathrm{cyc}\cong \mathbb I [[\Gamma _\mathrm{cyc}]]$. Consider the $\mathbb I$-algebra isomorphism

$$\begin{aligned} \mathrm{Tw}_{\theta _1}:\mathbb I [[\Gamma _\mathrm{cyc}]]\rightarrow \mathbb I [[\Gamma _\mathrm{cyc}]] \end{aligned}$$

(4.9)

given by $\mathrm{Tw}_{\theta _1}([\sigma ])=\epsilon ^{1/2}(\sigma )[\sigma ]$ for $\sigma \in \Gamma _\mathrm{cyc}$, where $\epsilon ^{1/2}$ is the unique square-root of the wild component of the cyclotomic character. By [30, Prop. 5.3] there exists an injective $\mathbb{I }\widehat{\otimes }_{{\mathbf{{Z}}}_p}\Lambda _\mathrm{cyc}$-linear map

$$\begin{aligned} \mathrm{Exp}_{{\fancyscript{F}}_w^+{\mathcal{T }^*}} :\mathcal ID \rightarrow H^1({\mathbf{{Q}}}_p,{\fancyscript{F}}_w^+{\mathcal{T }^*}) \end{aligned}$$

with cokernel killed by $\mathcal{I }$ which interpolates the Bloch-Kato exponential over the arithmetic primes of $\mathbb I$ and of $\Lambda _\mathrm{cyc}$. Notice that letting $\fancyscript{F}_w^+\mathcal T ^\dagger$ be the module $\fancyscript{F}_w^+\mathcal T ^*$ with the $\mathbb I [[\Gamma _\mathrm{cyc}]]$-action twisted by $\theta _1$, there is a Galois equivariant projection $\fancyscript{F}_w^+\mathcal T ^\dagger \rightarrow \fancyscript{F}_w^+\mathbb T ^\dagger$. The composition

$$\begin{aligned} \mathcal{ID }\xrightarrow {\mathrm{Exp}_{{\fancyscript{F}}_{w}^{+}\mathcal{T }^{*}}}H^{1}(\mathbf{Q }_{p},{\fancyscript{F}}_{w}^{+}\mathcal{T }^{*})\xrightarrow {\mathrm{Tw}_{}} H^{1}(\mathbf{Q }_{p},{\fancyscript{F}}_{w}^{+}\mathcal{T }^{\dagger })\xrightarrow {\mathrm{Cor}} H^{1}(\mathbf{Q }_{p},{\fancyscript{F}}_{w}^{+}\mathbb{T }^{\dagger })\nonumber \\ \end{aligned}$$

(4.10)

is an $\mathbb I$-linear map making for every $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ as in the statement the diagram

commutative, where $\mathrm{Sp}_{\nu ,\zeta }$ is given by the composition of (4.6) with the map

$$\begin{aligned} \mathcal{D }=\mathbb{D }\widehat{\otimes }_{{\mathbf{{Z}}}_p}\Lambda _\mathrm{cyc} \rightarrow \mathbb{D }_\nu \otimes D_\mathrm{dR}({\mathbf{{Q}}}_p(r_\nu ))\otimes {\mathbf{{Q}}}_{p,s} \end{aligned}$$

induced by specialization at $\nu$ on $\mathbb{D }$ and $\sigma \mapsto e_{\zeta }^{\otimes r_\nu }\otimes \zeta _s^\sigma$ ($\sigma \in \Gamma _\mathrm{cyc}$) on $\Lambda _\mathrm{cyc}$, and where the bottom horizontal arrow is given by:

$$\begin{aligned} (-1)^{r_\nu -1}(r_\nu -1)!\times \left\{ \begin{array}{l@{\quad }l} \left(1-\frac{p^{r_\nu -1}}{\nu ({\mathbf{{a}}}_p)}\right) \left(1-\frac{\nu ({\mathbf{{a}}}_p)}{p^{r_\nu }}\right)^{-1}\mathrm{exp}_{V_{{\mathbf{{f}}}_\nu }(r_\nu )}&\mathrm{if}\, \vartheta _\nu =\mathbb 1 ;\\ G(\vartheta _\nu ^{-1})\left(\frac{p^{r_\nu -1}}{\nu ({\mathbf{{a}}}_p)}\right)^{s_\nu } \mathrm{exp}_{s,V_{{\mathbf{{f}}}_\nu }(r_\nu )}&\mathrm{if}\, \vartheta _\nu \ne \mathbb 1 \end{array}\right. \end{aligned}$$

with $\mathrm{exp}_{V_{{\mathbf{{f}}}_\nu }(r_\nu )}$ (resp. $\mathrm{exp}_{s,V_{{\mathbf{{f}}}_\nu }(r_\nu )}$) the Bloch-Kato exponential map for $V_{{\mathbf{{f}}}_\nu }(r_\nu )$ over ${\mathbf{{Q}}}_p$ (resp. ${\mathbf{{Q}}}_{p,s}$). The map $\mathrm{Exp}_{\fancyscript{F}_w^+\mathbb T ^\dagger }$ factors through an injective $\mathbb I$-linear map

$$\begin{aligned} \mathrm{Exp}_{\fancyscript{F}_w^+\mathbb T ^\dagger }:\mathbb D ^\dagger \rightarrow H^1(\mathbf Q _p,\fancyscript{F}_w^+\mathbb T ^\dagger ), \end{aligned}$$

where $\mathbb D ^\dagger :=\mathcal ID \otimes _\mathbf{Z _p}\mathbb I [[\Gamma _\mathrm{cyc}]]/(\gamma _o^2-\gamma _o^{\prime })$ with $\gamma _o^{\prime }$ a topological generator of $\Gamma$. (Recall for the Introduction that $\Gamma$ acts on $\mathbb I$ via the diamond operators.)

Now if $\mathfrak{Y }\in H^1({\mathbf{{Q}}}_p,{\fancyscript{F}}_w^+\mathbb{T }^\dagger )$, then $\lambda \cdot \mathfrak{Y }$ lands in the image $\mathrm{Exp}_{{\fancyscript{F}}_w^+\mathbb{T }^\dagger }$ and so

$$\begin{aligned} \mathrm{Log}_{\mathbb{T }^\dagger }(\mathfrak{Y }):=\lambda ^{-1} \cdot (\mathrm{Exp}_{{\fancyscript{F}}_w^+\mathbb{T }^\dagger })^{-1}(\lambda \cdot \mathfrak{Y }) \in \mathbb{I }[\lambda ^{-1}]\otimes _\mathbb{I }\mathbb{D }^\dagger \end{aligned}$$

is well-defined. Thus defining $\mathrm{Log}^{(\eta )}_{\mathbb{T }^\dagger }(\mathfrak{Y })\in \mathbb{I }[\lambda ^{-1}]$ by the relation

$$\begin{aligned} \mathrm{Log}_{\mathbb{T }^\dagger }(\mathfrak{Y })=\mathrm{Log}^{(\eta )}_{\mathbb{T }^\dagger }(\mathfrak{Y })\cdot \eta \otimes 1 \end{aligned}$$

the result follows. $\square$

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 $\mathfrak{Z }$ (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 $\mathfrak{Z }$ taken up in the following.

5.1 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 $\mathfrak{N }\subset \mathcal{O }_K$ is a fixed cyclic $N$-ideal, i.e. such that $\mathcal{O }_K/\mathfrak{N }\cong {\mathbf{{Z}}}/N{\mathbf{{Z}}}$. We also assume that $p$ splits in $K$, and let $\mathfrak p$ be the prime of $K$ above $p$ induced by our fixed embedding $\imath _p$, and by $\bar{\mathfrak{p }}$ the other. Finally, $A$ is a fixed elliptic curve with CM by $\mathcal{O }_K$ defined over the Hilbert class field $H$ of $K$.

Let $R_0=\widehat{{\mathbf{{Z}}}}_p^\mathrm{nr}$ be the completion of the ring of integers of the maximal unramified extension of $\mathbf Q _p$, which we view as an overfield of $H$ via $\imath _p$. Since $p$ splits in $K$, $A$ admits a trivialization

$$\begin{aligned} \imath _A:\hat{A}\rightarrow \hat{\mathbf{G}}_m \end{aligned}$$

over $R_0$ with $\imath _A^{-1}(\varvec{\mu }_{p^s})=A[\mathfrak p ^s]$ for every $s>0$. Letting $\alpha _A$ be the cyclic $N$-isogeny on $A$ with kernel $A[\mathfrak{N }]$, the triple $(A,\alpha _A,\imath _A)$ thus defines a trivialized elliptic curve with $\Gamma _0(N)$-level structure defined over $R_0$.

Set $A_0:=A/A[\mathfrak p ^s]$ and let $(A_0,\alpha _{A_0},\imath _{A_0})$ be the trivialized elliptic curve deduced from $(A,\alpha _A,\imath _A)$ via the projection $A\rightarrow A_0$. Let $C\subset A_0[p^s]$ be any étale subgroup of order $p^s$, and set $A_s:=A_0/C$. Finally, let $(A_s,\alpha _{A_s},\imath _{A_s})$ be the trivialized elliptic curve with $\Gamma _0(N)$-level structure deduced from $(A_0,\alpha _{A_0},\imath _{A_0})$ via the projection $A_0\rightarrow A_s$, and consider the triple

$$\begin{aligned} h_s=(A_s,\alpha _{A_s},\imath _{A_s}(\zeta _s)), \end{aligned}$$

(5.1)

which defines an algebraic point on the modular curve $X_s$.

Write $p^*=(-1)^{\frac{p-1}{2}}p$, and let $\vartheta$ be the unique continuous character

$$\begin{aligned} \vartheta :G_{{\mathbf{{Q}}}(\sqrt{p^*})}\rightarrow {\mathbf{{Z}}}_p^\times /\{\pm {1}\} \end{aligned}$$

(5.2)

such that $\vartheta ^2=\varepsilon _\mathrm{cyc}$. Notice the inclusion $G_{H_{p^s}}\subset G_{{\mathbf{{Q}}}(\sqrt{p^*})}$ 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 $\mathcal{O }_{p^s}$ of $K$ of conductor $p^s$, and the point $h_s$ is rational over $L_{p^s}:=H_{p^s}(\varvec{\mu }_{p^s})$. In fact we have

$$\begin{aligned} h_s^\sigma =\langle \vartheta (\sigma )\rangle \cdot h_s\end{aligned}$$

(5.3)

for all $\sigma \in \mathrm{Gal}(L_{p^s}/H_{p^s})$.

Proof

The first assertion is clear, and immediately from the construction we also see that $\alpha _{A_s}$ is the cyclic $N$-isogeny on $A_s$ with kernel $A_s[\mathfrak{N }\cap \mathcal{O }_{p^s}]$. It follows that the point (5.1) gives rise to precisely the point $h_s\in X_s({\mathbf{{C}}})$ in [19, Eq. (4)]. The result thus follows from [loc.cit., Cor. 2.2.2]. $\square$

If $\nu$ is an arithmetic prime of $\mathbb{I }$, we let $\psi _\nu$ denote its wild character, defined as the composition of $\nu :\mathbb{I }\rightarrow \overline{{\mathbf{{Q}}}}_p$ with the structure map $\Gamma =1+p{\mathbf{{Z}}}_p\rightarrow \mathbb{I }^\times$. The nebentypus of ${\mathbf{{f}}}_\nu$ is then given by $\varepsilon _{{\mathbf{{f}}}_\nu }=\psi _\nu \omega ^{k-k_\nu }$, where $\omega :({\mathbf{{Z}}}/p{\mathbf{{Z}}})^\times \rightarrow \varvec{\mu }_{p-1} \subset {\mathbf{{Z}}}_p^\times$ is the Teichmüller character.

Recall the critical characters $\Theta$ and $\theta$ from Sect. 4, and for every $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ of weight 2, consider the $F_\nu ^\times$-valued Hecke character of $K$ given by

$$\begin{aligned} \chi _\nu (x)=\Theta _\nu (\mathrm{art}_{\mathbf{{Q}}}(\mathrm{N}_{K/{\mathbf{{Q}}}}(x))) \end{aligned}$$

(5.4)

for all $x\in \mathbb{A }_K^\times$. Notice that since $\chi _\nu$ has finite order, it may alternately be seen as character on $G_K$ via the Artin reciprocity map $\mathrm{art}_K:\mathbb{A }_K^\times \rightarrow G_K^\mathrm{ab}$.

Let $\mathcal{O }_{{\mathbf{{C}}}_p}$ be the ring of integers of the completion of $\overline{{\mathbf{{Q}}}}_p$. For every $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$, after fixing an embedding $F_\nu \rightarrow \overline{{\mathbf{{Q}}}}_p$, the form ${\mathbf{{f}}}_\nu \in S_{k_\nu }(X_{s_\nu })$ defines a $p$-adic modular form ${\mathbf{{f}}}_\nu \in \mathbf{M}(N)$. Finally, recall the dual form ${\mathbf{{f}}}_\nu ^*$ defined as in the paragraph before (2.20).

Lemma 5.2

Let $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ have weight 2 and non-trivial wild character, and let $s>1$ be the $p$-power of the conductor of $\psi _\nu$. Then

$$\begin{aligned} d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}\otimes \theta _\nu (A,\alpha _A,\imath _A) \!=\!\frac{u}{G(\theta _\nu ^{-1})}\sum _{\sigma \in \mathrm{Gal}(H_{p^s}/H)}\chi _\nu ^{-1}(\tilde{\sigma }) \cdot d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}(h_s^{\tilde{\sigma }}),\qquad \end{aligned}$$

(5.5)

where $u=\vert \mathcal{O }_K^\times \vert /2$, $G(\theta _\nu ^{-1})$ is the Gauss sum $\sum _{x\;\mathrm{mod}\;p^s}\theta _\nu ^{-1}(x)\zeta _s^{x}$, and for every $\sigma \in \mathrm{Gal}(H_{p^s}/H)$, $\tilde{\sigma }$ is any lift of $\sigma$ to $\mathrm{Gal}(L_{p^s}/H)$.

Proof

Notice that the expression in the right hand side of (5.5) does not depend on the choice of lifts $\tilde{\sigma }$. Indeed, as explained in [18, p. 808] the character $\chi _{0,\nu }:=\chi _\nu \vert _{\mathbb{A }_{\mathbf{{Q}}}^\times }$, seen as a Dirichlet character in the usual manner, is such that $\chi _{0,\nu }^{-1}=\theta _\nu ^2$. But since the weight of $\nu$ is 2, we have $\theta _\nu ^2=\varepsilon _{{\mathbf{{f}}}_\nu }=\varepsilon _{{\mathbf{{f}}}_\nu ^*}^{-1}$ (see [18, p. 806]), and our claim thus follows immediately from (5.3).

To compute the above value of the twist $d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}\otimes \theta _\nu ^{}$ we follow Definition 2.2. The integer $s>1$ in the statement is such that $\theta _\nu$ factors through $({\mathbf{{Z}}}/p^s{\mathbf{{Z}}})^\times$, therefore

$$\begin{aligned} d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}\!\otimes \!\theta _\nu (A,\alpha _A,\imath _A)\;=\;\sum _{a\;\mathrm{mod}\;p^s}\theta _\nu ^{}(a)\left(\int _{a+p^s{\mathbf{{Z}}}_p}d\mu _\mathrm{Gou}(x)\!\right)(d^{-1}{\mathbf{{f}}}_\nu ^{*[p]})(\!A,\alpha _A,\imath _A)\nonumber \\ \;=\;\frac{1}{p^s}\!\!\sum _{a\;\mathrm{mod}\;p^s}\theta _\nu ^{}(a)\!\!\sum _C\!\zeta _C^{-a}\!\cdot d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}(A_0/C,\alpha _C,\imath _C),\qquad \quad \end{aligned}$$

(5.6)

where as before $A_0:=A/\imath _A^{-1}(\varvec{\mu }_{p^s})=A/A[\mathfrak p ^s]$ and the sum is over the étale subgroups $C\subset A_0[p^s]$ of order $p^s$. Letting $\gamma _s$ be a generator of ${\mathbf{{Z}}}/p^s{\mathbf{{Z}}}$, these subgroups correspond bijectively with the cyclic subgroups $C_u=\langle \zeta _s^u.\gamma _s\rangle \subset \varvec{\mu }_{p^s}\times {\mathbf{{Z}}}/p^s{\mathbf{{Z}}}$, with $u$ running over the integers modulo $p^s$, and we set $\zeta _{C_u}=\zeta _s^u$.

Since $\theta _\nu$ does not factor through $({\mathbf{{Z}}}/p^{s-1}{\mathbf{{Z}}})^\times$, we have $\sum _{a\;\mathrm{mod}\;p^s}\theta _\nu (a)\zeta _s^{-ua}=0$ whenever $u\notin ({\mathbf{{Z}}}/p^s{\mathbf{{Z}}})^\times$. Continuing from (5.6), we thus obtain

$$\begin{aligned} d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}\!\otimes \!\theta _\nu ^{}(\!A,\alpha _A,\imath _A\!)\;=\;\frac{1}{p^s}\!\sum _{a\;\mathrm{mod}\;p^s}\theta _\nu ^{}(a)\!\sum _{u\;\mathrm{mod}\;p^s}\zeta _s^{-ua}\cdot d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}(A_{C_u},\alpha _{C_u},\imath _{C_u})\nonumber \\ \;=\;\frac{1}{p^s}\!\sum _{u\in {({\mathbf{{Z}}}/p^s{\mathbf{{Z}}})^\times }} d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}(A_{C_u},\alpha _{C_u},\imath _{C_u}) \!\sum _{a\;\mathrm{mod}\;p^s}\theta _\nu ^{}(a)\zeta _s^{-ua}\nonumber \\ \;=\;\frac{1}{G(\theta _\nu ^{-1})}\sum _{u\in {({\mathbf{{Z}}}/p^s{\mathbf{{Z}}})^\times }} \theta _\nu ^{-1}(u)\cdot d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}(A_{C_u}, \alpha _{C_u},\imath _{C_u}), \end{aligned}$$

with the last equality obtained by a change of variables. The result thus follows from the relation

$$\begin{aligned} \!\sum _{u\in ({\mathbf{{Z}}}/p^s{\mathbf{{Z}}})^\times }\theta _\nu ^{-1}(u)\cdot d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}(A_{C_u},\alpha _{C_u},\imath _{C_u}) \!=\!u\!\sum _{\sigma \in \mathrm{Gal}(H_{p^s}/H)}\chi _\nu ^{-1}(\!\tilde{\sigma }\!) \cdot d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}(h_s^{\tilde{\sigma }}),\nonumber \end{aligned}$$

where $u=\vert \mathcal{O }_K^\times \vert /2$, and for each $\sigma \in \mathrm{Gal}(H_{p^s}/H)$, $\tilde{\sigma }\in \mathrm{Gal}(L_{p^s}/H)$ lifts $\sigma$. $\square$

Keeping the above notations, let $\Delta _s\in J_s(L_{p^s})$ be the divisor class of $(h_s)-(\infty )$, and consider the element in $J_s(L_{p^s})\otimes _{{\mathbf{{Z}}}}F_\nu$ given by

$$\begin{aligned} \widetilde{Q}_{\chi _\nu }:=\sum _{\sigma \in \mathrm{Gal}(H_{p^s}/H)}\Delta _s^{\tilde{\sigma }} \otimes \chi _\nu ^{-1}(\tilde{\sigma }), \end{aligned}$$

(5.7)

where for every $\sigma \in \mathrm{Gal}(H_{p^s}/H)$, $\tilde{\sigma }$ is any lift to $\mathrm{Gal}(L_{p^s}/H)$.

Let $F_{s}$ be the completion of $\imath _p(L_{p^s})$, and consider the $p$-adic Abel-Jacobi map $\delta _{\mathbf{f}_{\nu}, F_{s}}^{(p)}$ defined in (2.9) which we extend by $F_\nu$-linearity to a map

$$\begin{aligned} \delta _{{\mathbf{{f}}}_\nu ,F_s}^{(p)}:J_s(L_{p^s})\otimes _{{\mathbf{{Z}}}}F_\nu \longrightarrow (\mathrm{Fil}^1D_\mathrm{dR}(V_{{\mathbf{{f}}}_\nu ^*}))^\vee . \end{aligned}$$

Proposition 5.3

Let $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ and $s>1$ be as in Lemma 5.2. Then

$$\begin{aligned} \sum _{\sigma \in \mathrm{Gal}(H_{p^s}/H)}\chi _\nu ^{-1}(\tilde{\sigma }) \cdot d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}(h_s^{\tilde{\sigma }})= \delta _{{\mathbf{{f}}}_\nu ,F_s}^{(p)}(\widetilde{Q}_{\chi _\nu })(\omega _{{\mathbf{{f}}}_\nu ^*}). \end{aligned}$$

(5.8)

Proof

The integer $s>1$ in the statement is so that the nebentypus $\varepsilon _{{\mathbf{{f}}}_\nu }$ of ${\mathbf{{f}}}_\nu$ is primitive modulo $p^s$. Moreover, since $p$ splits in $K$, we see from the construction that the point $h_s$ lies in the connected component $X_s(0)$ of the ordinary locus of $X_s$ containing the cusp $\infty$. Thus Proposition 2.9 applies, giving

$$\begin{aligned} \delta ^{(p)}_{{\mathbf{{f}}}_\nu ,F_s}(\Delta _s)(\omega _{{\mathbf{{f}}}_\nu ^{*}}) =F_{\omega _{{\mathbf{{f}}}_\nu ^*}}(h_s), \end{aligned}$$

where $F_{\omega _{{\mathbf{{f}}}_\nu ^*}}$ is the Coleman primitive of $\omega _{{\mathbf{{f}}}_\nu ^*}$ from Proposition 2.6 vanishing at $\infty$, and by linearity

$$\begin{aligned} \sum _{\sigma \in \mathrm{Gal}(H_{p^s}/H)}\chi _\nu ^{-1}(\tilde{\sigma }) \cdot F_{\omega _{{\mathbf{{f}}}_\nu }}(h_s^{\tilde{\sigma }})= \delta _{{\mathbf{{f}}}_\nu ,F_s}^{(p)}(\widetilde{Q}_{\chi _\nu })(\omega _{{\mathbf{{f}}}_\nu ^*}). \end{aligned}$$

(5.9)

Since $\phi$ lifts the Deligne–Tate map to $X_s$, we see that $\phi h_s$ is defined over the subfield $H_{p^{s-1}}(\zeta _s)\subset L_{p^s}$. If $b_p$ denotes the $U_p$-eigenvalue of ${\mathbf{{f}}}_\nu ^*$, by Corollary 2.8 we obtain

$$\begin{aligned} \sum _{\sigma }\chi _\nu ^{-1}(\tilde{\sigma })\cdot d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}(h_s^{\tilde{\sigma }})&= \sum _{\sigma }\chi _\nu ^{-1}(\tilde{\sigma })\cdot F_{\omega _{{\mathbf{{f}}}_\nu ^*}}(h_s^{\tilde{\sigma }}) -\frac{b_p}{p}\sum _{\sigma }\chi _\nu ^{-1}(\tilde{\sigma })\cdot F_{\omega _{{\mathbf{{f}}}_\nu ^*}}(\phi h_s^{\tilde{\sigma }})\nonumber \\&= \sum _{\sigma }\chi _\nu ^{-1}(\tilde{\sigma })\cdot F_{\omega _{{\mathbf{{f}}}_\nu ^*}}(h_s^{\tilde{\sigma }}), \end{aligned}$$

where all the sums are over $\sigma \in \mathrm{Gal}(H_{p^s}/H)$, and the second equality follows immediately from the fact $\theta _\nu$ is primitive modulo $p^s$. The result thus follows from (5.9). $\square$

Still with the same notations, recall Hida’s ordinary projector (1.2) and set $y_s:=e^\mathrm{ord}h_s$, which naturally lies in $e^\mathrm{ord}J_s(L_{p^s})$ (see [19, p.100]). Equation (5.3) then amounts to the fact that

$$\begin{aligned} y_s^\sigma =\Theta (\sigma )\cdot y_s\end{aligned}$$

(5.10)

for all $\sigma \in \mathrm{Gal}(L_{p^s}/H_{p^s})$, where $\Theta$ is the critical character (4.1). Denoting by $J_s^\mathrm{ord}(L_{p^s})^\dagger$ the module $e^\mathrm{ord}J_s(L_{p^s})$ with the Galois action twisted by $\Theta ^{-1}$, and by $y_s^\dagger$ the point $y_s$ seen in this new module, (5.10) translates into the statement that

$$\begin{aligned} y_s^\dagger \in H^0(H_{p^s},J_s^\mathrm{ord}(L_{p^s})^\dagger ). \end{aligned}$$

Lemma 5.4

(Howard) The classes

$$\begin{aligned} x_s:=\mathrm{Cor}_{H_{p^s}/H}(y_s^\dagger )\in H^0(H,J_s^\mathrm{ord}(L_{p^s})^\dagger ) \end{aligned}$$

(5.11)

are such that

$$\begin{aligned} \alpha _*x_{s+1}=U_p\cdot x_s,\quad \quad \mathrm{for \, all}\, s>0 \end{aligned}$$

under the Albanese maps induced from the degeneracy maps $\alpha :X_{s+1}\rightarrow X_s$.

Proof

This is shown in the course of the proof of [19, Lemma 2.2.4]. $\square$

Abbreviate by $\mathrm{Ta}_p^\mathrm{ord}(J_s)$ the module $e^\mathrm{ord}(\mathrm{Ta}_p(J_s)\otimes _{{\mathbf{{Z}}}_p}\mathcal{O })$ from the Introduction, and denote by $\mathrm{Ta}_p^\mathrm{ord}(J_s)^\dagger$ this same module with the Galois action twisted by $\Theta ^{-1}$. By the Galois and Hecke-equivariance of the twisted Kummer map

$$\begin{aligned} \mathrm{Kum}_s:H^0(H,J_s^\mathrm{ord}(L_{p^s})^\dagger )\rightarrow H^1(H,\mathrm{Ta}_p^\mathrm{ord}(J_s)^\dagger ) \end{aligned}$$

constructed in [19, p. 101], Lemma 5.4 implies that the cohomology classes $\mathfrak{X }_s:=\mathrm{Kum}_s(x_s)$ are such that $\alpha _*\mathfrak{X }_{s+1}=U_p\cdot \mathfrak{X }_s$, for all $s>0$.

Definition 5.5

(Howard) The big Heegner point of conductor one is the cohomology class $\mathfrak X$ given by the image of ${\mathop {\leftarrow s} \nolimits ^{\text{ lim}}}U_p^{-s}\cdot \mathfrak{X }_{s}$ under the natural map induced by the $\mathfrak{h }^\mathrm{ord}[G_{{\mathbf{{Q}}}}]$-linear projection ${\mathop {\leftarrow s} \nolimits ^{\text{ lim}}}(\mathrm{Ta}_p^\mathrm{ord}(J_s)^\dagger )\rightarrow \mathbb{T }^\dagger$.

Our object of study is in fact

$$\begin{aligned} \mathfrak{Z }:=\mathrm{Cor}_{H/K}(\mathfrak{X }), \end{aligned}$$

(5.12)

which [19, Conj. 3.4.1] predicts to be not $\mathbb{I }$-torsion. For $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ of weight 2, let $L(s,{\mathbf{{f}}}_\nu ,\chi _\nu )$ be the Rankin-Selberg convolution $L$-function of [20, §1]. In the spirit of the classical Gross-Zagier theorem, one has the following criterion.

Theorem 5.6

(Howard) If $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ has weight 2 and non-trivial nebentypus, then

$$\begin{aligned} \mathfrak{Z }_\nu \ne 0\quad \Longleftrightarrow \quad L^{\prime }(1,{\mathbf{{f}}}_{\nu },\chi _{\nu })\ne 0, \end{aligned}$$

(5.13)

and if the non-vanishing holds for at least one such $\nu$, then $\mathfrak{Z }$ is not $\mathbb{I }$-torsion.

Proof

See [18, Prop. 3] for the equivalence (5.13), and [loc.cit, Cor. 5] for the last implication. We outline the proof for future reference. For every $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ of weight 2 and non-trivial nebentypus, consider (with the same notations as above)

$$\begin{aligned} Q_{\chi _\nu }:=\sum _{\tau \in \mathrm{Gal}(L_{p^s}/K)}\Delta _s^{\tau }\otimes \chi _\nu ^{-1}(\tau ) \in J_s(L_{p^s})\otimes _{{\mathbf{{Z}}}}\overline{{\mathbf{{Q}}}}. \end{aligned}$$

(5.14)

If $e_{{\mathbf{{f}}}_\nu }$ denotes the idempotent of the Hecke algebra (tensored with $\overline{{\mathbf{{Q}}}}$) defined by the eigenform ${\mathbf{{f}}}_\nu$, the arguments in [18, pp. 809–810] show that

$$\begin{aligned} \mathfrak{Z }_\nu \ne 0\quad \Longleftrightarrow \quad e_{{\mathbf{{f}}}_\nu }Q_{\chi _\nu }\ne 0, \end{aligned}$$

(5.15)

and by the “twisted Gross-Zagier theorem” [20, Thm. 4.6.2], one has

$$\begin{aligned} e_{{\mathbf{{f}}}_\nu }Q_{\chi _\nu }\ne 0\quad \Longleftrightarrow \quad L^{\prime }(1,{\mathbf{{f}}}_\nu ,\chi _\nu )\ne 0. \end{aligned}$$

$\square$

Corollary 5.7

Assume that there is a $\nu ^{\prime }\in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ of weight 2 and non-trivial nebentypus such that $L^{\prime }(1,{\mathbf{{f}}}_{\nu ^{\prime }},\chi _{\nu ^{\prime }})\ne 0$. Then the localization map

$$\begin{aligned} \mathrm{loc}_\mathfrak p :H_{f}^1(K,\mathbb V _\nu ^\dagger ) \rightarrow H^1_f({\mathbf{{Q}}}_p,\mathbb V _\nu ^\dagger ) \end{aligned}$$

is injective at all but finitely many $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$.

Proof

By [18, Cor. 5], the assumption implies that $\mathfrak{Z }$ is nontorsion, and by [19, Cor. 3.4.3] that $\widetilde{H}_f^1(K,\mathbb{T }^\dagger )$ has rank 1 over $\mathbb{I }$. By [19, Lemma 2.1.7], it follows that

$$\begin{aligned} H_f^1(K,\mathbb V _\nu ^\dagger )=\mathfrak{Z }_\nu .F_\nu , \end{aligned}$$

for all but finitely many $\nu$ of weight 2 and non-trival nebentypus. On the other hand, since $\mathrm{dim}_{F_\nu }H_f^1({\mathbf{{Q}}}_p,\mathbb V _\nu ^\dagger )$ for every $\nu$ of weight 2 with non-trivial nebentypus, we see that it suffices to show that one has the implication

$$\begin{aligned} \mathfrak{Z }_\nu \ne 0\quad \Longrightarrow \quad \mathrm{loc}_\mathfrak p (\mathfrak{Z }_\nu )\ne 0 \end{aligned}$$

(5.16)

for every $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ of weight 2 and non-trivial nebentypus. (Indeed, (5.16) will show that $\mathrm{loc}_\mathfrak p$ is injective at infinitely many $\nu$, and by [19, Lemma 2.1.7] it will follow that the kernel of the localization map

$$\begin{aligned} \mathrm{loc}_\mathfrak p :\widetilde{H}_f^1(K,\mathbb{T }^\dagger )\rightarrow H^1({\mathbf{{Q}}}_p,\mathbb{T }^\dagger ) \end{aligned}$$

must be $\mathbb{I }$-torsion, hence contained in only finitely arithmetic primes).

The point $Q_{\chi _\nu }$ (5.14) defines a $K$-rational point on a twist $J_{\chi _\nu }$ of $J_s$ by the character $\chi _\nu ^{-1}$. Since the localization map

$$\begin{aligned} J_{\chi _\nu }(K)\otimes _{\mathbf{{Z}}}{\mathbf{{Q}}}\rightarrow J_{\chi _\nu }(K_\mathfrak p ) \end{aligned}$$

is injective, we thus see that (5.16) follows from (5.15), hence the result. $\square$

For any class $[\mathfrak a ]\in \mathrm{Pic}(\mathcal{O }_K)$, taking a representative $\mathfrak a \subset \mathcal{O }_K$ prime to $Np$, define

$$\begin{aligned} \mathfrak a *(A,\alpha _A,\imath _A):=(A_\mathfrak a ,\alpha _{A_\mathfrak a },\imath _{A_\mathfrak a }), \end{aligned}$$

where $A_\mathfrak a =A/A[\mathfrak{N }]$, $\alpha _{A_\mathfrak a }=A_\mathfrak a [\mathfrak{N }]$, and $\imath _{A_\mathfrak a }$ is the trivialization $\hat{A}_\mathfrak a \xrightarrow {\hat{\varphi }_\mathfrak a ^{-1}}\hat{A} \xrightarrow {\imath _{A}}\hat{\mathbf{G}}_m$ induced by the projection $\varphi _\mathfrak a :A\rightarrow A_\mathfrak a$.

Theorem 5.8

Let $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ have weight $2$ and non-trivial wild character $\psi _\nu$, and let $s>1$ be the $p$-power of the conductor of $\psi _\nu$. Then

$$\begin{aligned} \sum _{[\mathfrak a ]\in \mathrm{Pic}(\mathcal{O }_K)}d^{-1}{\mathbf{{f}}}_\nu ^{[p]} \otimes \theta _\nu ^{-1}(\mathfrak a *(A,\alpha _A,\imath _A)) \!=\!u\frac{\nu ({\mathbf{{a}}}_p)^s}{G(\theta _\nu ^{-1})}\mathrm{log}_{s,V_{{\mathbf{{f}}}_\nu }(1)}(\mathrm{loc}_\mathfrak p (\mathfrak{Z }_\nu ))(\omega _{{\mathbf{{f}}}_\nu ^*}),\nonumber \\ \end{aligned}$$

(5.17)

where $u=\vert \mathcal{O }_K^\times \vert /2$, and $G(\theta _\nu ^{-1})$ is the Gauss sum $\sum _{x\;\mathrm{mod}\;p^s}\theta _\nu ^{-1}(x)\zeta _s^{x}$.

Proof

Since clearly $d^{-1}{\mathbf{{f}}}_\nu ^{[p]}\otimes \theta _\nu ^{-1}=d^{-1}{\mathbf{{f}}}_\nu ^{*[p]} \otimes \theta _\nu$, letting $F_s$ be the completion of $\imath _p(L_{p^s})$ it suffices to establish the equality

$$\begin{aligned} d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}\otimes \theta _\nu (A,\alpha _A,\imath _A) =u\frac{\nu ({\mathbf{{a}}}_p)^s}{G(\theta _\nu ^{-1})}\mathrm{log}_{F_s,V_{{\mathbf{{f}}}_\nu }(1)}(\mathrm{loc}_\mathfrak p (\mathfrak{X }_\nu ))(\omega _{{\mathbf{{f}}}_\nu ^*}). \end{aligned}$$

(5.18)

Combining the formulas from Lemma 5.2 and Proposition 5.3, we have

$$\begin{aligned} d^{-1}{\mathbf{{f}}}_\nu ^{*[p]}\otimes \theta _\nu (A,\alpha _A,\imath _A) =\frac{u}{G(\theta _\nu ^{-1})} \delta _{{\mathbf{{f}}}_\nu ,F_s}^{(p)}(\widetilde{Q}_{\chi _\nu }). \end{aligned}$$

(5.19)

Now the integer $s>1$ is such that the natural map $\mathbb{T }\rightarrow \mathbb V _\nu$ can be factored as

$$\begin{aligned} \mathbb{T }\rightarrow \mathrm{Ta}_p^\mathrm{ord}(J_s)\rightarrow \mathbb V _\nu , \end{aligned}$$

(5.20)

and we have $\mathbb V _\nu ^\dagger \cong \mathbb V _\nu$ as $G_{L_{p^s}}$-modules. Tracing through the construction of $\mathfrak{X }$, we see that the image of $U_p^s\cdot \mathfrak{X }_\nu$ in $H^1(L_{p^s},\mathbb V _\nu ^\dagger )$ agrees with the image of $\widetilde{Q}_{\chi _\nu }$ under the composite map (where the unlabelled arrow is induced by (5.20))

$$\begin{aligned}&J_s(L_{p^s})\otimes _{{\mathbf{{Z}}}}F_\nu \xrightarrow {\mathrm{Kum}_s} H^1(L_{p^s},\mathrm{Ta}_p(J_s)\otimes _{{\mathbf{{Z}}}}F_\nu )\xrightarrow {e^\mathrm{ord}}H^1(L_{p^s},\mathrm{Ta}_p^\mathrm{ord}(J_s)\otimes _{{\mathbf{{Z}}}}F_\nu )\nonumber \\&\quad \longrightarrow H^1(L_{p^s},\mathbb V _\nu )\cong H^1(L_{p^s},\mathbb V _\nu ^\dagger ). \end{aligned}$$

(5.21)

Since $U_p$ acts on $\mathbb V _\nu ^\dagger$ as multiplication by $\nu ({\mathbf{{a}}}_p)$, we thus arrive at the equality

$$\begin{aligned} \mathrm{Kum}_s(e^\mathrm{ord}\widetilde{Q}_{\chi _\nu })=\nu ({\mathbf{{a}}}_p)^s\cdot \mathrm{res}_{L_{p^s}/H}(\mathfrak{X }_\nu )\in H^1(L_{p^s},\mathbb V _\nu ). \end{aligned}$$

(5.22)

By [32, Prop. 1.6.8], this shows that the restriction to $\mathrm{loc}_\mathfrak p (\mathfrak{X }_\nu )$ to $G_{F_s}$ is contained in the Bloch-Kato finite subspace $H_f^1(F_s,\mathbb V _\nu )\cong H_f^1(F_s,\mathbb V _\nu ^\dagger )$. Since the map $\delta _{{\mathbf{{f}}}_\nu ,F_s}^{(p)}$ is defined by the commutativity of the diagram

we thus see that (5.18) follows from (5.19) and (5.22). $\square$

Remark 5.9

The expression in the left hand side of (5.17) can be interpreted as the value of a certain $p$-adic Rankin $L$-series at a point outside the range of classical interpolation, and hence Theorem 5.8 may be seen as a $p$-adic analogue of the Gross-Zagier formula for the classes $\mathfrak{Z }_\nu$, in the same spirit as the main result of [1]. This interpretation, which does not play a direct role in this paper, is studied further in the companion paper [5].

5.2 Higher weight specializations

Now we can prove our main result. Recall from the Introduction that $f_o$ is a $p$-ordinary newform of level $N$ prime to $p$, even weight $k\ge 2$ and trivial nebentypus, that

$$\begin{aligned} {\mathbf{{f}}}=\sum _{n>0}{\mathbf{{a}}}_nq^n\in \mathbb{I }[[q]] \end{aligned}$$

is the Hida family passing through the ordinary $p$-stabilization of $f_o$, and that $K$ is an imaginary quadratic field such that every prime factor of $pN$ is split in $K$.

If ${\mathbf{{f}}}_\nu$ is the ordinary $p$-stabilization of a $p$-ordinary newform ${\mathbf{{f}}}_\nu ^\sharp$ of even weight $2r_\nu >2$ and trivial nebentypus, the Heegner cycle $\Delta _{A,r_\nu }^\mathrm{heeg}$ has been defined in Sect. 3, and by [28, Thm. (3.1)(i)] the class

$$\begin{aligned} \Phi ^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}_{{\mathbf{{f}}}_\nu ^\sharp ,K}(\Delta _{r_\nu }^\mathrm{heeg}) :=\mathrm{Cor}_{H/K}(\Phi ^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}_{{\mathbf{{f}}}_\nu ^\sharp ,H}(\Delta _{A,r_\nu }^\mathrm{heeg})) \end{aligned}$$

(5.23)

lies in the Bloch-Kato Selmer group $H_f^1(K,V_{{\mathbf{{f}}}_\nu ^\sharp }(r_\nu ))$.

On the other hand, by [19, Prop. 2.4.5], the big Heegner point $\mathfrak{X }$ lies in the strict Greenberg Selmer group $\mathrm{Sel}_\mathrm{Gr}(H,\mathbb{T }^\dagger )$ (defined in [loc.cit., Def. 2.4.2]), and since $\mathrm{Sel}_\mathrm{Gr}(K,\mathbb V _\nu ^\dagger )\cong H_f^1(K,\mathbb V _\nu ^\dagger )$ as explained in [19, p. 114]) and $\mathbb V _\nu ^\dagger \cong V_{{\mathbf{{f}}}_\nu ^\sharp }(r_\nu )$ by Lemma 4.2, the class

$$\begin{aligned} \mathfrak{Z }_\nu =\mathrm{Cor}_{H/K}(\mathfrak{X }_\nu ) \end{aligned}$$

naturally lies in $H_f^1(K,V_{{\mathbf{{f}}}_\nu ^\sharp }(r_\nu ))$ as well. Our main result relates these two classes.

Recall that the following hypotheses are being assumed throughout this paper.

Assumption 5.10

The residual representation $\bar{\rho }_{f_o}$ is irreducible, and the semi-simplification of $\overline{\rho }_{f_o}\vert _{G_{{\mathbf{{Q}}}_p}}$ is non-scalar.

Theorem 5.11

Let $\nu _o$ be the arithmetic prime of $\mathbb{I }$ such that ${\mathbf{{f}}}_{\nu _o}$ is the ordinary $p$-stabilization of $f_o$, and let $\mathbb{T }^\dagger =\mathbb{T }\otimes \Theta ^{-1}$ be the critical twist of $\mathbb{T }$ such that $\vartheta _{\nu _o}$ is the trivial character⁶. Assume that there is a $\nu ^{\prime }\in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ of weight 2 and non-trivial nebentypus such that

$$\begin{aligned} L^{\prime }(1,{\mathbf{{f}}}_{\nu ^{\prime }},\chi _{\nu ^{\prime }})\ne 0. \end{aligned}$$

(5.24)

Then for all but finitely many arithmetic primes $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ of weight $2r_\nu >2$ with $2r_{\nu }\equiv k\pmod {2(p-1)}$, we have

$$\begin{aligned} \langle \mathfrak{Z }_{\nu },\mathfrak{Z }_\nu \rangle _K =\left(1-\frac{p^{r_\nu -1}}{\nu ({\mathbf{{a}}}_p)}\right)^4 \frac{\langle \Phi _{{\mathbf{{f}}}_\nu ^\sharp ,K}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Delta ^\mathrm{heeg}_{r_\nu }), \Phi _{{\mathbf{{f}}}_\nu ^\sharp ,K}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Delta ^\mathrm{heeg}_{r_\nu })\rangle _K}{{u^2(4D)^{r_\nu -1}}}, \end{aligned}$$

(5.25)

where $\langle ,\rangle _K$ is the cyclotomic $p$-adic height pairing on $H_f^1(K,V_{{\mathbf{{f}}}_\nu ^\sharp }(r_\nu ))$, $u=\vert \mathcal{O }_K^\times \vert /2$, and $-D<0$ is the discriminant of $K$.

Proof

Since $\mathfrak{Z }\in \mathrm{Sel}_\mathrm{Gr}(K,\mathbb{T }^\dagger )$, the localization $\mathrm{loc}_\mathfrak p (\mathfrak{Z })$ lies in the kernel of the natural map

$$\begin{aligned} H^1({\mathbf{{Q}}}_p,\mathbb{T }^\dagger )\rightarrow H^1({\mathbf{{Q}}}_p,{\fancyscript{F}}_w^-\mathbb{T }^\dagger ), \end{aligned}$$

and since $H^0({\mathbf{{Q}}}_p,{\fancyscript{F}}_w^-\mathbb{T }^\dagger )=0$ by [19, Lemma 2.4.4], the class $\mathrm{loc}_\mathfrak p (\mathfrak{Z })$ can be seen as sitting inside $H^1({\mathbf{{Q}}}_p,{\fancyscript{F}}_w^+\mathbb{T }^\dagger )$. Thus upon taking an $\mathbb{I }$-basis ${\eta }$ of $\mathbb{D }$, we can form

$$\begin{aligned} \mathcal{L }_\mathfrak p ^\mathrm{arith}({\mathbf{{f}}}^\dagger ):=u\cdot \mathrm{Log}_{\mathbb{T }^\dagger }^{({\eta })}(\mathrm{loc}_\mathfrak p (\mathfrak{Z }))\in \mathbb{I }[\lambda ^{-1}]\quad \quad (\lambda :={\mathbf{{a}}}_p-1). \end{aligned}$$

On the other hand, consider the continuous function on $\mathrm{Spf}(\mathbb{I })(\overline{{\mathbf{{Q}}}}_p)$ given by

$$\begin{aligned} \mathcal{L }_\mathfrak p ^\mathrm{analy}({\mathbf{{f}}}^\dagger ):\nu \mapsto \sum _{[\mathfrak a ]\in \mathrm{Pic}(\mathcal{O }_K)}d^{-1}{\mathbf{{f}}}_\nu ^{[p]}\otimes \theta _\nu ^{-1} (\mathfrak a *(A,\alpha _A,\imath _A)). \end{aligned}$$

(Its continuity can be checked by staring at the $q$-expansion of $d^{-1}{\mathbf{{f}}}_\nu ^{[p]}\otimes \theta _\nu ^{-1}$ and appealing to the results in [14, § I.3.5], for example.)

By the specialization property (4.8) of the map $\mathrm{Log}_{\mathbb{T }^\dagger }^{({\eta })}$, we see that Theorem 5.8 can be reformulated as follows: For every $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ of weight 2 and non-trivial wild character, there exists a unit $\Omega _{\nu }^{(\eta )}\in \mathcal{O }_\nu ^\times$ such that

$$\begin{aligned} \nu \left(\mathcal{L }_\mathfrak p ^\mathrm{analy}({\mathbf{{f}}}^\dagger )\right)= \Omega _{\nu }^{(\eta )}\cdot \nu \left(\mathcal{L }_\mathfrak p ^\mathrm{arith}({\mathbf{{f}}}^\dagger )\right). \end{aligned}$$

(5.26)

In fact,

$$\begin{aligned} \Omega _{\nu }^{(\eta )}=\langle \eta _\nu \otimes {e_\zeta ^{\otimes r_\nu }},\omega _{{\mathbf{{f}}}_\nu ^*}\rangle _\mathrm{dR} \end{aligned}$$

(5.27)

under the pairing (4.7), so that $\omega _{{\mathbf{{f}}}_\nu ^*}=\Omega _{\nu }^{(\eta )}\cdot \eta _\nu ^{\prime }$ with $\eta _\nu ^{\prime }$ as defined in Theorem 4.4.⁷ Since both $\mathfrak{L }_\mathfrak p ^\mathrm{arith}({\mathbf{{f}}}^\dagger )$ and $\mathfrak{L }_\mathfrak p ^\mathrm{analy}({\mathbf{{f}}}^\dagger )$ are continuous functions of $\nu$, (5.26) shows that the map $\nu \mapsto \Omega _{\nu }^{(\eta )}$ is continuous, and hence (5.27) is valid for all $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$.

Now let $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$ be as in the statement. Then $\theta _{\nu }(z)=z^{r_\nu -1}\vartheta _{\nu }(z)=z^{r_\nu -1}$ as characters on ${\mathbf{{Z}}}_p^\times$, from where if follows that

$$\begin{aligned} \nu \left(\mathcal{L }_\mathfrak p ^\mathrm{analy}({\mathbf{{f}}}^\dagger )\right)&= \sum _{[\mathfrak a ]\in \mathrm{Pic}(\mathcal{O }_K)}d^{-1}{\mathbf{{f}}}_{\nu }^{[p]}\otimes \theta _{\nu }^{-1}(\mathfrak a *(A,\alpha _A,\imath _A))\nonumber \\&= \sum _{[\mathfrak a ]\in \mathrm{Pic}(\mathcal{O }_K)}d^{-r_\nu }{\mathbf{{f}}}_{\nu }^{[p]}(\mathfrak a *(A,A[\mathfrak{N }])). \end{aligned}$$

By Theorem 3.3, setting

$$\begin{aligned} \Delta _{r_\nu }^\mathrm{bdp}\!:=\!\mathrm{Norm}_{H/K}(\Delta _{\varphi _{(1)},r_\nu }^\mathrm{bdp}) \!=\!\sum _{[\mathfrak a ]\in \mathrm{Pic}(\mathcal{O }_K)}\mathrm{N}\mathfrak a ^{1-r}\cdot \Delta _{\varphi _\mathfrak a ,r_\nu }^\mathrm{bdp}\in \mathrm{CH}^{2r_\nu -1}(X_{r_\nu })_0(K),\nonumber \\ \end{aligned}$$

(5.28)

this shows that

$$\begin{aligned} \nu \left(\mathcal{L }_\mathfrak p ^\mathrm{analy}({\mathbf{{f}}}^\dagger )\right)&= \mathcal{E }_\nu (r_\nu )\mathcal{E }_\nu ^*(r_\nu ) \frac{(-1)^{r_\nu -1}}{(r_\nu -1)!} \mathrm{AJ}_{{\mathbf{{Q}}}_p}(\Delta _{r_\nu }^\mathrm{bdp})(\omega _{{\mathbf{{f}}}_\nu ^\sharp }\otimes e_\zeta ^{\otimes r_\nu -1})\nonumber \\&= \mathcal{E }_\nu (r_\nu )\mathcal{E }_\nu ^*(r_\nu ) \frac{(-1)^{r_\nu -1}}{(r_\nu -1)!}\mathrm{log}_\mathbb{V _\nu ^\dagger }(\mathrm{loc}_\mathfrak p (\Phi ^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}_{{\mathbf{{f}}}_\nu ^\sharp ,K} (\Delta _{r_\nu }^\mathrm{bdp})))(\omega _{{\mathbf{{f}}}_\nu ^\sharp }\otimes e_\zeta ^{\otimes r_\nu -1}),\nonumber \\ \end{aligned}$$

(5.29)

where

$$\begin{aligned} \mathcal{E }_{\nu }(r_\nu ):=\left(1-\frac{p^{r_\nu -1}}{\nu ({\mathbf{{a}}}_p)}\right),\quad \quad \mathcal{E }_{\nu }^*(r_\nu ):=\left(1-\frac{\nu ({\mathbf{{a}}}_p)}{p^{r_\nu }}\right), \end{aligned}$$

and $\Phi ^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}_{{\mathbf{{f}}}_\nu ^\sharp ,K} :=\pi _{{\mathbf{{f}}}_\nu ^\sharp ,\mathrm{N}^{r_\nu -1}}\circ \Phi _{K}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}$ with notations as in the diagram (3.3) defining $\mathrm{AJ}_{{\mathbf{{Q}}}_p}$.

On the other hand, by the specialization property of the map $\mathrm{Log}_{\mathbb{T }^\dagger }^{({\eta })}$ we have

$$\begin{aligned} \nu \left(\mathcal{L }_\mathfrak p ^\mathrm{arith}({\mathbf{{f}}}^\dagger )\right)=u\frac{(-1)^{r_{\nu }-1}}{(r_{\nu }-1)!}\mathcal{E }_{\nu }(r_\nu )^{-1}\mathcal{E }_{\nu }^*(r_\nu ) \mathrm{log}_\mathbb{V _{\nu }^\dagger }(\mathrm{loc}_\mathfrak p (\mathfrak{Z }_{\nu }))(\eta _\nu ^{\prime }). \end{aligned}$$

(5.30)

Comparing (5.30) and (5.29), we thus conclude form (5.26) that

$$\begin{aligned} \mathrm{log}_\mathbb{V _{\nu }^\dagger }(\mathrm{loc}_\mathfrak p (\mathfrak{Z }_{\nu }))(\omega _{{\mathbf{{f}}}_{\nu }^\sharp } \!\otimes \! e_\zeta ^{\otimes r_\nu -1}) \!=\! \frac{1}{u}\mathcal{E }_{\nu }(r_\nu )^2\mathrm{log}_\mathbb{V _{\nu }^\dagger }(\mathrm{loc}_\mathfrak p (\Phi ^{{\acute{\mathrm{e}}}\mathrm{t}}_{{\mathbf{{f}}}_{\nu }^\sharp ,K}(\Delta _{r_\nu }^\mathrm{bdp})))(\omega _{{\mathbf{{f}}}_{\nu }^\sharp }\!\otimes \! e_\zeta ^{\otimes r_\nu -1}). \end{aligned}$$

Since $\mathrm{Fil}^{1}D_\mathrm{dR}(V_{{\mathbf{{f}}}_\nu ^\sharp }(r_\nu -1))$ is spanned by $\omega _{{\mathbf{{f}}}_{\nu }^\sharp }\otimes e_\zeta ^{\otimes r_\nu -1}$, it follows that

$$\begin{aligned} \mathrm{log}_\mathbb{V _{\nu }^\dagger }(\mathrm{loc}_\mathfrak p (\mathfrak{Z }_{\nu }))=\frac{1}{u}\mathcal{E }_{\nu }(r_\nu )^2\mathrm{log}_\mathbb{V _{\nu }^\dagger }(\mathrm{loc}_\mathfrak p (\Phi ^{{\acute{\mathrm{e}}}\mathrm{t}}_{{\mathbf{{f}}}_{\nu }^\sharp ,K}(\Delta _{r_\nu }^\mathrm{bdp}))), \end{aligned}$$

and since $\mathrm{log}_\mathbb{V _{\nu }^\dagger }$ is an isomorphism, that

$$\begin{aligned} \text{ loc}_\mathfrak p (\mathfrak{Z }_{\nu })=\frac{1}{u}\mathcal{E }_{\nu }(r_\nu )^2\mathrm{loc}_\mathfrak p (\Phi ^{{\acute{\mathrm{e}}}\mathrm{t}}_{{\mathbf{{f}}}_{\nu }^\sharp ,K}(\Delta _{r_\nu }^\mathrm{bdp})). \end{aligned}$$

(5.31)

Our nonvanishing assumption (5.24) implies on the one hand, by Theorem 5.6, that $\mathfrak{Z }_\nu$ is non-zero for all but finitely many $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$, and on the other hand, by Corollary 5.7, that the localization map $\mathrm{loc}_\mathfrak p$ is injective for all but finitely many $\nu \in \mathcal{X }_\mathrm{arith}(\mathbb{I })$. In particular, we thus see from (5.31) that we have

$$\begin{aligned} \langle \mathfrak{Z }_{\nu },\mathfrak{Z }_\nu \rangle _K&= \frac{1}{u^2}\mathcal{E }_{r_\nu }(r_\nu )^4 \langle \Phi _{{\mathbf{{f}}}_\nu ^\sharp ,K}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Delta ^\mathrm{bdp}_{r_\nu }), \Phi _{{\mathbf{{f}}}_\nu ^\sharp ,K}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Delta ^\mathrm{bdp}_{r_\nu })\rangle _K\\&= \frac{1}{u^2}\mathcal{E }_{r_\nu }(r_\nu )^4 \frac{\langle \Phi _{{\mathbf{{f}}}_\nu ^\sharp ,K}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Delta ^\mathrm{heeg}_{r_\nu }), \Phi _{{\mathbf{{f}}}_\nu ^\sharp ,K}^{{{{\acute{\mathrm{e}}}}\mathrm{t}}}(\Delta ^\mathrm{heeg}_{r_\nu })\rangle _K}{{(4D)^{r_\nu -1}}} \end{aligned}$$

for all but finitely many $\nu$ as in the statement, where the last equality follows from Lemma 3.4 in light of the definitions (5.23) and (5.28). The result follows. $\square$