1.1 Digression: rigid analytic differentials on $\mathcal H_p$

This section briefly summarises a few key facts about rigid analytic differentials on $\mathcal H_p$ and their connection with harmonic cocycles on the Bruhat-Tits tree of $\mathbf { GL}_2(\mathbb {Q}_p)$ . A more thorough exposition can be found in [Reference Dasgupta and Teitelbaum8] for example.

Let ${\mathcal T} := {\mathcal T}_0 \sqcup {\mathcal T}_1$ denote the Bruhat–Tits tree of $\mathbf {SL}_2(\mathbb {Q}_p)$ , whose set ${\mathcal T}_0$ of vertices is in bijection with homothety classes of $\mathbb {Z}_p$ -lattices in $\mathbb {Q}_p^2$ , two lattices being joined by an edge in ${\mathcal T}_1\subset {\mathcal T}_0^2$ if they are represented by lattices contained one in the other with index p. There is a natural reduction map

$$ \begin{align*}{\mathrm {red}}\colon \mathcal H_p \longrightarrow {\mathcal T}\end{align*} $$

from $\mathcal H_p$ to ${\mathcal T}$ , which maps the standard affinoid

(4) $$ \begin{align} \mathcal A_{\circ} := \{ z \in {\mathcal O}_{\mathbb{C}_p} \mbox{ such that } |z-a|\ge 1, \mbox{ for all } a\in \mathbb{Z}_p \} \ \subset \ \mathbb{P}_1(\mathbb{C}_p) \end{align} $$

to the vertex attached to the lattice $v_\circ = [\mathbb {Z}_p^2]$ . The $p+1$ mod p residue discs in the complement of $\mathcal A_{\circ }$ are in natural bijection with $\mathbb {P}_1(\mathbb {F}_p)$ and contain the boundary annuli

(5) $$ \begin{align} \mathcal W_{\infty} &= \{ z\in \mathbb{P}_1(\mathbb{C}_p) \mbox{ such that } 1< |z|<p \}, \\ \nonumber \mathcal W_{j} &= \{ z\in \mathbb{P}_1(\mathbb{C}_p) \mbox{ such that } 1/p< |z-j|<1 \}, \mbox{ for } j=0,\ldots, p-1. \end{align} $$

The edges having $v_\circ $ as an endpoint are likewise in bijection with $\mathbb {P}_1(\mathbb {F}_p)$ by setting

(6) $$ \begin{align} e_\infty \leftrightarrow (\mathbb{Z}_p^2, \ \mathbb{Z}_p\cdot (1,0) + p \mathbb{Z}_p^2), \qquad e_j \leftrightarrow (\mathbb{Z}_p^2, \ \mathbb{Z}_p\cdot (j,1) + p \mathbb{Z}_p^2), \ \ \mbox{ for } j=0,\ldots, p-1. \end{align} $$

The preimage of the singleton $\{e_j\}$ under the reduction map is the annulus $\mathcal W_j$ , for each $j\in \mathbb {P}_1(\mathbb {F}_p)$ . The properties

$$ \begin{align*}\mathrm{red}^{-1}(\{v_\circ\}) = \mathcal A_{\circ}, \qquad \mathrm{red}^{-1}(\{e_j\}) = \mathcal W_j, \ \ \mbox{ for all } j\in \mathbb{P}_1(\mathbb{F}_p),\end{align*} $$

combined with the requirement of compatibility with the natural actions of $\mathbf {SL}_2(\mathbb {Q}_p)$ on $\mathcal H_p$ and on ${\mathcal T}$ , determine the reduction map uniquely. In particular, the preimage, denoted $\mathcal A_v$ , of the vertex $v\in {\mathcal T}_0$ is an affinoid obtained by taking the complement in $\mathbb {P}(\mathbb {C}_p)$ of $(p+1)$ mod p residue discs with $\mathbb {Q}_p$ -rational centers (relative to a coordinate on $\mathbb {P}(\mathbb {Q}_p)$ depending on v). Given an edge $e=(v_1,v_2) \in {\mathcal T}_1$ , the affinoids $\mathcal A_{v_1}$ and $\mathcal A_{v_2}$ are glued to each other along the p-adic annulus attached to e, denoted $\mathcal W_e$ . With just a modicum of artistic licence, the entire Drinfeld upper-half plane can be visualised as a tubular neighbourhood of ${\mathcal T},$ as in the figure below for $p=2$ .

Figure 1

The Drinfeld upper half plane and the Bruhat-Tits tree.

This picture suggests that $\mathcal H_p$ , unlike its Archimedean counterpart, is far from being simply connected and that its first cohomology is quite rich. For each edge $e\in {\mathcal T}_1$ , the de Rham cohomology of $\mathcal W_e$ is identified with $\mathbb {C}_p$ via the map that sends $\omega \in \Omega ^1_{\mathrm {rig}}(\mathcal W_e)$ to its p-adic annular residue, denoted ${\mathrm {res}}_{\mathcal W_e}(\omega )$ . This residue map is well defined up to a sign, which is determined by fixing an orientation on $\mathcal W_e$ , or, equivalently, viewing e as an ordered edge of ${\mathcal T}$ , having a source and target. Let $\mathcal E({\mathcal T})$ denote the set of such ordered edges, let $s,t:\mathcal E({\mathcal T}) \rightarrow {\mathcal T}_0$ denote the source and target maps, and write $\bar e$ for the edge e with its source and target interchanged.

Definition 1.1. A harmonic cocycle on ${\mathcal T}$ is a $\mathbb {C}_p$ -valued function

$$ \begin{align*}c: \mathcal E({\mathcal T}) \rightarrow \mathbb{C}_p\end{align*} $$

satisfying the following properties:

• • $ c(\bar e) =-c(e)$ , for all $e\in \mathcal E({\mathcal T})$ ;

• • for all vertices v of ${\mathcal T}$ ,

  $$ \begin{align*}\sum_{s(e)=v} c(e) = \sum_{t(e)=v} c(e)=0.\end{align*} $$

The $\mathbb {C}_p$ -vector space of $\mathbb {C}_p$ -valued harmonic cocycles on ${\mathcal T}$ is denoted ${\mathrm C}_{har}({\mathcal T}, \mathbb {C}_p)$ . The class of a rigid analytic differential $\omega \in \Omega ^1_{\mathrm {rig}}(\mathcal H_p)$ in the de Rham cohomology of $\mathcal H_p$ is encoded in the $\mathbb {C}_p$ -valued function $c_\omega $ on $\mathcal E({\mathcal T})$ defined by

$$ \begin{align*}c_\omega(e) = {\mathrm {res}}_{\mathcal W_e}(\omega).\end{align*} $$

That $c_\omega $ is a harmonic cocycle follows directly from the residue theorem for rigid differentials. The oriented edges of ${\mathcal T}$ are also in natural bijection with the compact open balls in $\mathbb {P}_1(\mathbb {Q}_p)$ , by assigning to $e\in \mathcal E({\mathcal T})$ the ball $U_e$ according to the following prescriptions:

$$ \begin{align*}U_{\bar e} \sqcup U_e = \mathbb{P}_1(\mathbb{Q}_p), \qquad U_{e_\infty} = \mathbb{P}_1(\mathbb{Q}_p) - \mathbb{Z}_p, \qquad U_{\gamma e} = \gamma U_e, \mbox{ for all } \gamma \in \Gamma,\end{align*} $$

where $e_\infty $ is the distinguished edge of $\mathcal E({\mathcal T})$ evoked in (6). The harmonic cocycle $c_\omega $ can therefore be parlayed into a $\mathbb {C}_p$ -valued distribution $\mu _\omega $ satisfying the defining property

$$ \begin{align*}\mu_\omega(U_e) = c_\omega(e),\end{align*} $$

where $U_e\subset \mathbb {P}_1(\mathbb {Q}_p)$ is the open ball corresponding to the ordered edge e, with zero total mass

$$\begin{align*}\mu_\omega(\mathbb{P}_1(\mathbb{Q}_p))=0. \end{align*}$$

The distribution $\mu _\omega $ belongs to the dual of the Steinberg representation (i.e., the dual space of locally constant $\mathbb {C}_p$ -valued functions on $\mathbb {P}_1(\mathbb {Q}_p)$ modulo constant functions).

For this paragraph, and this paragraph only, let $\Gamma \subset \mathbf {SL}_2(\mathbb {Q}_p)$ be a group acting discretely on $\mathcal H_p$ and for which the quotient graph $\Gamma \backslash {\mathcal T}$ is finite. If the rigid differential $\omega $ is $\Gamma $ -invariant, the harmonic cocycle $c_\omega $ takes on finitely many values and is therefore p-adically bounded. The distribution $\mu _\omega $ then extends to a $\mathbb {C}_p$ -valued measure, which can be integrated against continuous functions on $\mathbb {P}_1(\mathbb {Q}_p)$ . The differential $\omega $ can then be obtained from $\mu _\omega $ by the rule

(7) $$ \begin{align} \omega = \int_{\mathbb{P}_1(\mathbb{Q}_p)} \frac{\mathrm{d}z}{z-t}\hspace{1mm} \mathrm{d}\mu_\omega(t), \end{align} $$

a special case of Jeremy Teitelbaum’s p-adic Poisson Kernel formula [Reference Teitelbaum31] which recovers a rigid analytic modular form on $\mathcal H_p$ from its associated boundary distribution.

1.2 Modular form-valued harmonic cocycles as mock residues

Returning to the setting where $\Gamma = \mathbf {SL}_2(\mathbb {Z}[1/p])$ and to the dubious notion of a mock Hilbert modular form on $\Gamma \backslash (\mathcal H_p\times \mathcal H)$ proposed in (2), the discussion in the previous section suggests at least what its system of p-adic annular residues ought to look like:

Definition 1.2. A system of mock residues is a harmonic cocycle

$$ \begin{align*}c : {\mathcal E}({\mathcal T}) \longrightarrow \Omega^1({\mathcal H})\end{align*} $$

with values in the space $\Omega ^1({\mathcal H})$ of holomorphic differentials on $\mathcal H$ , satisfying

• • $c(e)$ is a weight two cusp form on the stabiliser $\Gamma _e$ of e in $\Gamma $ (i.e., a holomorphic differential on the standard compactification of $\Gamma _e\backslash \mathcal H$ );

• • for all $\gamma \in \Gamma $ and all $e \in \mathcal E({\mathcal T})$ , the following equality holds:

  $$ \begin{align*}\gamma^* c(\gamma e) = c(e).\end{align*} $$

Roughly speaking, a system of mock residues is what one might expect to obtain from the p-adic annular residues of a mock Hilbert modular form of parallel weight two. But unlike (2), Definition 1.2 is completely rigorous. Since $\Gamma $ acts transitively on the unordered edges of ${\mathcal T}$ , and because the Hecke congruence group $\Gamma _0(p)$ is the stabiliser in $\Gamma $ of the distinguished edge $e_\infty \in \mathcal E({\mathcal T})$ of (6), the map $c \mapsto c(e_\infty )$ identifies the complex vector space $\mathcal C_{\mathrm {har}}({\mathcal T},\Omega ^1(\mathcal H))^{\Gamma }$ of mock residues for $\Gamma $ with the space $S_2(\Gamma _0(p))^{p\mbox {-}\mathrm {new}}$ of weight two newforms of level p. It transpires that mock Hilbert modular forms – or at least, their systems of mock p-adic residues – are merely a slightly overwrought incarnation of classical modular forms of weight two on $\Gamma _0(p)$ .

1.3 $\mathbb {C}$ -valued distributions

Given a p-new weight two cusp form f on $\Gamma _0(p)$ , denote by $c_f$ the associated mock residue, and write $f_e:= c_f(e) \in S_2(\Gamma _e)$ . For any $x,y\in \mathcal H^* := \mathcal H \sqcup \mathbb {P}_1(\mathbb {Q})$ , the assignment

(8) $$ \begin{align} e \mapsto 2\pi i\int_x^y f_e(z)\hspace{0.5mm} \mathrm{d}z \end{align} $$

is a $\mathbb {C}$ -valued harmonic cocycle on ${\mathcal T}$ , denoted $c_f[x,y]$ . It determines a $\mathbb {C}$ -valued distribution $\mu _f[x,y]$ on $\mathbb {P}_1(\mathbb {Q}_p)$ , which can be integrated against locally constant complex-valued functions on $\mathbb {P}_1(\mathbb {Q}_p)$ . In order to integrate $\mu _f[x,y]$ against Teitelbaum’s p-adic Poisson kernel as in (7), the distribution $\mu _f[x,y]$ needs to be upgraded to a measure with suitable integrality properties.

1.4 Modular symbols

Suppose henceforth that f is a Hecke eigenform with rational fourier coefficients, and let $E_{/\mathbb {Q}}$ denote the corresponding strong Weil curve. The theory of modular symbols shows that the values of the harmonic cocycle $c_f[x,y]$ acquire good integrality properties when $x,y$ belong to the boundary $\mathbb {P}_1(\mathbb {Q})$ of the extended upper half plane. More precisely, Manin and Drinfeld have shown that the values

$$ \begin{align*} c_f[r,s](e) &= 2\pi i\int_r^s f_e(z) \hspace{0.5mm} \mathrm{d}z \qquad r,s\in \mathbb{P}_1(\mathbb{Q}) \\ &= 2\pi i\int_{\gamma r}^{\gamma s} f(z) \hspace{0.5mm} \mathrm{d}z, \qquad \mbox{where } \gamma\in \Gamma \mbox{ satisfies } \gamma e = e_{\infty} \end{align*} $$

belong to a lattice $\Lambda _f \subset \mathbb {C}$ which is commensurable with the period lattice $\Lambda _E$ of E. Restricting the function $(x,y) \mapsto c_f[x,y]$ to $\mathbb {P}_1(\mathbb {Q})\times \mathbb {P}_1(\mathbb {Q})$ leads to a modular symbol with values in the space of $\Lambda _f$ -valued harmonic cocycles on ${\mathcal T}$ . For economy of notation, the resulting $\Lambda _f$ -valued measures on $\mathbb {P}_1(\mathbb {Q}_p)$ will continue to be denoted

$$ \begin{align*}\mu_f[r,s] \ \in \ {\mathrm {Meas}}(\mathbb{P}_1(\mathbb{Q}_p), \Lambda_f).\end{align*} $$

The $\Lambda _f$ -valued measures $\mu _f[r,s]$ are intimately connected to special values of the Hasse–Weil L-series attached to E, via the formulae

(9) $$ \begin{align} \mu_f[0,\infty](\mathbb{Z}_p) = L(E,1), \qquad \mu_f[0,\infty](\mathbb{Z}_p^\times) = (1-a_p(E)) \cdot L(E,1), \end{align} $$

where $a_p(E)= 1$ or $-1$ , depending on whether E has split or non-split multiplicative reduction at p. The Mazur–Swinnerton-Dyer p-adic L-function attached to E (viewed as taking values in $\mathbb {Q}_p\otimes \Lambda _f$ ) is the Mellin–Mazur transform of $\mu _f[0,\infty ]$ restricted to $\mathbb {Z}_p^\times $ :

(10) $$ \begin{align} L_p(E,s) = \int_{\mathbb{Z}_p^\times} \langle x\rangle^{s-1} d\mu_f[0,\infty](x). \end{align} $$

More generally, if $\chi $ is a primitive Dirichlet character of conductor c prime to p, the twisted L-values $L(E,\chi ,1)$ can be obtained analogously from the measures $\mu _f[a/c,\infty ]$ :

$$ \begin{align*} \sum_{a\in (\mathbb{Z}/c\mathbb{Z})^\times} \overline{\chi}(a) \cdot\mu_f[-a/c,\infty](\mathbb{Z}_p) &= \frac{c}{\tau(\chi)}\cdot L(E,\chi,1), \\ \sum_{a\in (\mathbb{Z}/c\mathbb{Z})^\times} \overline{\chi}(a) \cdot\mu_f[-a/c,\infty](\mathbb{Z}_p^\times) &= \frac{c}{\tau(\chi)}\cdot\big(1-\overline{\chi}(p) a_p(E)\big) \cdot L(E,\chi,1), \end{align*} $$

where $\tau (\chi )=\sum _{a\in (\mathbb {Z}/c\mathbb {Z})^\times }\chi (a)e^{2\pi i \frac {a}{c}}$ is the Gauss sum attached to $\chi $ , while the Mazur–Swinnerton-Dyer p-adic L-function can be defined by setting

(11) $$ \begin{align} L_p(E,\chi,s) := \sum_{a\in (\mathbb{Z}/c\mathbb{Z})^\times} \chi(a) \int_{\mathbb{Z}_p^\times} \langle x\rangle^{s-1} d\mu_f[-a/c,\infty](x). \end{align} $$

Even more importantly for the constructions that will follow, a system of $\Lambda _f \otimes \mathbb {C}_p$ -valued rigid differentials on the p-adic upper half-plane can also be obtained by integrating the measures $\mu _f[r,s]$ against Teitelbaum’s Poisson kernel:

(12) $$ \begin{align} \omega_f[r,s] \ := \ \int_{\mathbb{P}_1(\mathbb{Q}_p)} \frac{\mathrm{d}z}{z-t}\hspace{1mm} \mathrm{d}\mu_f[r,s](t) \ \ \ \in \ \ \Omega^1_{\mathrm{rig}}(\mathcal H_p) \otimes \Lambda_f. \end{align} $$

The assignment $\omega _f: (r,s)\mapsto \omega _f[r,s]$ determines a $\Gamma $ -equivariant modular symbol with values in $\Omega ^1_{\mathrm {rig}}(\mathcal H_p) \otimes \Lambda _f$ , satisfying

$$ \begin{align*}\gamma^* \omega_f[\gamma r,\gamma s] = \omega_f[r,s], \qquad \mbox{ for all } \gamma \in \Gamma.\end{align*} $$

1.5 The Mazur–Tate–Teitelbaum conjecture

Write $K_p$ for the quadratic unramified extension of $\mathbb {Q}_p$ , let $\mathcal A^\times $ denote the multiplicative group of nowhere vanishing rigid analytic functions on $\mathcal H_p$ endowed with the $\Gamma $ -action induced by Möbius transformations, and let $\mathcal A^\times /K_p^\times $ be the quotient by the subgroup of constant $K_p^\times $ -valued functions. The logarithmic derivative $F \mapsto \mathrm {d}F/F$ gives a $\Gamma $ -equivariant map from $ \mathcal A^\times /K_p^\times $ to $\Omega ^1_{\mathrm {rig}}(\mathcal H_p)$ , whose image contains the rigid differentials $\omega _f[r,s]$ :

Lemma 1.4. The differentials $\omega _f[r,s]$ are in the image of the logarithmic derivative map; that is, there are elements $F_f[r,s]\in (\mathcal A^\times /K_p^\times ) \otimes \Lambda _f$ satisfying

(13) $$ \begin{align} \mathrm{dlog}\hspace{0.5mm}F_f[r,s] = \omega_f[r,s]. \end{align} $$

Proof. A partitioning of $\mathbb {P}_1(\mathbb {Q}_p)$ is a collection

(14) $$ \begin{align} \mathcal C = \{ (C_1,t_1), \ldots, (C_m,t_m)\}, \end{align} $$

where the $C_j$ are compact open subsets of $\mathbb {P}_1(\mathbb {Q}_p)$ which are mutually disjoint and satisfy

$$ \begin{align*}\mathbb{P}_1(\mathbb{Q}_p) = C_1 \sqcup \cdots \sqcup C_m, \qquad t_j\in C_j \mbox{ for } j=1,\ldots, m.\end{align*} $$

The set of partitionings of $\mathbb {P}_1(\mathbb {Q}_p)$ is equipped with a natural partial ordering in which $ \mathcal C \le \mathcal C'$ if each of the compact open subsets involved in $\mathcal C'$ is contained in one of the compact open subsets arising in $\mathcal C$ . Let $\mu $ be a $\mathbb {Z}$ -valued mesure of total measure zero on $\mathbb {P}_1(\mathbb {Q}_p)$ . Each partitioning of $\mathbb {P}_1(\mathbb {Q}_p)$ gives rise to a system of degree zero divisors on $\mathbb {P}_1(\mathbb {Q}_p)$ by associating to the partitioning $\mathcal C$ of (14) the divisor

$$ \begin{align*}{\mathscr{D}}_{\mathcal C} := \sum_{j=1}^m \mu(C_j) \cdot [t_j].\end{align*} $$

Fix a base point $z_0\in \mathcal H_p(K_p)$ and let $F_{\mathcal C}$ be the unique rational function satisfying

$$ \begin{align*}\mathrm{Divisor}(F_{\mathcal C}) = {\mathscr{D}}_{\mathcal C}, \qquad F_{\mathcal C}(z_0)= 1,\end{align*} $$

which exists because the divisor ${\mathscr {D}}_{\mathcal C}$ is supported on $\mathbb {P}_1(\mathbb {Q}_p)$ . The limit

$$ \begin{align*}F_\mu := \lim_{\mathcal C} F_{\mathcal C} \in \mathcal A^\times,\end{align*} $$

taken over any maximal chain in the set of partitionings, is a well-defined element of $\mathcal A^\times $ which depends only on $\mu $ and on the base point $z_0$ , and whose image in $\mathcal A^\times /K_{p}^\times $ does not depend on the choice of $z_0$ . Extending this construction to $\Lambda _f$ -valued measures in the obvious way and applying it to the measures $\mu _f[r,s]$ , it is readily verified that

$$ \begin{align*}F_f[r,s]:= F_{\mu_f[r,s]} \in (\mathcal A^\times/K_{p}^\times)\otimes \Lambda_f\end{align*} $$

satisfies (13).

The assignment $(r,s) \mapsto F_f[r,s]$ defines a $\Gamma $ -invariant modular symbol with values in the $\Gamma $ -module $(\mathcal A^\times /K_p^\times )\otimes \Lambda _f$ , that is,

$$ \begin{align*}F_f \in \mathrm{MS}\big((\mathcal A^\times/K_p^\times)\otimes \Lambda_f\big)^\Gamma,\end{align*} $$

where $\mathrm {MS}(\Omega )$ denotes the $\Gamma $ -module of modular symbols with values in a $\Gamma $ -module $\Omega $ . The obstruction to lifting $F_f$ to $\mathrm {MS}(\mathcal A^\times \otimes \Lambda _f)^\Gamma $ is intimately tied with the p-adic uniformisation of the elliptic curve E which has multiplicative reduction at p. Namely, let $q \in \mathbb {Q}_p^\times $ be the p-adic Tate period of E. The following theorem is a consequence of the conjecture of Mazur, Tate and Teitelbaum [Reference Mazur, Tate and Teitelbaum23] and its proof by Greenberg and Stevens [Reference Greenberg and Stevens15]:

Proof. The functor $\mathrm {MS}(-)$ of modular symbols is exact, and taking $\Gamma $ -cohomology gives the exact sequence

where $\ker (\eta )$ is finite because so is the abelianization of $\Gamma $ ([Reference Serre30], II, 1.4). The obstruction to lifting $F_f$ to $\mathrm {MS}\big ((\mathcal A^\times /q^{\mathbb {Z}})\otimes \Lambda _{f}\big )^\Gamma $ is encoded by the class $c_f=\delta (F_f)$ which can be represented by the $1$ -cocycle

$$\begin{align*}\tilde{c}_f(\gamma)=\frac{\gamma\cdot\tilde{F_f}}{\tilde{F_f}}, \end{align*}$$

where $\tilde {F_f}\in \mathrm {MS}(\mathcal {A}^\times \otimes \Lambda _f)$ is any lift of $F_f$ . Let $\log _q$ be the branch the p-adic logarithm satisfying $\log _q(q)=0$ which induces a map $K_p^\times /q^{\mathbb {Z}} \to K_p$ with finite kernel. At the cost of replacing the lattice $\Lambda _f$ with $\Lambda _f'=\frac {1}{t}\Lambda _f$ for some $t\in \mathbb {Z}$ , the claim of the theorem reduces to the equality

$$\begin{align*}\mathrm{log}_q(c_f)=0 \end{align*}$$

in $H^1\big (\Gamma , \mathrm {MS}(K_p\otimes \Lambda _{f}')\big )$ , or equivalently to

$$\begin{align*}\log(c_f)=\frac{\log(q)}{\mathrm{ord}_p(q)}\cdot \mathrm{ord}_p(c_f). \end{align*}$$

Now, [Reference Darmon6, Corollary 3.3 $\&$ Lemma 3.4] imply that $\mathrm {ord}_p(c_f)$ is nontrivial and that the two classes $\log (c_f)$ and $\mathrm {ord}_p(c_f)$ are proportional. The factor of proportionality is obtained by producing a suitable triple $(\gamma ,r,s)\in \Gamma \times \mathbb {P}_1(\mathbb {Q})^2$ such that any $1$ -coboundary b for $\Gamma $ with values in $\mathrm {MS}((K_p^\times /q^{\mathbb {Z}})\otimes \Lambda _{f}')$ satisfies $b(\gamma )[r,s]=0$ and

(15) $$ \begin{align} \log(c_f)(\gamma)[r,s]=\frac{\log(q)}{\mathrm{ord}_p(q)}\cdot \mathrm{ord}_p(c_f)(\gamma)[r,s]\qquad\&\qquad \mathrm{ord}_p(c_f)(\gamma)[r,s]\not=0. \end{align} $$

Note that the first requirement is satisfied when $\gamma \in \Gamma $ fixes r and s. The stabiliser in $\Gamma $ of any pair $(r,s)\in \mathbb {P}_1(\mathbb {Q})^2$ is generated (up to torsion) by a hyperbolic matrix $\gamma _{r,s}$ which has powers of p as its eigenvalues and fixes the differential $\omega _f[r,s]$ . The multiplicative period

$$ \begin{align*}J_f[r,s]: = \tilde{c}_f(\gamma_{r,s})[r,s](z)= \frac{\tilde F_f[r,s](\gamma_{r,s}z)}{\tilde F_f[r,s](z)}, \quad \mbox{satisfying} \quad \log( J_f[r,s]) = \int_{z}^{\gamma_{r,s} z} \omega_f[r,s]\end{align*} $$

does not depend on the base point $z\in \mathcal H_p(K_p)$ and belongs to $\mathbb {Q}_p^\times \otimes \Lambda _f'$ [Reference Darmon6, Prop. 2.7]. When $(r,s)= (0,\infty )$ , the period is related to the central critical value $L(E,1)$ and to the first derivative of the Mazur–Swinnerton-Dyer p-adic L-function $L_p(E,s)$ attached to E in (10):

$$ \begin{align*}{\mathrm{ord}}_p(J_f[0,\infty]) = \delta_p(E) \cdot L(E,1), \qquad \log(J_f[0,\infty]) = \delta_p(E) \cdot L_p'(E,1), \end{align*} $$

where $ \delta _p(E) = 1$ if $a_p(E)=1$ and $\delta _p(E)=0$ if $a_p(E) =-1$ (cf. [Reference Darmon6, §2.2 $\&$ 2.3]). As the Mazur–Tate–Teitelbaum conjecture asserts that

$$ \begin{align*}L_p'(E,1) = \frac{\log(q)}{{\mathrm{ord}}_p(q)}\cdot L(E,1),\end{align*} $$

we deduce that

$$\begin{align*}\log(J_f[0,\infty])=\frac{\log(q)}{{\mathrm{ord}}_p(q)}\cdot\mathrm{ord}_p(J_f[0,\infty]). \end{align*}$$

More generally, the valuations and logarithms of the periods $J_f[\infty ,a/c]$ with $\gcd (a,c)=1$ can be expressed in terms the special values (resp. derivatives) of partial L-series (resp. partial p-adic L-series) whose linear combinations give all the twisted values $L(E,\chi ,1)$ and $L_p'(E,\chi ,1)$ defined in (11), as $\chi $ ranges over all primitive Dirichlet characters of conductor c for which $\chi (p)= a_p(E)$ [Reference Darmon6, §2.2 $\&$ 2.3]. The Mazur–Tate–Teitelbaum conjecture for these L-series and the nonvanishing result of [Reference Darmon6, Lemma 2.17] then imply that (15) can be achieved.

Explicitly, Theorem 1.5 ensures the existence of a lattice $\Lambda _f'\supset \Lambda _f$ such that $F_f$ is a $\Gamma $ -invariant modular symbol with values in $(\mathcal A^\times /q^{\mathbb {Z}})\otimes \Lambda _f'$ satisfying

• • $\mathrm {dlog}\hspace {0.5mm}F_f[r,s] = \omega _f[r,s]$ , for all $r,s\in \mathbb {P}_1(\mathbb {Q})$ ;

• • $ F[ \gamma r, \gamma s](\gamma z) = F[r,s](z)\ \pmod {q^{\mathbb {Z}}\otimes \Lambda _f'}, \quad \mbox { for all } \gamma \in \Gamma , \ r,s\in \mathbb {P}_1(\mathbb {Q}), \mbox { and } z\in \mathcal H_p.$

The statement that the (multiplicative) periods $J_f[r,s]$ of the ‘mock Hilbert modular form’ attached to E lie in a lattice commensurable with $q^{\mathbb {Z}}\otimes \Lambda _E$ resonates with Oda’s period conjecture for Hilbert modular surfaces. The emergence of the Tate period in what had, up to now, been a rather formal sequence of constructions provides the first inkling that the point of view of mock Hilbert modular surfaces opens genuinely new perspectives on arithmetic questions related to f and its associated elliptic curve E.

3.1 $E(\mathbb {C})$ -valued harmonic cocycles

To parlay the system $c_f$ of mock residues attached to f into a rigorous evaluation of the plectic invariant attached to $\tau $ , it is natural to replace the $\Gamma $ -stable subset $\mathbb {P}_1(\mathbb {Q})\subset \mathcal H^*$ of Section 1.4 by the $\Gamma $ -orbit

$$ \begin{align*}\Sigma := \Gamma \tau_\infty\end{align*} $$

of $\tau _\infty $ in $\mathcal H$ . For each pair $(x,y)\in \Sigma ^2$ , one obtains a $\mathbb {C}$ -valued harmonic cocycle on ${\mathcal T}$ via (8), denoted $c_f[x,y]$ . The collection of $c_f[x,y]$ as $x,y$ vary over $\Sigma $ satisfies the $\Gamma $ -equivariance property

$$ \begin{align*}c_f[\gamma x,\gamma y](\gamma e) = c_f[x,y](e), \qquad \mbox{ for all } \gamma \in \Gamma, \ x,y\in \Sigma, \mbox{ and } e \in \mathcal E({\mathcal T}).\end{align*} $$

Recall that $\Lambda _f$ is a lattice in $\mathbb {C}$ containing all the periods of the form $2\pi i\int _r^s f(z) dz$ with $r,s \in \mathbb {P}_1(\mathbb {Q})$ , and that E has been replaced by the isogenous curve with period lattice $\Lambda _f$ , which is possible by the Manin-Drinfeld theorem. The theory of modular symbols can be invoked to obtain a $\Gamma $ -equivariant collection $\{c_f[x]\}_{x\in \Sigma }$ of harmonic cocycles, indexed by a single $x\in \Sigma $ , but with values in $ \mathbb {C}/\Lambda _f = E(\mathbb {C})$ , satisfying

$$ \begin{align*}c_f[x,y] = c_f[y] - c_f[x]\quad \pmod{\Lambda_f}, \qquad \mbox{ for all } x,y \in \Sigma.\end{align*} $$

This is done by setting

(17) $$ \begin{align} \begin{aligned}c_f[x](e) &= \lvert E(\mathbb{Q})_{\mathrm{tors}}\rvert\cdot2\pi i\int_{i\infty}^x f_e(z) \hspace{0.5mm}\mathrm{d}z \quad\pmod{\Lambda_f} \\ &= \lvert E(\mathbb{Q})_{\mathrm{tors}}\rvert\cdot2\pi i \int_{i \infty} ^{\gamma x} f(z)\hspace{0.5mm}\mathrm{d}z \quad\pmod{\Lambda_f} \end{aligned}\qquad\mbox{where } \gamma e = e_{\infty} \mbox{ for } \gamma\in \Gamma.\end{align} $$

As in Sections 1.1 and 1.3, the harmonic cocycle $c_f[x]$ gives rise to an $E(\mathbb {C})$ -valued distribution on $\mathbb {P}_1(\mathbb {Q}_p)$ with zero total mass, denoted $\mu _f[x]$ , which can only be integrated against locally constant $\mathbb {Z}$ -valued functions on $\mathbb {P}_1(\mathbb {Q}_p)$ .

3.2 The isogeny tree of a CM curve

The eventual upgrading of $\mu _f[x]$ to a measure is based on the observation that the values in (17) can be interpreted as Heegner points on $E_{/\mathbb {Q}}$ attached to CM points of p-power conductor on the modular curve $X_0(p)$ . These points are defined over the anticyclotomic extension

$$ \begin{align*}K_\infty = \bigcup_{n=0}^\infty K_n,\end{align*} $$

where $K_n$ is the ring class field of K of conductor $p^n$ . This field is totally ramified at the (unique) prime of K above p. Write

$$ \begin{align*}G_n = \mathrm{Gal}(K_n/K), \qquad G_\infty = \mathrm{Gal}(K_\infty/K) = \lim_{\leftarrow} G_n.\end{align*} $$

Global class field theory identifies $G_\infty $ with $K_{p,1}^\times $ the group of norm one elements,

$$\begin{align*}\mathrm{rec}\colon K_{p,1}^\times\overset{\sim}{\longrightarrow}G_\infty. \end{align*}$$

Let A be the elliptic curve over $\overline {\mathbb {Q}}$ with complex multiplication by the maximal order ${\mathcal O}_K$ . It is unique up to isomorphism over $\overline {\mathbb {Q}}$ and has a model over $\mathbb {Q}$ because of the running class number one assumption. Let ${\mathcal T}_A$ be the p-isogeny graph of A, whose vertices are elliptic curves over $\overline {\mathbb {Q}}$ related to A by a cyclic isogeny of p-power degree, and whose edges correspond to p-isogenies. This graph is a tree of valency $(p+1)$ with a distinguished vertex $v_A$ attached to A. Since the elliptic curves that are p-power isogenous to A are all defined over $K_\infty $ , the Galois group $G_\infty = K_{p,1}^\times $ acts on ${\mathcal T}_A$ in the natural way. This action fixes $v_A$ and transitively permutes all the vertices (or edges) that lie at a fixed distance from $v_A$ . More precisely, for every $n\ge 0$ , the subgroup $\mathscr {U}_n\le K_{p,1}^\times $ attached to the ring class field $K_n$ under the Galois correspondence stabilizes all the vertices of ${\mathcal T}_A$ at distance n from $v_A$ . Choose a sequence of adjacent vertices $\{v_n\}_{n\ge 0}$ satisfying

$$\begin{align*}\mathscr{U}_n=\mathrm{Stab}_{K_{p,1}^\times}(v_n). \end{align*}$$

For every $n\ge 1$ , the oriented edge $e_n=(v_{n-1},v_{n})$ from $v_{n-1}$ to $v_{n}$ satisfies

$$\begin{align*}\mathscr{U}_n=\mathrm{Stab}_{K_{p,1}^\times}(e_n). \end{align*}$$

The vertices (resp. edges) of ${\mathcal T}_A$ at distance n from $v_A$ are in bijection with the $G_n=K^\times _{p,1}/\mathscr {U}_n$ -orbit of $v_n$ (resp. $e_n$ ). It is convenient to interpret each vertex of ${\mathcal T}_A$ as a point on the j-line $X_0(1)$ , and to view each edge as a point on the modular curve $X_0(p)$ , the coarse moduli space of pairs of elliptic curves related by a p-isogeny. For $n\ge 1$ , let $P_n\in X_0(p)(K_{n})$ be the point corresponding to the oriented edge $e_n$ . Since $\mathscr {U}_n/\mathscr {U}_{n+1}$ acts simply transitively on the set of edges at distance $n+1$ from $v_A$ having $v_n$ as an endpoint, it follows that

$$\begin{align*}\mathrm{Tr}_{K_{n+1}/K_n}(P_{n+1})= U_p(P_n) \qquad\forall\ n\ge1. \end{align*}$$

Fix a trivialisation $H_1(A(\mathbb {C}),\mathbb {Z}_p) \simeq \mathbb {Z}_p^2$ . The choice of a complex embedding $\iota _\infty : K_\infty \hookrightarrow \mathbb {C}$ together with Shimura’s reciprocity law determine a $K_{p,1}^\times $ -equivariant graph isomorphism

$$ \begin{align*}j_A: {\mathcal T}_A\overset{\sim}{{\longrightarrow}} {\mathcal T},\end{align*} $$

which sends the vertex attached to $A'$ to the lattice $ H_1(A'(\mathbb {C}),\mathbb {Z}_p) \subseteq H_1(A(\mathbb {C}),\mathbb {Q}_p) = \mathbb {Q}_p^2$ , after viewing $A'$ as a curve over $\mathbb {C}$ via $\iota _\infty $ . In particular, $j_A$ maps the distinguished vertex $v_A$ to $v_\circ $ . The identification $j_A$ allows the harmonic cocycle $c_f[\tau _\infty ]$ to be viewed as taking values in ${E(K_\infty )}$ . More precisely, [Reference Darmon6, Lemma 1.5] and equation (17) give

$$\begin{align*}c_f[\tau_\infty](\alpha\cdot e_n)= y_n^{\mathrm{rec}(\alpha)},\qquad \forall\ n\ge1,\ \alpha\in K_{p,1}^\times. \end{align*}$$

Moreover, as $K_{p,1}^\times $ acts transitively on the edges at a fixed distance from $v_A$ , we deduce that

(18) $$ \begin{align} c_f[\tau_\infty](\alpha\cdot e)= \mathrm{rec}(\alpha)\cdot c_f[\tau_\infty](e),\qquad \forall\ e\in\mathcal{E}(\mathcal{T}),\ \alpha\in K_{p,1}^\times. \end{align} $$

The general case of $x=\gamma ^{-1}\tau _\infty \in \Sigma $ is dealt with by the formula

$$\begin{align*}c_f[x](e)=c_f[\tau_\infty](\gamma e)\qquad\forall\ e\in\mathcal{E}({\mathcal T}). \end{align*}$$

3.3 Measures and the Poisson transform

In order to integrate continuous functions with respect to the measure $\mu _f[\tau _\infty ]$ , the value group $E(K_\infty )$ needs to be p-adic completed. We will denote the p-adic completion of the (infinitely generated) Mordell-Weil group $E(K_\infty )$ by

$$ \begin{align*}\widehat{E(K_\infty)} := \varprojlim_n\hspace{1mm} E(K_\infty) \otimes \mathbb{Z}/p^n\mathbb{Z}.\end{align*} $$

Viewing $\mu _f[x]$ (for $x\in \Sigma $ ) as an $\widehat {E(K_\infty )}$ -valued measure on $\mathbb {P}_1(\mathbb {Q}_p)$ , the Teitelbaum transform of $\mu _f[x]$ gives a collection of elements

(19) $$ \begin{align} \omega_f[x] := \int_{\mathbb{P}_1(\mathbb{Q}_p)} \frac{\mathrm{d}z}{z-t}\hspace{1mm} \mathrm{d}\mu_f[x](t) \ \ \in \ \ \Omega^1_{\mathrm{ rig}}(\mathcal H_p) {\widehat \otimes} {\widehat{ E(K_\infty)}}. \end{align} $$

For any $x\in \Sigma $ , let

$$ \begin{align*}\iota_x: K \longrightarrow M_2(\mathbb{Q})\end{align*} $$

be the algebra embedding that sends K to the fraction field of the order ${\mathcal O}_x$ . The group $\Gamma $ acts on $\Omega ^1_{\mathrm { rig}}(\mathcal H_p) {\widehat \otimes } {\widehat { E(K_\infty )}}$ by translation on $\mathcal H_p$ , and the Galois group $ G_\infty $ acts via its natural action on ${\widehat { E(K_\infty )}}$ .

Proof. The first part of the proposition follows from the $\Gamma $ -invariance of the measure $\mu _f[x]$ , the triviality of its total mass, and the equality

$$\begin{align*}\frac{1}{\gamma z-\gamma t}=\frac{(cz+d)^2}{z-t}+\frac{1}{\gamma z-\gamma\infty}. \end{align*}$$

To prove the second claim, note that the first part of the proposition implies

$$\begin{align*}\iota_x(\alpha)^\ast (\omega_f[x]) = \omega_f[\iota_x(\alpha)^{-1}x]. \end{align*}$$

Thus, it suffices to show that

$$\begin{align*}c_f[\iota_x(\alpha)^{-1}x](e)={\mathrm {rec}}(\alpha)\cdot c_f[x](e)\qquad\forall\ e\in \mathcal{E}(\mathcal{T}). \end{align*}$$

As $x\in \Sigma $ , we can write $x=\gamma ^{-1}\tau _\infty $ for some $\gamma \in \Gamma $ and obtain $\gamma \cdot \iota _x(-)=\iota _{\tau _\infty }(-)\cdot \gamma $ . Then, the $\Gamma $ -invariance of the harmonic cocycle together with (18) allow us to establish the claim:

$$\begin{align*}\begin{aligned} c_f[\iota_x(\alpha)^{-1}x](e)&=c_f[(\gamma\iota_x(\alpha))^{-1}\tau_\infty](e)\\ &=c_f[\tau_\infty](\alpha\cdot \gamma e)\\ &={\mathrm {rec}}(\alpha)\cdot c_f[\tau_\infty](\gamma e)\\ &={\mathrm {rec}}(\alpha)\cdot c_f[x](e).\\[-19pt] \end{aligned}\end{align*}$$

3.4 The mock plectic invariant

Following the same ideas as in the proof of Lemma 1.4, a well-defined system of multiplicative primitives

$$ \begin{align*}F_f[x] \in (\mathcal A^\times/K_p^\times) {\hat \otimes}\widehat{E(K_\infty)}\end{align*} $$

can be attached to the elements $\omega _f[x]$ , satisfying

$$ \begin{align*}\mathrm{dlog}(F_f[x]) = \omega_f[x] \qquad \mbox{for all } x\in \Sigma.\end{align*} $$

The torus $\iota _\tau (K_p^\times )$ has two fixed points $\tau _p$ , $\overline {\tau }_p$ acting on $\mathcal H_p$ , and they are interchanged by the action of ${\mathrm {Gal}}(K_p/\mathbb {Q}_p)$ . This circumstance leads to the definition of the multiplicative Neková $\check {\text {r}}$ –Scholl mock plectic invariant attached to the CM point $\tau $ , by setting

$$ \begin{align*}\mathcal{Q}^\times(\tau) := \frac{F_f[\tau_\infty](\tau_p)}{F_f[\tau_\infty](\overline{\tau}_p)} \ \ \in \ \ \widehat{E(K_\infty)} {\widehat \otimes} K_{p,1}^\times.\end{align*} $$

Since p is inert in K, the group $K_{p,1}^\times $ consists of p-adic units and the invariant $\mathcal {Q}^\times (\tau )$ is determined, up to $(p^2-1)$ -torsion, by its p-adic logarithm

$$ \begin{align*}\mathcal{Q}(\tau) := \log \mathcal{Q}^\times(\tau)= \int_{\overline{\tau}_p}^{\tau_p} \!\omega_f[\tau_\infty] \ \ \in \ \ \widehat{E(K_\infty)} {\widehat \otimes} K_{p}.\end{align*} $$

Proof. For all $\alpha \in K_{p,1}^\times $ , we have

$$ \begin{align*}{\mathrm {rec}}(\alpha) \mathcal{Q}(\tau) = \int_{\overline{\tau}_p}^{\tau_p} {\mathrm {rec}}(\alpha) \omega_f[\tau] = \int_{\overline{\tau}_p}^{\tau_p} \iota_\tau(\alpha)^\ast \omega_f[\tau] = \mathcal{Q}(\tau),\end{align*} $$

where the penultimate equality follows from the second assertion in Proposition 3.2, and the last from the change of variables formula and the fact that $\iota _\tau (K_p^\times )$ fixes both $\tau _p$ and  $\overline {\tau }_p$ .

3.5 Anticyclotomic p-adic L-functions

We will now give a formula for the mock plectic invariant $\mathcal {Q}(\tau )$ in terms of the first derivatives of certain ‘anticyclotomic p-adic L-functions’ in the sense of [Reference Bertolini and Darmon1, Section 2.7].

Recall that the CM point $\tau =(\tau _p,\tau _\infty )$ determines an embedding $K_p\subseteq \mathrm {M}_2(\mathbb {Q}_p)$ and hence an action of $K_p^\times $ on $\mathbb {P}_1(\mathbb {Q}_p)$ . Let

(20) $$ \begin{align} A\colon \mathbb{P}_1(\mathbb{Q}_p) {\longrightarrow} K_{p,1}^\times, \qquad A(x) = \frac{x-\tau_p}{x-{\overline \tau}_p} \end{align} $$

be the Möbius transformation that sends $(\tau _p,\bar \tau _p,\infty )$ to $(0,\infty ,1)$ , and let $\mu _{f,K}$ be the pushforward of the measure $\mu _f[\tau _\infty ]$ to $K_{p,1}^\times $ via A:

$$ \begin{align*}\mu_{f,K} := A_\ast \mu_f[\tau_\infty].\end{align*} $$

The anticyclotomic p-adic L-function attached to $(E,K)$ is the Mazur-Mellin transform of the measure $\mu _{f,K}$ :

(21) $$ \begin{align} L_p(E,K,s) := \int_{K_{p,1}^\times} \langle \alpha\rangle^{s-1} d\mu_{f,K}(\alpha). \end{align} $$

It can be viewed as a p-adic analytic function from $1+p\mathbb {Z}_p$ to ${\widehat {E(K_\infty )}} \widehat \otimes K_p$ .

Theorem 3.4. The p-adic L-function $L_p(E,K,s)$ vanishes at $s=1$ and

$$ \begin{align*}\mathcal{Q}(\tau) = L_p'(E,K,1).\end{align*} $$

Proof. The vanishing of $L_p(E,K,1)$ follows from the fact that $\mu _{f}[\tau _\infty ]$ , and hence also $\mu _{f,K}$ , have total measure zero. By the definition of $\mathcal {Q}(\tau )$ combined with (19),

$$ \begin{align*}\mathcal{Q}(\tau) = \int_{\bar \tau_p}^{\tau_p} \omega_f[\tau_\infty] = \int_{\bar \tau_p}^{\tau_p} \left(\int_{\mathbb{P}_1(\mathbb{Q}_p)} \frac{1}{z-t} d\mu_f[\tau_\infty](t) \right) dz.\end{align*} $$

Interchanging the order of integration and integrating with respect to z gives

$$ \begin{align*}\mathcal{Q}(\tau) = \int_{\mathbb{P}_1(\mathbb{Q}_p)} \log \left( \frac{\tau_p-t}{\bar\tau_p-t}\right) d\mu_f[\tau_\infty](t) = \int_{\mathbb{P}_1(\mathbb{Q}_p)} \log A(t) \cdot d\mu_f[\tau_\infty](t).\end{align*} $$

The change of variables $\alpha =A(t)$ can be used to rewrite this last expression as an integral over $K_{p,1}^\times $ ,

$$ \begin{align*}\mathcal{Q}(\tau) = \int_{K_{p,1}^\times} \log(\langle\alpha \rangle) \cdot d\mu_{f,K}(\alpha) = L_p'(E,K,1),\end{align*} $$

where the last equality follows directly from (21).

As explained in [Reference Bertolini and Darmon2], the quantity $L_p'(E,K,1)$ is directly related to the Kolyvagin derivative of the norm-compatible collection $y_n= c_f[\tau _\infty ](e_n)\in E(K_n)$ of Heegner points, where $\{e_n\}_{n\ge 1}\subset \mathcal {E}({\mathcal T})$ is the sequence of adjacent edges that was fixed in Section 3.2. More precisely, for every $n\ge 1$ , the collection $\{\alpha \cdot U_{e_n}\}_{\alpha \in K_{p,1}^\times }$ is a finite covering of $\mathbb {P}_1(\mathbb {Q}_p)$ by compact open subsets, which are equal to the cosets $\{\alpha \cdot \mathscr {U}_n\}_{\alpha \in K_{p,1}^\times }$ under the homeomorphism (20). Rewriting $L_p'(E,K,1)$ as the limit when $n\rightarrow \infty $ of the Riemann sums attached to these coverings, equation (18) gives

(22) $$ \begin{align} \begin{aligned} \mathcal{Q}(\tau) &=\int_{K_{p,1}^\times} \log(\alpha) d\mu_{f,K}(\alpha)\\ &=\underset{n\rightarrow\infty}{\lim}\ \sum_{\alpha\in K_{p,1}^\times/\mathscr{U}_n}c_f[\tau_\infty](\alpha\cdot e_n)\otimes \log(\langle \alpha\rangle)\\ &=\underset{n\rightarrow \infty}{\lim}\ \sum_{\alpha\in K_{p,1}^\times/\mathscr{U}_n}y_n^{\mathrm{rec}(\alpha)} \otimes\log(\langle\alpha\rangle) \end{aligned} \end{align} $$

(see [Reference Bertolini and Darmon3, §6] for related discussions).

3.6 Kolyvagin’s cohomology classes

After choosing a topological generator of $K_{p,1}^\times $ , Lemma 3.3 allows the mock plectic invariant $\mathcal {Q}(\tau )$ to be viewed (non-canonically) as an element of $\widehat {E(K_\infty )}$ fixed by $G_\infty $ . Consider the natural injective map

$$ \begin{align*}E(K)\otimes \mathbb{Z}_p \longrightarrow \widehat{E(K_\infty)}^{G_\infty}.\end{align*} $$

It need not be surjective in general: the group $\widehat {E(K_\infty )}$ fails to satisfy the principle of Galois descent. Moreover, while it is relatively straightforward to establish the infinitude of $\widehat {E(K_\infty )}^{G_\infty }$ , the Birch and Swinnerton-Dyer conjecture predicts the finitude of $E(K)$ when $L(E/K,1)\ne 0$ . Even when $L(E/K,s)$ vanishes at $s=1$ , and hence has a zero of order $\ge 2$ , establishing that $E(K)\otimes \mathbb {Z}_p$ is infinite seems very hard to do unconditionally.

A useful handle on the group $\widehat {E(K_\infty )}^{G_\infty }$ is obtained by relating it to Galois cohomology. Let $H^1(K_m, E[p^n])$ be the first Galois cohomology of $K_m$ with values in the module of $p^n$ -division points of E, and let

$$ \begin{align*}H^1(K_m, T_p(E)) = \varprojlim_n\hspace{1mm} H^1(K_m, E[p^n]),\end{align*} $$

the inverse limit being taken relative to the multiplication by p maps $E[p^{n+1}] {\longrightarrow } E[p^n]$ .

Lemma 3.6. The restriction map

$$ \begin{align*}H^1(K, E[p^n]) {\longrightarrow} H^1(K_m, E[p^n])^{G_m}\end{align*} $$

is an isomorphism.

Proof. The kernel and cokernel in the inflation-restriction sequence

are trivial by Lemma 3.5, and the claim follows.

Denote by $\delta _n$ the mod $p^n$ Kummer map

$$ \begin{align*}\delta_n: E(K_m)/p^n E(K_m) {\longrightarrow} H^1(K_m, E[p^n]),\end{align*} $$

and by

$$ \begin{align*}\delta_\infty: E(K_m)\otimes \mathbb{Z}_p {\longrightarrow} H^1(K_m, T_p(E))\end{align*} $$

the map induced on the inverse limits. Let $P_\infty = \{ P_n \}_{n\ge 1} \in {\widehat {E(K_\infty )}}^{G_\infty }$ , where

$$ \begin{align*}P_n \in (E(K_{m_n})\otimes \mathbb{Z}/p^n\mathbb{Z})^{G_{m_n}}\end{align*} $$

for suitable integers $m_n\ge 1$ . Lemma 3.6 shows that each $\delta _n(P_n)\in H^1(K_{m_n}, E[p^n])^{G_{m_n}}$ is the restriction of a unique class $\kappa _n \in H^1(K, E[p^n])$ . The $\kappa _n$ are compatible under the natural multiplication by p maps from $E[p^{n+1}]$ to $E[p^n]$ , and the assignment $(P_n)_{n\ge 1} \mapsto (\kappa _n)_{n\ge 1}$ determines a canonical inclusion

It will be convenient for the rest of this note to identify $\mathcal {Q}(\tau )$ with its image in $H^1(K,T_p(E))$ under this map.

3.7 Local properties of the mock plectic invariant

For each place v of K, let

$$ \begin{align*}\mathrm{res}_v: H^1(K,T_p(E)) {\longrightarrow} H^1(K_v, T_p(E))\end{align*} $$

be the local restriction map to a decomposition group at v. The local cohomology group $H^1(K_p,T_p(E))$ is equipped with a natural two-step filtration

$$ \begin{align*}0 {\longrightarrow} H^1_f(K_p, T_p(E)) {{\longrightarrow}} H^1(K_p,T_p(E)) {\longrightarrow} H^1_{\mathrm{sing}}(K_p, T_p(E)) {\longrightarrow} 0,\end{align*} $$

where $H^1_f(K_p, T_p(E)):= \delta _\infty (E(K_p) \otimes \mathbb {Z}_p)$ .

Definition 3.7. The pro-p-Selmer group of E is the group, denoted $H^1_f(K, T_p(E))$ , of global classes $\kappa \in H^1(K,T_p(E))$ satisfying

$$ \begin{align*}\mathrm{res}_p(\kappa) \in H^1_f(K_p, T_p(E)).\end{align*} $$

The reader will note that in the setting of pro-p Selmer groups, no local conditions need to be imposed at the places $v\ne p$ because the Kummer map $\delta _{v}: E(K_v) \otimes \mathbb {Z}_p {\longrightarrow } H^1(K_v, T_p(E))$ is an isomorphism. The only possible obstruction for a global class to lie in the pro-p Selmer group therefore occurs at the place p. Let

$$ \begin{align*}\mathcal{Q}_p(\tau):= \mathrm{res}_p(\mathcal{Q}(\tau)) \in H^1(K_p,T_p(E))\end{align*} $$

denote the restriction of the global class $\mathcal {Q}(\tau )$ to the decomposition group at p. The first important result about the mock plectic invariant is that it is non-Selmer precisely when $L(E/K,1) \ne 0$ .

Proof. Recall that the ring class field $K_n$ of K of conductor $p^n$ is a cyclic Galois extension of K with Galois group $G_n\cong K_{p,1}^\times /\mathscr {U}_n$ . The Kolyvagin derivative of the Heegner point $y_n\in E(K_n)$ is defined as

$$\begin{align*}D_ny_n:=\sum_{\alpha\in K_{p,1}^\times/\mathscr{U}_n}y_n^{\mathrm{rec}(\alpha)}\otimes\alpha\ \ \in \ \ E(K_n)\otimes K_{p,1}^\times/\mathscr{U}_n \end{align*}$$

and is fixed by the action of $G_n$ because $\mathrm {Tr}_{K_n/K}(y_n)=0$ (cf. [Reference Bertolini and Darmon1, (8)] and [Reference Gross16, Proposition 3.6]). Then, just as in equation (22), one sees that

$$\begin{align*}\mathcal{Q}^\times(\tau) =\varprojlim_n\hspace{1mm} D_ny_n. \end{align*}$$

By choosing a topological generator of $K_{p,1}^\times $ , we can consider the image of $\mathcal {Q}^\times (\tau )$ in $H^1(K,T_p(E))$ and study its local properties via the following diagram taken from [Reference Gross16, (4.2)]:

By construction, the image of $D_ny_n$ in $H^1(K,E)$ belongs to the image of $H^1(G_n,E(K_n))$ under inflation, and it can be represented by the following $1$ -cocycle [Reference McCallum25, Lemma 4.1]:

(23) $$ \begin{align} G_n\ni\sigma\mapsto-\frac{(\sigma-1)D_n(y_n)}{p^{n-1}}, \end{align} $$

where $\frac {(\sigma -1)D_n(y_n)}{p^{n-1}}$ denotes the unique $p^{n-1}$ -th root of $(\sigma -1)D_n(y_n)$ in $E(K_n)$ . Now, we can use the explicit description (23) to study the image $\partial _p\mathcal {Q}^\times (\tau )$ of $\mathcal {Q}^\times (\tau )$ in $H^1_{\mathrm {sing}}(K_p,T_p(E))$ .

Denote by $\Phi _n$ the group of connected components of the special fiber of the Néron model of E over the p-adic completion $K_{n,p}$ of $K_n$ . By [Reference Bertolini and Darmon2, Lemma 6.7], the reduction map $E(K_{n,p})\to \Phi _n$ produces an injection

where $H^1(G_n,\Phi _n)=\mathrm {Hom}_{\mathbb {Z}}(G_n,\Phi _n)$ is identified with $\Phi _n$ by evaluation at a generator of $G_n$ . A direct computation using (23) and [Reference Gross16, (3.5)] then shows that $ \partial _p\mathcal {Q}^\times (\tau ) $ vanishes exactly when the compatible collection of Heegner points $\{y_n\}_{n\ge 1}$ maps to a finite order element in the projective limit $\{\Phi _n\}_{n\ge 1}$ . By adapting the proof of [Reference Bertolini and Darmon2, Theorem 5.1], one sees that this happens precisely when the L-value $L(E/K,1)$ vanishes.

Interestingly, when $L(E/K,1)\ne 0$ , the mock plectic invariant $\mathcal {Q}(\tau )$ suffices to prove that the Mordell-Weil group of $E'$ – the quadratic twist of E attached to K – is finite.

Sketch of proof.

We follow the proof of [Reference Bertolini and Darmon2, Corollary 7.2]. By Theorem 3.8, we know that $\mathcal {Q}(\tau )$ is a global class ramified only at p. The claim is then that the localization map $E'(\mathbb {Q})\otimes \mathbb {Q}_p\hookrightarrow E'(\mathbb {Q}_p)\otimes \mathbb {Q}_p$ – which is always injective – is the zero morphism.

Under our assumptions, the hypothesis $L(E/K,1)\not =0$ also implies that E has split multiplicative reduction at p; hence, the class $\mathcal {Q}(\tau )$ lives in the minus-eigenspace for the action of complex conjugation [Reference Bertolini and Darmon2, Prop. 6.5], and its image in $H^1_{\mathrm {sing}}(K_p, V_p(E))$ satisfies $H^1_{\mathrm {sing}}(\mathbb {Q}_p, V_p(E'))=\mathbb {Q}_p\cdot \mathcal {Q}_p(\tau )$ . The claim then follows because local Tate duality induces the identification

$$\begin{align*}E'(\mathbb{Q}_p)\otimes\mathbb{Q}_p\overset{\sim}{\longrightarrow}H^1_{\mathrm{sing}}(\mathbb{Q}_p, V_p(E'))^\vee\qquad Q\mapsto \big\langle Q, -\big\rangle_p, \end{align*}$$

and Poitou–Tate duality implies that any point $P\in E(K)\otimes \mathbb {Q}_p$ satisfies

$$\begin{align*}0=\sum_\ell\big\langle \mathrm{res}_\ell(P), \mathrm{res}_\ell (\mathcal{Q}(\tau))\big\rangle_\ell=\big\langle \mathrm{res}_p(P), \mathcal{Q}_p(\tau)\big\rangle_p. \end{align*}$$

(See [Reference Bertolini and Darmon2, Prop. 6.8].)

3.8 Elliptic curves of rank two

When $L(E/K,1)=0$ , the local class $\mathcal {Q}_p(\tau )$ belongs to $H^1_f(K_p,T_p(E))\otimes K_p$ (i.e., the global class $\mathcal {Q}(\tau )$ lies in the pro-p Selmer group of E over K). It is not expected to be trivial in general: in fact, as we now proceed to explain, it should provide a nontrivial Selmer class in settings where $L(E/K,s) = L(E,s) L(E',s)$ has a double zero at $s=1$ , and even when the factor $L(E',s)$ admits such a double zero.

Since the first derivative of the p-adic L-function $L_p(E,K,s)$ computes the mock plectic invariant $\mathcal {Q}(\tau )$ according to Theorem 3.4, we can predict when $L(E/K,1)=0$ and $\mathcal {Q}(\tau )$ does not vanish by analyzing the anticyclotomic Birch and Swinnerton-Dyer conjecture of [Reference Bertolini and Darmon1, Conj.4.1]. In the terminology of [Reference Bertolini and Darmon1, Introduction $\&$ Definition 2.3], the triple $(E,K,p)$ falls into the non-split exceptional indefinite case, where the adjective non-split exceptional refers to the fact that the elliptic curve has multiplicative reduction at the prime p inert in K (and not to the fact that E has necessarily non-split multiplicative reduction at p). If we set $E^+ :=E$ , denote by $E^- = E'$ the quadratic twist of E attached to K, and define

$$\begin{align*}\delta^\pm=\begin{cases} 1&\text{if}\ a_p(E^\pm)=+1\\ 0&\text{if}\ a_p(E^\pm)=-1, \end{cases} \end{align*}$$

then [Reference Bertolini and Darmon1, Conjecture 4.1] suggests that the order of vanishing of $L_p(E,K,s)$ at $s=1$ is

(24) $$ \begin{align} \varrho=\mathrm{max}\big\{1,\hspace{1mm} r_{\mathrm{alg}}(E^\pm/\mathbb{Q})+\delta^\pm -1\big\}, \end{align} $$

which satisfies $2\varrho \ge \mathrm {max}\big \{2,\hspace {1mm} r_{\mathrm {alg}}(E/K)\big \}$ .

We deduce that, when $L(E/K,1)=0$ , we should expect $\mathcal {Q}(\tau )$ to be nontrivial only when $r_{\mathrm {alg}}(E/K)=2$ and the rank is distributed in the two eigenspaces for the action of complex conjugation according to a rule depending on reduction type at the prime p:

Conjecture 3.12. Suppose that $L(E/K,1)= 0$ . Then

$$\begin{align*}\mathcal{Q}(\tau)\not=0\quad\iff\quad \begin{cases} r_{\mathrm{alg}}(E/\mathbb{Q})=0, \quad r_{\mathrm{alg}}(E'/\mathbb{Q})=2\quad &\mathrm{ if}\hspace{2mm} a_p(E)=+1,\\ r_{\mathrm{alg}}(E/\mathbb{Q})=1,\quad r_{\mathrm{alg}}(E'/\mathbb{Q})=1\quad &\mathrm{ if}\hspace{2mm} a_p(E)=-1. \end{cases} \end{align*}$$

In order to make explicit the relation between the mock plectic invariant $ \mathcal {Q}(\tau )$ and global points in $E(K)$ , denote by $\log _{E}^{a_p}\colon E(K_p)\to K_p$ the composition of the p-adic logarithm of E with the endomorphism $(1-a_p(E)\cdot \sigma _p)$ of $E(K_p)$ , where $\sigma _p\in \text {Gal}(K_p/\mathbb {Q}_p)$ denotes the nontrivial involution.

Conjecture 3.13. Suppose that $L(E/K,1)= 0$ . Then $\mathcal {Q}(\tau )$ is in the image of the regulator $ \wedge ^2E(K)\to H^1_f(K,T_p(E))\otimes _{\mathbb {Z}_p} K_p$ given by

$$\begin{align*}P\wedge Q\mapsto \delta_\infty(P)\otimes \log_{E}^{a_p}(Q)-\delta_\infty(Q)\otimes\log_{E}^{a_p}(P). \end{align*}$$

The fact that the plectic invariant is forced to lie in a specific eigenspace for complex conjugation and can sometimes vanish for trivial reasons (for example, when $E(\mathbb {Q})$ has rank two) suggests that it is only ‘part of the story’ and represents the projection of a more complete invariant which should be nontrivial in all scenarios when $E(K)$ has rank two. The authors believe that a full mock analogue of plectic Stark–Heegner points can be obtained by exploiting the p-adic deformations of f arising from Hida theory, and hope to treat this idea in future work. These ‘mock plectic points’ should control the arithmetic of elliptic curves of rank two over quadratic imaginary fields, and be unaffected by the degeneracies of anticyclotomic height pairings that plague [Reference Bertolini and Darmon1, Conjecture 4.1].

The approach of this article might also be exploited to upgrade the plectic Heegner points of [Reference Fornea and Gehrmann11] from tensor products of p-adic points to global cohomology classes. Such an improvement would take care of the degeneracies of the original construction (evoked, for example, in [Reference Fornea and Gehrmann11, Remark 1.1]) and, in light of Theorems 3.8 and 3.9, it would help explain the arithmetic meaning of plectic Stark–Heegner points in the setting considered in [Reference Fornea and Gehrmann11, Remark 1.3]. (See also the paragraph before [Reference Fornea, Guitart and Masdeu13, Conjecture 1.6].)