On exceptional zeros of Garrett–Hida p-adic L-functions

Abstract

This article proves a case of the p-adic Birch and Swinnerton-Dyer conjecture for Garrett p-adic L-functions of [6], in the exceptional zero setting of extended analytic rank 2.

Résumé

Cet article prouve un cas de la conjecture p-adique de Birch et Swinnerton-Dyer pour les fonctions L p-adiques de Garrett formulée dans [6], dans le cadre de zéros exceptionnels de rang analytique étendu égal à 2.

1 Introduction

Let A be an elliptic curve defined over $\mathbf{Q}$, having ordinary reduction at a rational prime $p>3$. Let $\varrho _{1}$ and $\varrho _{2}$ be odd, irreducible, two-dimensional Artin representations of the absolute Galois group of $\mathbf{Q}$, which are unramified at p and satisfy the self-duality condition

$$\begin{aligned} \det (\varrho _{1})=\det (\varrho _{2})^{-1}. \end{aligned}$$

By modularity, the triple $(A,\varrho _{1},\varrho _{2})$ arises from a triple (f, g, h) of cuspidal p-ordinary newforms of weights $w_{o}=(2,1,1)$. Let $f_{\alpha }$ be the ordinary p-stabilisation of f, and fix p-stabilisations $g_{\alpha }$ and $h_{\alpha }$ of g and h respectively. Set $\varrho =\varrho _{1}\otimes {}\varrho _{2}$. In the recent paper [6] we proposed a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for the leading term at $w_{o}$ of the 3-variable Garrett–Hida p-adic L-function $L_{p}^{\alpha \alpha }(A,\varrho )=L_{p}(\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha })$ associated with the triple $(\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha })$ of Hida families specialising to $(f_{\alpha },g_{\alpha },h_{\alpha })$ at $w_{o}$. In this article we verify our conjecture in the analytic rank-zero exceptional cases, viz. when the complex Garrett L-function $L(A,\varrho ,s)=L(f\otimes {}g\otimes {}h,s)$ does not vanish at $s=1$ and $L_{p}^{\alpha \alpha }(A,\varrho )$ has an exceptional zero at $w_{o}$ in the sense of Mazur–Tate–Teitelbaum (cf. Theorem 2.1 and Sect. 2.1 below). Moreover, when $L(A,\varrho ,1)=0$ and $L_{p}^{\alpha \alpha }(A,\varrho )$ has an exceptional zero, we propose a conjecture relating the value at $w_{o}$ of the fourth partial derivative of $L_{p}^{\alpha \alpha }(A,\varrho )$ along the $\varvec{f}$-direction to the p-adic logarithms of two global points on A rational over the number field cut out by $\varrho$ (cf. Conjecture 2.3).

2 Setting and notations

Fix algebraic closures ${\bar{\mathbf{Q }}}$ and ${\bar{\mathbf{Q }}}_{p}$ of $\mathbf{Q}$ and $\mathbf{Q} _{p}$ respectively, and field embeddings and . With the notations of the Introduction, let

$$\begin{aligned} \xi =\sum _{n\geqslant {}1}a_{n}(\xi )\cdot {}q^{n}\in {}S_{u}(N_{\xi },\chi _{\xi })_{{\bar{\mathbf{Q }}}} \end{aligned}$$

denote one of the cuspidal newforms f, g and h. Here u and $N_{\xi }$ are the weight and the conductor of $\xi$ respectively, and $S_{u}(N_{\xi },\chi _{\xi })_{F}$ is the space of cuspidal modular forms of level $\Gamma _{1}(N_{\xi })$, weight u, character $\chi _{\xi }$ and Fourier coefficients in the subfield F of ${\bar{\mathbf{Q }}}_{p}$. Fix a number field $\mathbf{Q} (\varrho )$ containing for any $\xi$ the Fourier coefficients $a_{n}(\xi )$, as well as the roots $\alpha _{\xi }$ and $\beta _{\xi }$ of the pth Hecke polynomials $P_{\xi ,p}=X^{2}-a_{p}(\xi )\cdot {}X+\chi _{\xi }(p)\cdot {}p$. Let $V_{\varrho _{i}}$ be a two-dimensional $\mathbf{Q} (\varrho )$-vector space affording the representation $\varrho _{i}$, and let $K_{\varrho }$ be a Galois number field such that $\varrho _{i}$ factors through $\text {Gal}(K_{\varrho }/\mathbf{Q} )$. Set

$$\begin{aligned} V_{\varrho }=V_{\varrho _{1}}\otimes _\mathbf{Q (\varrho )}V_{\varrho _{2}}\ \ \text {and}\ \ V_{p}(A,\varrho )=V_{p}(A)\otimes _\mathbf{Q }V_{\varrho }, \end{aligned}$$

where is the p-adic Tate module of A with $\mathbf{Q} _{p}$-coefficients. Throughout this note we make the following

Assumption 1.1

• 1. (Self-duality) The characters $\chi _{g}$ and $\chi _{h}$ are inverse to each other.

• 2. (Local signs) The conductors $N_{g}$ and $N_{h}$ are coprime to $p\cdot {}N_{f}$.

• 3. (Étaleness) The forms g and h are cuspidal, p-regular and do not have RM by a real quadratic field in which p splits.

The first condition is a reformulation of the self-duality condition mentioned in the Introduction, namely $\det (\varrho _{1})=\text {det}(\varrho _{2})^{-1}$. Recall that the form $\xi$ is p-regular if $P_{\xi ,p}$ has distinct roots. Moreover, one says that a weight-one eigenform has RM (real multiplication) if it is the theta series associated with a ray class character of a real quadratic field. Assumption 1.1.3 is equivalent to require that $V_{\varrho _{i}}$ is irreducible, not isomorphic to $\text {Ind}^\mathbf{Q }_{K}\chi$ for a finite order character $\chi : G_{K}{\mathop {\longrightarrow }\limits ^{}}\mathbf{Q} (\varrho )^{*}$ of a real quadratic field K in which p splits, and that an arithmetic Frobenius at p acts on $V_{\varrho _{i}}$ with distinct eigenvalues. For $\xi =g,h$, this assumption guarantees that the p-adic Coleman–Mazur–Buzzard eigencurve of tame level $N_{\xi }$ is étale over the weight space at the points corresponding to the p-stabilisations of $\xi$ (cf. [2]). It is used in [6] to construct the Garrett–Nekovář height which appears in the main result of this note. To explain the relevance of Assumptions 1.1.1 and 1.1.2, let $\alpha _{f}$ be the unit root of $P_{f,p}$ and fix roots $\alpha _{g}$ and $\alpha _{h}$ of $P_{g,p}$ and $P_{h,p}$ respectively. Fix a finite extension L of $\mathbf{Q} _{p}$ containing $\mathbf{Q} (\varrho )$ and the roots of unity of order $\text {lcm}(N_{f},N_{g},N_{h})$. Let $\xi$ be one of f, g and h, and let $u_{o}$ be the weight of $\xi$. According to the results of [2, 10, 18], there exists a unique Hida family

$$\begin{aligned} \varvec{\xi }_{\alpha }=\sum _{n\geqslant {}1}a_{n}(\varvec{\xi }_{\alpha })\cdot {}q^{n}\in {}\mathscr {O}_{\varvec{\xi }}[\![q]\!] \end{aligned}$$

which specialises at $u_{o}$ to the p-stabilised newform

$$\begin{aligned} \xi _{\alpha }=\xi (q)-\frac{\chi _{\xi }(p)p^{u-1}}{\alpha _{\xi }}\cdot {}\xi (q^{p})\in {}S_{u_{o}}(p\cdot {}M_{\xi },\chi _{\xi })_{L}. \end{aligned}$$

Here $M_{\xi }=N_{\xi }/p^{\text {ord}_{p}(N_{\xi })}$ is the tame level of $\xi$ (so that $M_{\xi }=N_{\xi }$ if $\xi =g,h$), and $\mathscr {O}_{\varvec{\xi }}$ is the ring of bounded analytic functions on a (sufficiently small) connected open disc $U_{\varvec{\xi }}$ in the p-adic weight space over L. For each classical weight u in $U_{\varvec{\xi }}\cap {}\mathbf{Z} _{\geqslant {}3}$, the weight-u specialisation $\varvec{\xi }_{\alpha ,u}=\sum _{n\geqslant {}1}a_{n}(\varvec{\xi }_{\alpha })(u)\cdot {}q^{n}\in {}L[\![q]\!]$ of $\varvec{\xi }_{\alpha }$ is the q-expansion of the ordinary p-stabilisation of a newform $\xi _{u}$ in $S_{u}(M_{\xi },\chi _{\xi })_{L}$. Since f has a unique p-ordinary p-stabilisation $f_{\alpha }$, we simply write $\varvec{f}$ for $\varvec{f}_{\!\alpha }$.

Assumption 1.1.1 guarantees that for each classical triple $w=(k,l,m)$ in the set

$$\begin{aligned} \Sigma =U_{\!\varvec{f}}\times {}U_{\varvec{g}}\times {}U_{\varvec{h}}\cap {}\mathbf{Z} _{\geqslant {}1}^{3} \end{aligned}$$

the complex Garrett L-function $L(f_{k}\otimes {}g_{l}\otimes {}h_{m},s)$ admits an analytic continuation to all of $\mathbf{C}$ and satisfies a functional equation relating its values at s and $k+l+m-2-s$, with root number $\varepsilon (w)=\prod _{\ell \leqslant \infty }\varepsilon _{\ell }(w)$ equal to $+1$ or to $-1$. Assumption 1.1.2 implies that all the local signs $\varepsilon _{\ell }(w)$ are equal to $+1$ for every w in the f-unbalanced region $\Sigma _{f}=\{w=(k,l,m)\in {}\Sigma : k\geqslant {}l+m\}$ (cf. [11]). Under these assumptions, [12] associates with $(\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha })$ an analytic function

$$\begin{aligned} \mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )=\mathscr {L}_{p}(\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha }) \end{aligned}$$

in the ring $\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}=\mathscr {O}_{\!\varvec{f}}{\hat{\otimes }}_{L}\mathscr {O}_{\varvec{g}}{\hat{\otimes }}_{L}\mathscr {O}_{\varvec{h}}$, whose square

$$\begin{aligned} L_{p}^{\alpha \alpha }(A,\varrho )=L_{p}(\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha })=\mathscr {L}_{p}(\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha })^{2} \end{aligned}$$

satisfies the following interpolation property. For each $w=(k,l,m)$ in $\Sigma _{f}$, the value of $L_{p}^{\alpha \alpha }(A,\varrho )$ at w is an explicit non-zero complex multiple of

$$\begin{aligned} \left( 1-\frac{\beta _{k}\alpha _{l}\alpha _{m}}{p^{c_{w}}}\right) ^{2} \left( 1-\frac{\beta _{k}\beta _{l}\alpha _{m}}{p^{c_{w}}}\right) ^{2} \left( 1-\frac{\beta _{k}\alpha _{l}\beta _{m}}{p^{c_{w}}}\right) ^{2} \left( 1-\frac{\beta _{k}\beta _{l}\beta _{m}}{p^{c_{w}}}\right) ^{2}\cdot {}L(f_{k}\otimes {}g_{l}\otimes {}h_{m},c_{w}). \end{aligned}$$

(1)

Here $c_{w}=\frac{k+l+m-2}{2}$, and for $\varvec{\xi }=\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha }$ one denotes by $\alpha _{u}$ the unit root of $P_{\xi _{u},p}$ and sets $\beta _{u}\cdot {}\alpha _{u}=\chi _{\xi }^{\prime }(p)\cdot {}p^{u-1}$, where $\chi _{\xi }^{\prime }$ is the prime-to-p part of $\chi _{\xi }$ (so that $\chi _{\xi }^{\prime }=\chi _{\xi }$ for $\xi =g,h$, and $\chi _{f}^{\prime }$ is the trivial character modulo $M_{f}$). We refer to Theorem A of loc. cit. for the precise interpolation formula. We call $L_{p}^{\alpha \alpha }(A,\varrho )=L_{p}(\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha })$ the Garrett–Hida p-adic L-function associated with $(A,\varrho )$ (or with $(\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha })$).

3 Exceptional zero formulae

The p-adic variant of the Birch and Swinnerton-Dyer conjecture formulated in [6] predicts that the leading term of $L_{p}^{\alpha \alpha }(A,\varrho )$ at $w_{o}=(2,1,1)$ is encoded by the discriminant of the Garrett– Nekovář height pairing

(2)

constructed in Section 2 of loco citato, where $\mathscr {I}$ is the ideal of functions in $\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}$ which vanish at $w_{o}$ and the p-extended Mordell–Weil group $A^{\dag }(K_{\varrho })^{\varrho }$ is defined as follows. When A has good reduction at p, one sets $A^{\dag }(K_{\varrho })^{\varrho }=A(K_{\varrho })^{\varrho }$, where $A(K_{\varrho })^{\varrho }$ is a shorthand for the $\text {Gal}(K_{\varrho }/\mathbf{Q} )$-invariants of $A(K_{\varrho })\otimes _\mathbf{Z }V_{\varrho }$. If A has multiplicative reduction at p, then $\alpha _{f}=a_{p}(f)=\pm 1$ and the maximal p-unramified quotient $V_{p}(A)^{-}$ of $V_{p}(A)$ is a 1-dimensional $\mathbf{Q} _{p}$-vector space on which an arithmetic Frobenius acts as multiplication by $\alpha _{f}$. Let $q_{A}$ in $p\mathbf{Z} _{p}$ be the p-adic Tate period of the base change $A_\mathbf{Q _{p}}$ of A to $\mathbf{Q} _{p}$ (cf. Chapter V of [15]), and let $\mathbf{Q} _{p^{2}}$ be the quadratic unramified extension of $\mathbf{Q} _{p}$. The Tate uniformisation yields a rigid analytic morphism

$$\begin{aligned} \wp _{\text {Tate}} : \mathbf{G} _{m,\mathbf{Q} _{p^{2}}}^{rig}{\mathop {\longrightarrow }\limits ^{}}A_\mathbf{Q _{p^{2}}} \end{aligned}$$

with kernel $q_{A}^\mathbf{Z }$ and unique up to sign. Set

$$\begin{aligned} q(A)=p^{-}\big ((\wp _{\text {Tate}}(\!\!\root p^{n} \of {q_{A}}\,))_{n\geqslant {}1}\big )\in {}V_{p}(A)^{-}, \end{aligned}$$

where $p^{-}$ denotes the projection $V_{p}(A){\mathop {\longrightarrow }\limits ^{}}V_{p}(A)^{-}$ and $(\!\!\root p^{n} \of {q_{A}}\,)_{n\geqslant {}1}$ is any compatible system of $p^{n}$-th roots of $q_{A}$, and define

$$\begin{aligned} A^{\dag }(K_{\varrho })^{\varrho }=A(K_{\varrho })^{\varrho }\oplus {} {\mathcal {Q}}_{p}(A,\varrho ) \end{aligned}$$

to be the direct sum of $A(K_{\varrho })^{\varrho }$ and the $\mathbf{Q} (\varrho )$-submodule

$$\begin{aligned} {\mathcal {Q}}_{p}(A,\varrho )=H^{0}(\mathbf{Q} _{p},\mathbf{Q} (\varrho )\cdot {}q(A)\otimes _\mathbf{Q (\varrho )}V_{\varrho }) \end{aligned}$$

of $H^{0}(\mathbf{Q} _{p},V_{p}(A)^{-}\otimes _\mathbf{Q }V_{\varrho })$. The Garrett–Nekovář height depends on the choice of suitably normalised $G_\mathbf{Q }$-equivariant embeddings

(3)

where $V(\xi )=V(\varvec{\xi }_{\alpha })\otimes _{1}L$ (for $\xi =g,h$) is the weight-one specialisation of the big Galois representation $V(\varvec{\xi }_{\alpha })$ associated with $\varvec{\xi }_{\alpha }$. (We refer to Sect. 3.1 below for precise definitions.) More precisely, denote by V(f) the $f_{\alpha }$-isotypic component of the cohomology group , where $X_{1}(N_{f},p)_{{\bar{\mathbf{Q }}}}$ is the base change to ${\bar{\mathbf{Q }}}$ of the compact modular curve $X_{1}(N_{f},p)$ of level $\Gamma _{1}(N_{f})\cap {}\Gamma _{0}(p)$ over $\mathbf{Q}$, and set

$$\begin{aligned} V(f,g,h)=V(f)\otimes _\mathbf{Q _{p}}V(g)\otimes _{L}V(h). \end{aligned}$$

Section 2 of [6] constructs a canonical Garrett–Nekovář p-adic height pairing

(4)

on the naive extended Selmer group of V(f, g, h) over $\mathbf{Q}$, defined as the direct sum of the Bloch–Kato Selmer group $\text {Sel}(\mathbf{Q} ,V(f,g,h))$ of V(f, g, h) over $\mathbf{Q}$ and the module $H^{0}(\mathbf{Q} _{p},V(f,g,h)^{-})$ of $G_\mathbf{Q _{p}}$-invariants of the maximal p-unramified quotient $V(f,g,h)^{-}$ of V(f, g, h). (The definition of is briefly recalled in Sect. 3.2.3 below.) Fix a modular parametrisation $\wp _{\infty } : X_{1}(N_{f},p){\mathop {\longrightarrow }\limits ^{}}A$, under which one identifies V(f) and $V_{p}(A)$. The embeddings $\gamma _{g}$ and $\gamma _{h}$ and the global Kummer map on $A(K_{\varrho })$ then induce an embedding . The pairing (2) is defined to be composition of the canonical Garrett–Nekovář height and $\gamma _{gh}^{\otimes {}2}$. The pairings (2) and (4) are skew-symmetric, and the discriminant of (2) in $(\mathscr {I}^{r^{\dag }(A,\varrho )}/\mathscr {I}^{r^{\dag }(A,\varrho )+1})/\mathbf{Q} (\varrho )^{*2}$, where $r^{\dag }(A,\varrho )=\dim _\mathbf{Q (\varrho )}A^{\dag }(K_{\varrho })^{\varrho }$, is independent of the choice of $\wp _{\infty }$, $\gamma _{g}$ and $\gamma _{h}$. We refer to [6] for more details.

If $\xi$ denotes either g or h, then the restriction to $G_\mathbf{Q _{p}}$ of the Artin representation $V(\xi )$ is the direct sum of the submodules $V(\xi )_{\alpha }$ and $V(\xi )_{\beta }$ on which an arithmetic Frobenius acts as multiplication by $\alpha _{\xi }$ and $\beta _{\xi }$ respectively (cf. Assumption 1.1.3). The $G_\mathbf{Q _{p}}$-representation $V(f,g,h)^{-}$ then decomposes as the direct sum of the subspaces

$$\begin{aligned} V(f)^{-}_{ij}=V(f)^{-}\otimes _\mathbf{Q _{p}}V(g)_{i}\otimes _{L}V(h)_{j}, \end{aligned}$$

where (i, j) is a pair of elements of $\{\alpha ,\beta \}$. If $\xi$ denotes either g or h, Sect. 3.1.1 below recalls the definition of canonical weight-one differentials

$$\begin{aligned} \omega _{\xi _{\alpha }}\in {}(V(\xi )_{\alpha }\otimes _\mathbf{Q _{p}}\mathbf{Q} _{p}^{\text {nr}})^{G_\mathbf{Q _{p}}}\ \ \ \text {and}\ \ \ \eta _{\xi _{\alpha }}\in {}(V(\xi )_{\beta }\otimes _\mathbf{Q _{p}}\mathbf{Q} _{p}^{\text {nr}})^{G_\mathbf{Q _{p}}}, \end{aligned}$$

(5)

where $\mathbf{Q} _{p}^{\text {nr}}$ is the maximal unramified extension of $\mathbf{Q} _{p}$. If A is multiplicative at p, set

$$\begin{aligned} q(f)=\wp _{\infty }^{-1}(q(A))\in {}V(f)^{-}, \end{aligned}$$

where one denotes again by $\wp _{\infty } : V(f)^{-}\simeq {}V_{p}(A)^{-}$ the isomorphism arising form the fixed modular parametrisation $\wp _{\infty } : X_{1}(N_{f},p){\mathop {\longrightarrow }\limits ^{}}A$.

Under the running assumptions, the $\mathbf{Q} (\varrho )$-module ${\mathcal {Q}}_{p}(A,\varrho )$ (resp., the L-module $H^{0}(\mathbf{Q} _{p},V(f,g,h)^{-})$) is non-zero precisely A is multiplicative at p and

$$\begin{aligned} \alpha _{f}=\alpha _{g}\cdot {}\alpha _{h}\ \ \ \text {or}\ \ \ \alpha _{f}=\beta _{g}\cdot {}\alpha _{h}, \end{aligned}$$

in which case it has dimension 2 and one says that $(A,\varrho )$ is exceptional at p. More precisely, note that $\alpha _{g}\not =\beta _{g}$ by Assumptions 1.1.3, hence only one of the previous identities can be satisfied. Moreover $\alpha _{f}=\alpha _{g}\cdot {}\alpha _{h}$ (resp., $\alpha _{f}=\beta _{g}\cdot {}\alpha _{h}$) if and only if $\alpha _{f}=\beta _{g}\cdot {}\beta _{h}$ (resp., $\alpha _{f}=\alpha _{g}\cdot {}\beta _{h}$) by Assumption 1.1.1. Fix an auxiliary integer $m_{p}$ such that p splits (resp., is inert) in $\mathbf{Q} \left[ \sqrt{m_{p}}\right]$ if $\alpha _{f}=+1$ (resp., $\alpha _{f}=-1$), so that $G_\mathbf{Q _{p}}$ acts trivially on $\sqrt{m_{p}}\cdot {}q(f)$ in $V(f)^{-}\otimes _\mathbf{Q _{p}}\mathbf{Q} _{p}^{\text {nr}}$. If $\alpha _{f}=\alpha _{g}\cdot {}\alpha _{h}$, then $G_\mathbf{Q _{p}}$ acts trivially on $V(f)^{-}_{\alpha \alpha }$ and $V(f)^{-}_{\beta \beta }$, hence the p-adic periods

$$\begin{aligned} q_{\alpha \alpha }=\sqrt{m_{p}}\cdot {}q(f)\otimes \omega _{g_{\alpha }}\otimes \omega _{h_{\alpha }}\ \ \ \text {and}\ \ \ q_{\beta \beta }=\sqrt{m_{p}}\cdot {}q(f) \otimes \eta _{g_{\alpha }}\otimes \eta _{h_{\alpha }} \end{aligned}$$

can naturally be viewed as elements of $V(f)^{-}_{\alpha \alpha }$ and $V(f)^{-}_{\beta \beta }$ respectively, which generate $H^{0}(\mathbf{Q} _{p},V(f,g,h)^{-})$. Similarly, if $\alpha _{f}=\beta _{g}\cdot {}\alpha _{h}$, then the periods

$$\begin{aligned} q_{\alpha \beta }=\sqrt{m_{p}}\cdot {}q(f)\otimes \omega _{g_{\alpha }}\otimes \eta _{g_{h}}\ \ \ \text {and}\ \ \ q_{\beta \alpha }=\sqrt{m_{p}}\cdot {}q(f)\otimes \eta _{g_{\alpha }}\otimes \omega _{h_{\alpha }} \end{aligned}$$

can naturally be viewed as generators of $H^{0}(\mathbf{Q} _{p},V(f,g,h)^{-})$.

Equation (1) shows that the value of the square-root Garrett–Hida L-function $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )$ at $w_{o}$ is a non-zero multiple of

$$\begin{aligned} \left( 1-\frac{\alpha _{g}\alpha _{h}}{\alpha _{f}}\right) \left( 1-\frac{\beta _{g}\alpha _{h}}{\alpha _{f}}\right) \left( 1-\frac{\alpha _{g}\beta _{h}}{\alpha _{f}}\right) \left( 1-\frac{\beta _{g}\beta _{h}}{\alpha _{f}}\right) \cdot {}\sqrt{L(A,\varrho ,1)}, \end{aligned}$$

where $L(A,\varrho ,s)=L(f\otimes {}g\otimes {}h,s)$. The previous discussion then shows that $(A,\varrho )$ is exceptional at p precisely if one of the Euler factors which appear in the previous expression is zero, id est if $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )$ (or $L_{p}^{\alpha \alpha }(A,\varrho )$) has an exceptional zero in the sense of Mazur–Tate–Teitelbaum [13]. In this case Lemma 9.8 of [7] proves that the restriction $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )|_{\mathsf {L}}$ of $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )$ to the improving line $\mathsf {L}$ defined by the equations $\varvec{m}=1$ and $\varvec{k}=l+1$ admits the factorisation

$$\begin{aligned} \mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )|_{\mathsf {L}}=\mathscr {E}_{f}\cdot {}\mathscr {E}_{g}\cdot {}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )^{\star } \end{aligned}$$

in the ring ${\mathcal {O}}(\mathsf {L})$ of analytic functions on $\mathsf {L}$, where

$$\begin{aligned} \mathscr {E}_{f}=\left. 1-\frac{a_{p}(\varvec{f})}{a_{p}(\varvec{g}_{\alpha })\cdot {}a_{p}(\varvec{h}_{\alpha })}\right| _{\mathsf {L}}\ \ \text {and}\ \ \mathscr {E}_{g}=\left. 1-\chi _{h}(p)\cdot {}\frac{a_{p}(\varvec{g}_{\alpha })}{a_{p}(\varvec{f})\cdot {}a_{p}(\varvec{h}_{\alpha })}\right| _{\mathsf {L}}. \end{aligned}$$

Moreover, the value at $w_{o}$ of the improved p-adic L-function $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )^{\star }$ is an explicit algebraic number in $\mathbf{Q} (\varrho )$, equal to zero precisely if $L(A,\varrho ,s)$ vanishes at $s=1$. We refer to the proof of Proposition 8.3 of [12] for details.

The following is the main result of this note.

Theorem 2.1

Assume that $(A,\varrho )$ is exceptional at p. Let $(q_{\flat },q_{\natural })$ denote either the pair $(q_{\alpha \alpha },q_{\beta \beta })$ or $(q_{\alpha \beta },q_{\beta \alpha })$, depending on whether $\alpha _{f}=\alpha _{g}\cdot {}\alpha _{h}$ or $\alpha _{f}=\beta _{g}\cdot {}\alpha _{h}$ respectively. Then the following equality holds in $\mathscr {I}/\mathscr {I}^{2}$ up to sign.

Theorem 2.1 is proved in Sect. 4 below. More precisely, Sects. 3.3 and 3.4 below prove that the following equality holds in $\mathscr {I}/\mathscr {I}^{2}$ up to sign:

(6)

where $\varepsilon =+1$ if $\alpha _{f}=\alpha _{g}\cdot {}\alpha _{h}$ and $\varepsilon =-1$ if $\alpha _{f}=\beta _{g}\cdot {}\beta _{h}$, and where

$$\begin{aligned} -\frac{1}{2}\cdot {}{\mathfrak {L}}_{\varvec{\xi }}^{\text {an}}=d\log a_{p}(\varvec{\xi })_{\varvec{u}=u_{o}} \end{aligned}$$

(7)

is the value at the centre $u_{o}$ of $U_{\varvec{\xi }}$ of the logarithmic derivative of the p-th Fourier coefficient of the Hida family $\varvec{\xi }=\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha }$. In Sect. 4 we then deduce Theorem  2.1 from Eq. (6) and the study carried out in [7, Section 9] of the linear term of $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )$ at $w_{o}$ in the exceptional case.

It should be possible to extend Theorem 2.1 (and Conjecture 2.3 below) to the case of p-new eigenforms of even weight $k\geqslant {}2$ and trivial character (cf. Section 1.1 of [6]). We have not checked the details.

3.1 The rank-zero exceptional case of [6, Conjecture 1.1]

Assume in this section that $(A,\varrho )$ is exceptional at p, and that the Garrett complex L-function $L(A,\varrho ,s)=L(f\otimes {}g\otimes {}h,s)$ does not vanish at $s=1$:

$$\begin{aligned} L(A,\varrho ,1)\not =0. \end{aligned}$$

According to the main result of [8] (see also Theorem B of [3]), one has

$$\begin{aligned} A(K_{\varrho })^{\varrho }=0, \end{aligned}$$

hence $A^{\dag }(K_{\varrho })^{\varrho }={\mathcal {Q}}_{p}(A,\varrho )$. The Garrett– Nekovář p-adic regulator $R_{p}^{\alpha \alpha }(A,\varrho )$, viz. the discriminant of the p-adic height on $A^{\dag }(K_{\varrho })^{\varrho }$, is then given by

in $(\mathscr {I}^{2}/\mathscr {I}^{3})/\mathbf{Q} (\varrho )^{*2}$, where $(q_{1},q_{2})$ is a $\mathbf{Q} (\varrho )$-basis of ${\mathcal {Q}}_{p}(A,\varrho )$.

Let be the $G_\mathbf{Q }$-equivariant embedding defined by the tensor product of the isomorphism $V_{p}(A)^{-}\simeq {}V(f)^{-}$ induced by $\wp _{\infty }$, $\gamma _{g}$ and $\gamma _{h}$ (cf. Eq. (3)). The normalisation imposed on the embeddings $\gamma _{g}$ and $\gamma _{h}$ (and described in Sect. 3.1.1 below) implies that the matrix M in $\text {GL}_{2}(L)$ defined by the identity $(q_{\flat } \ q_{\natural })\cdot {}M=(\gamma _{gh}(q_{1})\ \gamma _{gh}(q_{2}))$ has determinant in $\mathbf{Q} (\varrho )^{*}$. In light of the above discussion, Theorem  2.1 then proves the following corollary, which together with Eq. (6) establishes [6, Conjecture 1.1] in the present setting.

Corollary 2.2

If $L(A,\varrho ,s)$ does not vanish at $s=1$, then $A^{\dag }(K_{\varrho })^{\varrho }={\mathcal {Q}}_{p}(A,\varrho )$ and the following equality holds in the quotient of $\mathscr {I}^{2}/\mathscr {I}^{3}$ by the action of $\mathbf{Q} (\varrho )^{*2}$.

$$\begin{aligned} L_{p}^{\alpha \alpha }(A,\varrho )\!\!\!\pmod {\mathscr {I}^{3}}=R_{p}^{\alpha \alpha }(A,\varrho ) \end{aligned}$$

3.2 Exceptional zeros and rational points (cf. [14])

Assume in this section that $(A,\varrho )$ is exceptional at p, and that the Garrett complex L-function $L(A,\varrho ,s)$ vanishes at the central critical point $s=1$:

$$\begin{aligned} L(A,\varrho ,1)=0. \end{aligned}$$

Set $\{\flat ,\natural \}=\{\alpha \alpha ,\beta \beta \}$ of $\{\flat ,\natural \}=\{\alpha \beta ,\beta \alpha \}$, depending on whether

$$\begin{aligned} \alpha _{f}=\alpha _{g}\cdot {}\alpha _{h} \hbox {or} \alpha _{f}=\beta _{g}\cdot {}\alpha _{h}. \end{aligned}$$

The p-adic L-function $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )$ belongs to $\mathscr {I}^{2}$ (cf. Theorem 2.1) and Conjecture 2.3 of [6] predicts that its image in $(\mathscr {I}^{2}/\mathscr {I}^{3})/\mathbf{Q} (\varrho )^{*}$ equals

for two rational points P and Q in $A(K_{\varrho })^{\varrho }$. (Recall that the p-adic height is skew-symmetric, hence the previous expression is a square root of its discriminant on the $\mathbf{Q} (\varrho )$-submodule of $A^{\dag }(K_{\varrho })^{\varrho }$ generated by $q_{\flat }, q_{\natural }, P$ and Q.) One has

by Eq. (6). Moreover, Sect. 3.5 below proves that

(8)

for each Selmer class x in $\text {Sel}(\mathbf{Q} ,V(f,g,h))$, where

$$\begin{aligned} \log _{\flat }=\langle \log _{p}(\cdot ),q_{\natural }\rangle _{fgh} : H^{1}_{\text {fin}}(\mathbf{Q} _{p},V(f,g,h)){\mathop {\longrightarrow }\limits ^{}}L. \end{aligned}$$

Here $\log _{p} : H^{1}_{\text {fin}}(\mathbf{Q} _{p},V(f,g,h))\simeq {}D_{\text {dR}}(V(f,g,h))/\text {Fil}^{0}$ is the Bloch–Kato p-adic logarithm (cf. Lemma 9.1 of [7]), and $\left<{}\cdot ,\cdot \right>_{fgh} : D_{\text {dR}}(V(f,g,h))^{\otimes {}2}{\mathop {\longrightarrow }\limits ^{}}L$ is the pairing induced by the natural Kummer duality $\pi _{fgh} : V(f,g,h)^{\otimes 2}{\mathop {\longrightarrow }\limits ^{}}L(1)$ defined in Sect. 3.1.1 below (cf. Eq. (11)). We are then led to the following

Conjecture 2.3

Assume that $A(K_{\varrho })^{\varrho }$ is a 2-dimensional $\mathbf{Q} (\varrho )$-vector space. Then for any $\mathbf{Q} (\varrho )$-basis (P, Q) of $A(K_{\varrho })^{\varrho }$, the equality

$$\begin{aligned} \frac{\partial ^{2}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )}{\partial \varvec{k}^{2}}(w_{o})=\log _{\flat }(P)\cdot {}\log _{\natural }(Q)- \log _{\natural }(P)\cdot {}\log _{\flat }(Q) \end{aligned}$$

holds in L up to multiplication by a non-zero scalar in $\mathbf{Q} (\varrho )^{*}$.

As explained in [5], the main result of [1] can be used to prove cases of Conjecture 2.3 when g and h are theta series associated with certain ray class characters of the same imaginary quadratic field in which p is inert (and P and Q are Heegner points). By combining this with an extension of the height computations carried out in [16, 17], the article [4] proves instances of Conjecture 1.1 of [6] in this setting.

Remark 2.4

In light of the aforementioned results of [5], Rivero proposes in [14, Conjecture 4.5] a variant of Conjecture 2.3. He also asks (cf. Question 5.3 of [14]) if one can expect a similar description of $\frac{\partial ^{2}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )}{\partial \varvec{k}^{2}}(w_{o})$ when A has good reduction at p. The previous discussion places Rivero’s conjecture within a conceptual framework and sheds some light on this question.

4 Height computations

Throughout the rest of this note we assume that $(A,\varrho )$ is exceptional at p. In particular A has multiplicative reduction at p, id est p divides exactly $N_{f}$.

4.1 Setting and notations

This subsection briefly recalls the needed definitions and notations from our previous articles [6, 7].

4.1.1 Galois representations

Set $N=\text {lcm}(N_{f},N_{g},N_{h})$ and let $G_\mathbf{Q ,N}$ be the Galois group of the maximal extension of $\mathbf{Q}$ contained in ${\bar{\mathbf{Q }}}$ and unramified outside $N\infty$. If $\varvec{\xi }$ denotes one of $\varvec{f},\varvec{g}_{\alpha }$ and $\varvec{h}_{\alpha }$, let $V(\varvec{\xi })$ be the big Galois representation associated with $\varvec{\xi }$ (cf. Section 5 of [7]). It is a free $\mathscr {O}_{\varvec{\xi }}$-module of rank two, equipped with a continuous linear action $G_\mathbf{Q ,N}$. For each u in $U_{\varvec{\xi }}\cap {}\mathbf{Z} _{\geqslant {}2}$ the base change $V(\varvec{\xi })\otimes _{u}L$ of $V(\varvec{\xi })$ along evaluation at u on $\mathscr {O}_{\varvec{\xi }}$ is canonically isomorphic to the homological p-adic Deligne representation of $\varvec{\xi }_{u}$ with coefficients in L (cf. loco citato for more details). In particular if $\varvec{\xi }=\varvec{f}$ and $u=2$ there is a natural specialisation isomorphism $\rho _{2} : V(\varvec{f})\otimes _{2}L\simeq {}V(f)$. If $\varvec{\xi }=\varvec{g}_{\alpha },\varvec{h}_{\alpha }$ and $u=1$ set $V(\xi )=V(\varvec{\xi })\otimes _{1}L$ (cf. Sect. 1). It is a two-dimensional L-vector space affording the dual of the p-adic Deligne–Serre representation of $\xi =g,h$ with coefficients in L. In order to have a uniform notation, in this case one defines $\rho _{1} : V(\varvec{\xi })\otimes _{1}L{\mathop {\longrightarrow }\limits ^{}}V(\xi )$ to be the identity.

The restriction of $V(\varvec{\xi })$ to $G_\mathbf{Q _{p}}$ (via the embedding $i_{p}$ fixed at the outset) fits into a short exact sequence of $\mathscr {O}_{\varvec{\xi }}[G_\mathbf{Q _{p}}]$-modules with $V(\varvec{\xi })^{\pm }$ free of rank one over $\mathscr {O}_{\varvec{\xi }}$. More precisely, let $\chi _{\text {cyc}} : G_\mathbf{Q }{\mathop {\longrightarrow }\limits ^{}}\mathbf{Z} _{p}^{*}$ be the p-adic cyclotomic character, and let ${\check{a}}_{p}(\varvec{\xi }) : G_\mathbf{Q _{p}}{\mathop {\longrightarrow }\limits ^{}}\mathscr {O}_{\varvec{\xi }}^{*}$ be the unramified character sending an arithmetic Frobenius to the p-th Fourier coefficients $a_{p}(\varvec{\xi })$ of $\varvec{\xi }$. Then

$$\begin{aligned} V(\varvec{\xi })^{+}\simeq {}\mathscr {O}_{\varvec{\xi }}\big (\chi _{\text {cyc}}^{\varvec{u}-1}\cdot {}\chi _{\xi }{\check{a}}_{p}(\varvec{\xi })^{-1}\big )\ \ \ \text {and}\ \ \ V(\varvec{\xi })^{-}\simeq {}\mathscr {O}_{\varvec{\xi }}({\check{a}}_{p}(\varvec{\xi })), \end{aligned}$$

(9)

where $\chi _{\text {cyc}}^{\varvec{u}-1} : G_\mathbf{Q }{\mathop {\longrightarrow }\limits ^{}}\mathscr {O}_{\varvec{\xi }}^{*}$ satisfies $\chi _{\text {cyc}}^{\varvec{u}-1}(\sigma )(u)=\chi _{\text {cyc}}(\sigma )^{u-1}$ for each u in $U_{\varvec{\xi }}\cap {}\mathbf{Z}$. (The freeness of $V(\varvec{\xi })^{\pm }$ is guaranteed by Assumption 1.1.3, cf. Section 5 of [7].) If $\varvec{\xi }=\varvec{f}$ and $u=2$ the specialisation isomorphism $\rho _{2}$ identifies $V(\varvec{f})^{-}\otimes _{2}L$ with the maximal unramified quotient $V(f)^{-}$ of V(f). If $\varvec{\xi }=\varvec{g}_{\alpha },\varvec{h}_{\alpha }$ and $u=1$ we set $V(\xi )_{\beta }=V(\varvec{\xi })^{+}\otimes _{1}L$ and $V(\xi )_{\alpha }=V(\varvec{\xi })^{-}\otimes _{1}L$. One has $V(\xi )=V(\xi )_{\alpha }\oplus {}V(\xi )_{\beta }$, where $V(\xi )_{\gamma }=V(\xi )^{\text {Frob}_{p}=\gamma _{\xi }}$ for $\gamma =\alpha ,\beta$ is the submodule of $V(\xi )$ on which an arithmetic Frobenius $\text {Frob}_{p}$ acts as multiplication by $\gamma _{\xi }=\alpha _{\xi },\beta _{\xi }$ (cf. Assumption 1.1.3).

There is a natural $G_\mathbf{Q }$-equivariant skew-symmetric perfect pairing

$$\begin{aligned} \pi _{\varvec{\xi }} : V(\varvec{\xi })\otimes _{\mathscr {O}_{\varvec{\xi }}}V(\varvec{\xi }){\mathop {\longrightarrow }\limits ^{}}\mathscr {O}_{\varvec{\xi }}(\chi _{\xi }\cdot {}\chi _{\text {cyc}}^{\varvec{u}-1}), \end{aligned}$$

inducing perfect dualities $\pi _{\varvec{\xi }} : V(\varvec{\xi })^{\pm }\otimes _{\mathscr {O}_{\varvec{\xi }}}V(\varvec{\xi })^{\mp }{\mathop {\longrightarrow }\limits ^{}}\mathscr {O}_{\varvec{\xi }}(\chi _{\xi }\cdot {}\chi _{\text {cyc}}^{\varvec{u}-1})$. (See Section 5 cf. [7] for the definitions).

Denote by $\Xi _{\varvec{f}\varvec{g}\varvec{h}}=\chi _{\text {cyc}}^{(4-\varvec{k}-l-\varvec{m})/2} : G_\mathbf{Q }{\mathop {\longrightarrow }\limits ^{}}\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}^{*}$ the character whose composition with evaluation at (k, l, m) in $U_{\!\varvec{f}}\times {}U_{\varvec{g}}\times {}U_{\varvec{h}}\cap {}\mathbf{Z} ^{3}$ on $\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}$ equals $\chi _{\text {cyc}}^{(4-k-l-m)/2}$. If $\cdot$ denotes one of the symbols $\emptyset ,+$ and −, define

$$\begin{aligned} \varvec{V}^{\cdot }=V(\varvec{f})^{\cdot }{\hat{\otimes }}_{L}V(\varvec{g}_{\alpha }){\hat{\otimes }}V(\varvec{h}_{\alpha })\otimes _{\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}}\Xi _{\varvec{f}\varvec{g}\varvec{h}}. \end{aligned}$$

Then $\varvec{V}=V(\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha })$, resp. $\varvec{V}^{\pm }=V(\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha })^{\pm }$ is a free $\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}$-module of rank 8, resp. 4, equipped with a continuous action of $G_\mathbf{Q ,N}$, resp. $G_\mathbf{Q _{p}}$. As $\chi _{g}\cdot {}\chi _{h}=1$ (cf. Assumption 1.1), the product of the perfect dualities $\pi _{\varvec{\xi }}$, for $\varvec{\xi }=\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha }$, yields a perfect skew-symmetric Kummer duality $\varvec{\pi } : \varvec{V}\otimes _{\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}}\varvec{V}{\mathop {\longrightarrow }\limits ^{}}\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}(1)$, inducing a perfect local Kummer duality $\varvec{\pi } : \varvec{V}^{\pm }\otimes _{\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}}\varvec{V}^{\mp }{\mathop {\longrightarrow }\limits ^{}} \mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}(1)$. After setting

$$\begin{aligned} V^{\cdot }=V(f,g,h)^{\cdot }=V(f)^{\cdot }\otimes _{L}V(g)\otimes _{L}V(h) \end{aligned}$$

and $w_{o}=(2,1,1)$, the product $\rho _{w_{o}}=\rho _{2}{\hat{\otimes }}\rho _{1}{\hat{\otimes }}\rho _{1}$ gives natural isomorphisms

$$\begin{aligned} \rho _{w_{o}} : \varvec{V}^{\cdot }\otimes _{w_{o}}L\simeq {}V^{\cdot } \end{aligned}$$

(10)

(where $\cdot \otimes _{w_{o}}L$ denotes the base change along evaluation at $w_{o}$ on $\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}$). Let

$$\begin{aligned} \pi _{fgh} : V\otimes _{L}V{\mathop {\longrightarrow }\limits ^{}}L(1) \end{aligned}$$

(11)

be the specialisation of $\varvec{\pi }$ via $\rho _{w_{o}}$, and define $\pi : V^{\pm }\otimes _{L}V^{\mp }{\mathop {\longrightarrow }\limits ^{}}L(1)$ similarly.

4.1.1.1 Weight one differentials

Define $D(\varvec{\xi })^{-}=H^{0}(\mathbf{Q} _{p},V(\varvec{\xi })^{-}{\hat{\otimes }}_\mathbf{Q _{p}}{\hat{\mathbf{Q }}}_{p}^{\text {nr}})$, where ${\hat{\mathbf{Q }}}_{p}^{\text {nr}}$ is the p-adic completion of the maximal unramified extension of $\mathbf{Q} _{p}$ (and as usual $\varvec{\xi }$ denotes one of $\varvec{f}$, $\varvec{g}_{\alpha }$ and $\varvec{h}_{\alpha }$). For each u in $U_{\varvec{\xi }}\cap {}\mathbf{Z} _{\geqslant {}2}$ there is a natural comparison isomorphism between $D(\varvec{\xi })^{-}\otimes _{u}L$ and the $\varvec{\xi }_{u}$-isotypic component of the space of cuspidal modular forms of weight u, level $\Gamma _{1}(N_{\xi }p)$ and Fourier coefficients in L. Assumption 1.1.3 guarantees that $D(\varvec{\xi })^{-}$ is free (of rank one) over $\mathscr {O}_{\varvec{\xi }}$, and admits a basis $\omega _{\varvec{\xi }}$ whose image in $D(\varvec{\xi })^{-}\otimes _{u}L$ corresponds to $\varvec{\xi }_{u}$ under the aforementioned comparison isomorphism, for each u in $U_{\varvec{\xi }}\cap {}\mathbf{Z} _{\geqslant {}2}$. (We refer to Section 3.1 of [6] and the references therein for more details.)

For $\varvec{\xi }=\varvec{g}_{\alpha },\varvec{h}_{\alpha }$, the holomorphic weight-one differential

$$\begin{aligned} \omega _{\xi _{\alpha }}\in {}(V(\xi )_{\alpha }\otimes _\mathbf{Q _{p}}\mathbf{Q} _{p}^{\text {nr}})^{G_\mathbf{Q _{p}}} \end{aligned}$$

mentioned in Eq. (5) is defined to be the weight-one specialisation of $\omega _{\varvec{\xi }}$, viz. the image of $\omega _{\varvec{\xi }}$ in the quotient $D(\varvec{\xi })^{-}\otimes _{1}L=D(\xi )_{\alpha }$. The weight-one specialisation of $\pi _{\varvec{\xi }}$ yields a perfect $G_\mathbf{Q }$-equivariant skew-symmetric pairing

$$\begin{aligned} \pi _{\xi } : V(\xi )\otimes _{L}V(\xi ){\mathop {\longrightarrow }\limits ^{}}L(\chi _{\xi }). \end{aligned}$$

Let c be the common conductor of $\chi _{g}$ and $\chi _{h}$, and identify $(L(\chi _{\xi })\otimes _\mathbf{Q _{p}}\mathbf{Q} _{p}^{\text {nr}})^{G_\mathbf{Q _{p}}}$ with L via the Gauß sum $G(\chi _{\xi })=(-c)^{i_{\xi }}\sum _{a\in {}(\mathbf{Z} /c\mathbf{Z} )^{*}}\chi _{\xi }(a)^{-1}\otimes {}e^{2\pi {}ia/c}$, where $i_{g}=0$ and $i_{h}=1$ (so that $G(\chi _{g})\cdot {}G(\chi _{h})=1$ by Assumption 1.1.1). The pairing $\pi _{\xi }$ then induces a perfect duality $\left<{}\cdot ,\cdot \right>_{\xi } : D(\xi )_{\alpha }\otimes _{L}D(\xi )_{\beta }{\mathop {\longrightarrow }\limits ^{}}L$, where $D(\xi )_{\gamma }=(V(\xi )_{\gamma }\otimes _\mathbf{Q _{p}}\mathbf{Q} _{p}^{\text {nr}})^{G_\mathbf{Q _{p}}}$. One defines the antiholomorphic weight-one differential (cf. Eq. (5))

$$\begin{aligned} \eta _{\xi _{\alpha }}\in {}(V(\xi )_{\beta }\otimes _\mathbf{Q _{p}}\mathbf{Q} _{p}^{\text {nr}})^{G_\mathbf{Q _{p}}} \end{aligned}$$

to be the dual of $\omega _{\xi _{\alpha }}$ under $\left<{}\cdot ,\cdot \right>_{\xi }$, viz. the element satisfying $\left<{}\omega _{\xi _{\alpha }},\eta _{\xi _{\alpha }}\right>_{\xi }=1$.

4.1.1.2 The embeddings $\gamma _{g}$ and $\gamma _{h}$

With the notations of Sect. 1, set $V_{g}=V_{\varrho _{1}}$ and $V_{h}=V_{\varrho _{2}}$. Let $\xi$ denote either g or h. As recalled above, the Artin representation $V(\xi )=V(\varvec{\xi })\otimes _{1}L$ affords the dual of the p-adic Deligne representation of $\xi$ with coefficients in L, id est is isomorphic to $V_{\xi }\otimes _\mathbf{Q (\varrho )}L$. Enlarging L if necessary, we normalise the $G_\mathbf{Q }$-equivariant embedding $\gamma _{\xi } : V_{\xi }{\mathop {\longrightarrow }\limits ^{}}V(\xi )$ (introduced in Eq. (3)) by requiring that the composition $\pi _{\xi }\circ {}(\gamma _{\xi }\otimes {}\gamma _{\xi })$ takes values in the number field $\mathbf{Q} (\varrho )$ (via the embedding fixed at the outset).

4.1.2 Selmer complexes

Let $\mathbf{R} {\tilde{\Gamma }}_{f}(\mathbf{Q} ,V)$ be the Nekovář Selmer complex associated with $(V,V^{+})$ (cf. Section 2.2 of [6]). It is an element of the derived category $\text {D}_{\text {ft}}^{b}(L)$ of cohomologically bounded complexes of L-modules with cohomology of finite type over L, sitting is an exact triangle

$$\begin{aligned} \mathbf{R} \Gamma _{\text {cont}}(G_\mathbf{Q ,N},V){\mathop {\longrightarrow }\limits ^{p^{-}\circ {}\text {res}_{p}}}\mathbf{R} \Gamma _{\text {cont}}(G_\mathbf{Q _{p}},V^{-}){\mathop {\longrightarrow }\limits ^{}}\mathbf{R} {\tilde{\Gamma }}_{f}(\mathbf{Q} ,V)[1], \end{aligned}$$

(12)

where $\mathbf{R} \Gamma _{\text {cont}}(G,\cdot )$ is the complex of continuous non-homogeneous cochains of G with values in $\cdot$, $\text {res}_{p}$ is the restriction map (induced by the embedding fixed at the outset) and $p^{-}$ is the map induced by the projection $V{\mathop {\longrightarrow }\limits ^{}}V^{-}$. Denote by

$$\begin{aligned} \tilde{H}^{\cdot }_{f}(\mathbf{Q} ,V)=H^{\cdot }(\mathbf{R} {\tilde{\Gamma }}_{f}(\mathbf{Q} ,V)) \end{aligned}$$

the cohomology of $\mathbf{R} {\tilde{\Gamma }}(\mathbf{Q} ,V)$, let $\text {Sel}(\mathbf{Q} ,V)$ be the Bloch–Kato Selmer group of V over $\mathbf{Q}$, and let $i^{+} : V^{+}{\mathop {\longrightarrow }\limits ^{}}V$ be the natural inclusion. Then there is a commutative and exact diagram of L-vector spaces (cf. loc. cit.)

(13)

where the first line arises from the exact triangle (12). In addition there is a unique section $\imath _{\text {ur}} : \text {Sel}(\mathbf{Q} ,V){\mathop {\longrightarrow }\limits ^{}}\tilde{H}^{1}_{f}(\mathbf{Q} ,V)$ of the above projection such that $\imath _{\text {ur}}(x)^{+}$ belongs to the Bloch–Kato finite subspace $H^{1}_{\text {fin}}(\mathbf{Q} _{p},V^{+})$ for each x in $\text {Sel}(\mathbf{Q} ,V)$. We often use $\jmath$ and $\imath _{\text {ur}}$ to identify Nekovář’s extended Selmer group $\tilde{H}^{1}_{f}(\mathbf{Q} ,V)$ with the naive extended Selmer group $\text {Sel}^{\dag }(\mathbf{Q} ,V)=H^{0}(\mathbf{Q} _{p},V^{-})\oplus {}\text {Sel}(\mathbf{Q} ,V)$ (cf. Sect. 1).

One similarly associates with $(\varvec{V},\varvec{V}^{+})$ a Selmer complex

$$\begin{aligned} \mathbf{R} {\tilde{\Gamma }}_{f}(\mathbf{Q} ,\varvec{V})\in {}\text {D}_{\text {ft}}^{b}(\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}) \end{aligned}$$

sitting in an exact triangle analogous to (12). (We refer to loc. cit. for more details.)

4.2 Preliminary lemmas

This section gives a concrete description of the functionals for q in $H^{0}(\mathbf{Q} _{p},V^{-})$ (cf. Lemma 3.4 below).

4.2.1 Bockstein maps

Let $(\varvec{{\mathcal {C}}},{\mathcal {C}})$ denote one of the pairs

$$\begin{aligned} (\mathbf{R} \Gamma _{p}(\varvec{V}^{-}),\mathbf{R} \Gamma _{p}(V^{-})), (\mathbf{R} \Gamma (\varvec{V}),\mathbf{R} \Gamma (V)) \quad {\mathrm {and}}\quad (\mathbf{R} {\tilde{\Gamma }}_{f} (\mathbf{Q} ,\varvec{V}),\mathbf{R} {\tilde{\Gamma }}_{f}(\mathbf{Q} ,V)), \end{aligned}$$

where $\mathbf{R} \Gamma _{p}(\cdot )$ and $\mathbf{R} \Gamma (\cdot )$ are shorthands for $\mathbf{R} \Gamma _{\text {cont}}(\mathbf{Q} _{p},\cdot )=\mathbf{R} \Gamma _{\text {cont}}(G_\mathbf{Q _{p}},\cdot )$ and $\mathbf{R} \Gamma _{\text {cont}}(G_\mathbf{Q ,N},\cdot )$ respectively (cf. Sect. 3.1.2). The specialisation maps $\rho _{w_{o}}$ (cf. Eq. (10)) induce isomorphisms

$$\begin{aligned} \rho _{w_{o}} : \varvec{{\mathcal {C}}}\otimes _{\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}},w_{o}}^\mathbf{L }L\simeq {}{\mathcal {C}}\ \ \ \text {and}\ \ \ \rho _{w_{o}}\otimes \text {id} : \varvec{{\mathcal {C}}}\otimes _{\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}}^\mathbf{L }\mathscr {I}/\mathscr {I}^{2}[1]\simeq {}{\mathcal {C}}\otimes _{L}\mathscr {I}/\mathscr {I}^{2}[1]. \end{aligned}$$

(14)

Applying $\varvec{{\mathcal {C}}}\otimes _{\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}}^\mathbf{L }\cdot$ to the exact triangle

$$\begin{aligned} \mathscr {I}/\mathscr {I}^{2}{\mathop {\longrightarrow }\limits ^{}}\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}/\mathscr {I}^{2}{\mathop {\longrightarrow }\limits ^{}}L{\mathop {\longrightarrow }\limits ^{}}\mathscr {I}/\mathscr {I}^{2}[1] \end{aligned}$$

(arising from evaluation on $w_{o}$) then yields a derived Bockstein map

$$\begin{aligned} \varvec{\beta }_{\varvec{{\mathcal {C}}}/{\mathcal {C}}} : {\mathcal {C}}{\mathop {\longrightarrow }\limits ^{}}{\mathcal {C}}\otimes _{L}\mathscr {I}/\mathscr {I}^{2}[1], \end{aligned}$$

which in turn induces in cohomology a Bockstein map

$$\begin{aligned} \beta _{\varvec{{\mathcal {C}}}/{\mathcal {C}}} : H^{i}({\mathcal {C}}){\mathop {\longrightarrow }\limits ^{}}H^{i+1}({\mathcal {C}})\otimes _{L}\mathscr {I}/\mathscr {I}^{2}. \end{aligned}$$

If no risk of confusion arises, we simply write $\beta$ for $\beta _{\varvec{{\mathcal {C}}}/{\mathcal {C}}}$. Let

$$\begin{aligned} \jmath : H^{i}(\mathbf{Q} _{p},V^{-}){\mathop {\longrightarrow }\limits ^{}}\tilde{H}^{i+1}_{f}(\mathbf{Q} ,V) \end{aligned}$$

be the maps arising from the exact triangle (12).

Lemma 3.1

The following diagram commutes.

Proof

For $M=V,\varvec{V}$ one has an exact triangle (cf. Equation (12))

$$\begin{aligned} \Delta _{M} : \mathbf{R} \Gamma _{\text {cont}}(G_\mathbf{Q ,N},M)[-1]{\mathop {\longrightarrow }\limits ^{p^{-}\circ {}\text {res}_{p}\,}}\mathbf{R} \Gamma _{\text {cont}}(\mathbf{Q} _{p},M^{-})[-1]{\mathop {\longrightarrow }\limits ^{\varvec{\jmath }_{M}\,}} \mathbf{R} {\tilde{\Gamma }}_{f}(\mathbf{Q} ,M). \end{aligned}$$

Moreover $\Delta _{V}$ is obtained by applying $\cdot {}\otimes _{\mathscr {O}_{\varvec{f}\varvec{g}\varvec{h}},w_{o}}^\mathbf{L }L$ to $\Delta _{\varvec{V}}$ (cf. Eq. (14)). It follows from the definition of the derived Bockstein maps $\varvec{\beta }^{-}$ and $\varvec{\beta }$ on $\mathbf{R} \Gamma _{\text {cont}}(\mathbf{Q} _{p},V^{-})$ and $\mathbf{R} {\tilde{\Gamma }}(\mathbf{Q} ,V)$ respectively that $\varvec{\jmath }_{V}\otimes \mathscr {I}/\mathscr {I}^{2}[1]\circ {}\varvec{\beta }^{-}$ is equal to $\varvec{\beta }\circ {}\varvec{\jmath }_{V}$. Since by definition the maps $\jmath$ are the ones induced in cohomology by $\varvec{\jmath }_{V}$, the lemma follows. $\square$

The following lemma gives a concrete description of $\beta _{\varvec{{\mathcal {C}}}/{\mathcal {C}}}$.

Lemma 3.2

Let $(\varvec{{\mathcal {C}}},{\mathcal {C}})$ be as above, let z be a 1-cocycle in ${\mathcal {C}}$, let Z be a 1-cochain in $\varvec{{\mathcal {C}}}$, and let $Z_{\varvec{k}}, Z_{l}$ and $Z_{\varvec{m}}$ be 2-cochains in $\varvec{{\mathcal {C}}}$ such that

$$\begin{aligned} \rho _{w_{o}}(Z)=z\ \ \text {and}\ \ dZ=Z_{\varvec{k}}\cdot {}(\varvec{k}-2)+Z_{l}\cdot {}(l-1)+Z_{\varvec{m}}\cdot {}(\varvec{m}-1). \end{aligned}$$

Then $z_{\cdot }=\rho _{w_{o}}(Z_{\cdot })$ is a 2-cocycle for $\cdot =\varvec{k},l,\varvec{m}$, and one has the equality

$$\begin{aligned} -\beta _{\varvec{{\mathcal {C}}}/{\mathcal {C}}}(cl(z))=cl(z_{\varvec{k}})\cdot {}(\varvec{k}-2)+cl(z_{l})\cdot {}(l-1)+cl(z_{\varvec{m}})\cdot {}(\varvec{m}-1) \end{aligned}$$

in $H^{2}({\mathcal {C}})\otimes _{L}\mathscr {I}/\mathscr {I}^{2}$, where $cl(\cdot )$ is the class in $H^{i}({\mathcal {C}})$ represented by the i-cocycle $\cdot$.

Proof

The proof is very similar to that of [16, Lemma 5.5]. We omit it. $\square$

4.2.2 Local and global duality

Nekovář’s generalised Poitou–Tate duality associates with the perfect duality $\pi _{fgh}$ introduced in Eq. (11) a global cup-product pairing (cf. Section 2.4 of [6])

$$\begin{aligned} \left<{}\cdot ,\cdot \right>_{\text {Nek}} : \tilde{H}^{2}_{f}(\mathbf{Q} ,V)\otimes _{L}\tilde{H}^{1}_{f}(\mathbf{Q} ,V){\mathop {\longrightarrow }\limits ^{}}L. \end{aligned}$$

(15)

The pairing $\pi _{fgh}$ induces a Kummer duality $V^{-}\otimes _{L}V^{+}{\mathop {\longrightarrow }\limits ^{}}L(1)$ and we denote by

$$\begin{aligned} \left<{}\cdot ,\cdot \right>_{\text {Tate}} : H^{1}(\mathbf{Q} _{p},V^{-})\otimes _{L}H^{1}\left( \mathbf{Q} _{p},V^{+}\right) {\mathop {\longrightarrow }\limits ^{}}L \end{aligned}$$

(16)

the induced local Tate duality pairing. Recall finally the map

$$\begin{aligned} \cdot ^{+} : \tilde{H}_{f}^{1}(\mathbf{Q} ,V){\mathop {\longrightarrow }\limits ^{}}H^{1}(\mathbf{Q} _{p},V^{+}) \end{aligned}$$

introduced in diagram (13).

Lemma 3.3

For each $\zeta$ in $H^{1}(\mathbf{Q} _{p},V^{-})$ and $\xi$ in $\tilde{H}^{1}_{f}(\mathbf{Q} ,V)$ one has

$$\begin{aligned} \left<{}\jmath (\zeta ),\xi \right>_{\text {Nek}}=\langle \zeta ,\xi ^{+}\rangle _{\text {Tate}}. \end{aligned}$$

Proof

This is proved as in [16, Lemma 5.7]. $\square$

4.2.3 The Garrett–Nekovář p-adic height pairing

Set

$$\begin{aligned} {\tilde{\beta }}_{\varvec{f}\varvec{g}_{\alpha }\varvec{h}_{\alpha }} =\beta _\mathbf{R {\tilde{\Gamma }}_{f}(\mathbf{Q} ,\varvec{V})/\mathbf{R} {\tilde{\Gamma }}_{f}(\mathbf{Q} ,V)} : \tilde{H}^{1}_{f}(\mathbf{Q} ,V){\mathop {\longrightarrow }\limits ^{}}\tilde{H}^{2}_{f}(\mathbf{Q} ,V)\otimes _{L}\mathscr {I}/\mathscr {I}^{2}. \end{aligned}$$

After identifying $\tilde{H}^{1}_{f}(\mathbf{Q} ,V)$ with $\text {Sel}^{\dag }(\mathbf{Q} ,V)$ (cf. Sect. 3.1.2), the canonical height introduced in Sect.  is defined by (cf. [6, Section 2])

for each x and y in $\tilde{H}^{1}_{f}(\mathbf{Q} ,V)$, where we write again $\left<{}\cdot ,\cdot \right>_{\text {Nek}}$ for the $\mathscr {I}/\mathscr {I}^{2}$-base change of Nekovář’s cup-product (15). Lemmas 3.1 and 3.3 give the following

Lemma 3.4

For each q in $H^{0}(\mathbf{Q} _{p},V^{-})$ one has

as $\mathscr {I}/\mathscr {I}^{2}$-valued maps on $\tilde{H}^{1}_{f}(\mathbf{Q} ,V)$, where $\beta _{\varvec{f}\varvec{g}_{\alpha }\varvec{h}_{\alpha }}^{-}=\beta _\mathbf{R \Gamma _{p}(\varvec{V}^{-})/\mathbf{R} \Gamma _{p}(V^{-})}$ (and we write again $\left<{}\cdot ,\cdot \right>_{\text {Tate}}$ for the $\mathscr {I}/\mathscr {I}^{2}$-base change of the local Tate pairing (16)).

4.3 Computation of

Assume in this subsection $\alpha _{f}=\alpha _{g}\cdot {}\alpha _{h}$, so that $H^{0}(\mathbf{Q} _{p},V^{-})$ is generated over L by the periods

$$\begin{aligned} q_{\alpha \alpha }=\sqrt{m_{p}}\cdot {}q(f)\otimes \omega _{g_{\alpha }}\otimes \omega _{h_{\alpha }}\ \ \ \text {and}\ \ \ q_{\beta \beta }=\sqrt{m_{p}}\cdot {}q(f)\otimes \eta _{g_{\alpha }}\otimes \eta _{h_{\alpha }}. \end{aligned}$$

Recall that $\chi _{\text {cyc}} : G_\mathbf{Q }{\mathop {\longrightarrow }\limits ^{}}\mathbf{Z} _{p}^{*}$ denotes the p-adic cyclotomic character. Fix a lift $\varvec{q}_{\beta \beta }$ in $\varvec{V}^{-}$ of $q_{\beta \beta }$ under $\rho _{w_{o}}$. Since (cf. Sect. 3.1.1)

and $V(\xi )_{\beta }=V(\varvec{\xi }_{\alpha })^{+}\otimes _{1}L$ for $\xi =g,h$, we can choose $\varvec{q}_{\beta \beta }$ in the $G_\mathbf{Q _{p}}$-submodule

(cf. Sect. 3.1.1). By Eq. (9) one has

$$\begin{aligned} d\varvec{q}_{\beta \beta }=\Phi \cdot {}\varvec{q}_{\beta \beta }, \end{aligned}$$

(17)

where d denotes the differentials of the complex $\mathbf{R} \Gamma _{\text {cont}}(\mathbf{Q} _{p},\varvec{V}^{-})$ and

$$\begin{aligned} \Phi =\frac{{\check{a}}_{p}(\varvec{f})}{{\check{a}}_{p}(\varvec{g}_{\alpha })\cdot {}{\check{a}}_{p}(\varvec{h}_{\alpha })}\cdot {}\chi _{\text {cyc}}^{(l+\varvec{m}-\varvec{k})/2}-1 : G_\mathbf{Q _{p}}{\mathop {\longrightarrow }\limits ^{}}\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}. \end{aligned}$$

The assumption $\alpha _{f}=\alpha _{g}\cdot {}\alpha _{h}$ implies that $\Phi$ takes value in $\mathscr {I}$, and that its composition $\Phi ^{\prime }$ with the projection $\mathscr {I}{\mathop {\longrightarrow }\limits ^{}}\mathscr {I}/\mathscr {I}^{2}$ is of the form

$$\begin{aligned} \Phi ^{\prime }=\varphi _{\varvec{k}}\cdot {}(\varvec{k}-2)+\varphi _{l}\cdot {}(l-1)+\varphi _{\varvec{m}}\cdot {}(\varvec{m}-1) \end{aligned}$$

with $\varphi _{\varvec{u}}$ in $H^{1}(\mathbf{Q} _{p},\mathbf{Q} _{p})$ for $\varvec{u}=\varvec{k},l,\varvec{m}$. Identify $H^{1}(\mathbf{Q} _{p},\mathbf{Q} _{p})$ with the $\mathbf{Q} _{p}$-vector space $\text {Hom}_{}(\mathbf{Q} _{p}^{*},\mathbf{Q} _{p})$ of continuous morphisms of groups from $\mathbf{Q} _{p}^{*}$ to $\mathbf{Q} _{p}$ via the local reciprocity map $\text {rec}_{p} : \mathbf{Q} _{p}^{*}{\mathop {\longrightarrow }\limits ^{}}G_\mathbf{Q _{p}}^{\text {ab}}$, normalised by requiring $\text {rec}_{p}(p^{-1})$ to be an arithmetic Frobenius. By local class field theory, for each p-adic unit u one has

$$\begin{aligned} \varphi _{\varvec{k}}(u)=\frac{\partial }{\partial \varvec{k}}\left. \left( \left<{}u\right>^{(l+\varvec{m}-\varvec{k})/2}-1\right) \right| _{w_{o}}=-\frac{1}{2}\cdot {}\log _{p}(u), \end{aligned}$$

where $\left<{}\cdot \right> : \mathbf{Z} _{p}^{*}{\mathop {\longrightarrow }\limits ^{}}1+p\mathbf{Z} _{p}$ denotes the projection to principal units, and

$$\begin{aligned} \varphi _{\varvec{k}}(p)=\frac{\partial }{\partial \varvec{k}}\left. \left( \frac{a_{p}(\varvec{g}_{\alpha })\cdot {}a_{p}(\varvec{h}_{\alpha })}{a_{p}(\varvec{f})}-1\right) \right| _{w_{o}}= \frac{1}{2}\cdot {}{\mathfrak {L}}_{\varvec{f}}^{\text {an}} \end{aligned}$$

(cf. Eq. (7)). As a consequence $-2\cdot {}\varphi _{\varvec{k}}$ is equal to

$$\begin{aligned} \log _{\varvec{f}}=\log _{p}-{\mathfrak {L}}_{\varvec{f}}^{\text {an}}\cdot {}\text {ord}_{p}\in {}H^{1}(\mathbf{Q} _{p},\mathbf{Q} _{p}) \end{aligned}$$

(where the p-adic valuation $\text {ord}_{p} : \mathbf{Q} _{p}^{*}{\mathop {\longrightarrow }\limits ^{}}\mathbf{Q} _{p}$ is normalised by $\text {ord}_{p}(p)=1$). Similarly one shows that $2\cdot {}\varphi _{l}$ and $2\cdot {}\varphi _{m}$ are equal to the logarithms $\log _{\varvec{g}_{\alpha }}=\log _{p}-{\mathfrak {L}}_{\varvec{g}_{\alpha }}^{\text {an}}\cdot {}\text {ord}_{p}$ and $\log _{\varvec{h}_{\alpha }}=\log _{p}-{\mathfrak {L}}_{\varvec{g}_{\alpha }}^{\text {an}}\cdot {}\text {ord}_{p}$. It then follows from Eq. (17) and Lemma 3.2 that

$$\begin{aligned} 2\cdot {}\beta _{\varvec{f}\varvec{g}_{\alpha }\varvec{h}_{\alpha }}^{-}(q_{\beta \beta })=\left( \log _{\varvec{f}}\cdot {}(\varvec{k}-2)-\log _{\varvec{g}_{\alpha }}\cdot {}(l-1)-\log _{\varvec{h}_{\alpha }}\cdot {}(\varvec{m}-1)\right) \otimes {}q_{\beta \beta } \end{aligned}$$

(18)

in $H^{1}(\mathbf{Q} _{p},V^{-})\otimes _{L}\mathscr {I}/\mathscr {I}^{2}$, where (with the notations introduced in Sect. 3.2.1) one writes $\beta ^{-}_{\varvec{f}\varvec{g}_{\alpha }\varvec{h}_{\alpha }}$ for the Bockstein map $\beta _{\varvec{{\mathcal {C}}}/{\mathcal {C}}}$ associated with $\varvec{{\mathcal {C}}}=\mathbf{R} \Gamma _{p}(\varvec{V}^{-})$. Note that

$$\begin{aligned} V(f)^{-}_{\beta \beta }=V(f)^{-}\otimes _\mathbf{Q _{p}}V(g)_{\beta }\otimes _{L}V(h)_{\beta } \end{aligned}$$

is an $L[G_\mathbf{Q _{p}}]$-direct summand of $V^{-}$ on which $G_\mathbf{Q _{p}}$ acts trivially, so that $\log _{\varvec{\xi }}\otimes {}q_{\beta \beta }$ (for $\varvec{\xi }=\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha }$) belongs to the direct summand

$$\begin{aligned} H^{1}(\mathbf{Q} _{p},V(f)^{-}_{\beta \beta })=H^{1}(\mathbf{Q} _{p},\mathbf{Q} _{p})\otimes _\mathbf{Q _{p}}V(f)^{-}_{\beta \beta } \end{aligned}$$

of the local cohomology group $H^{1}(\mathbf{Q} _{p},V^{-})$. Similarly

$$\begin{aligned} V(f)^{+}_{\alpha \alpha }=V(f)^{+}\otimes _\mathbf{Q _{p}}V(g)_{\alpha }\otimes _{L}V(h)_{\alpha } \end{aligned}$$

is an $L[G_\mathbf{Q _{p}}]$-direct summand of $V^{+}$ isomorphic to $\mathbf{Q} _{p}(1)$, hence

$$\begin{aligned} H^{1}(\mathbf{Q} _{p},V(f)^{+}_{\alpha \alpha })=H^{1}(\mathbf{Q} _{p},\mathbf{Q} _{p}(1))\otimes _\mathbf{Q _{p}}V(f)^{+}_{\alpha \alpha }(-1) \end{aligned}$$

(19)

is a direct summand of $H^{1}(\mathbf{Q} _{p},V^{+})$. The local Tate pairing $\left<{}\cdot ,\cdot \right>_{\text {Tate}}$ introduced in Sect. 3.2.2 induces a perfect duality (denoted by the same symbol) between $H^{1}(\mathbf{Q} _{p},V(f)^{-}_{\beta \beta })$ and $H^{1}(\mathbf{Q} _{p},V(f)^{+}_{\alpha \alpha })$, and identifying $H^{1}(\mathbf{Q} _{p},\mathbf{Z} _{p}(1))$ with the p-adic completion ${\hat{\mathbf{Q }}}_{p}^{*}$ of $\mathbf{Q} _{p}^{*}$ via the local Kummer map, local class field theory gives

$$\begin{aligned} \langle \varphi \otimes {}v^{-},u\otimes {}v^{+}\rangle _{\text {Tate}}=\varphi (u)\cdot {}\pi _{fgh}(-1)(v^{+}\otimes {}v^{-}) \end{aligned}$$

(20)

for each $\varphi$ in $H^{1}(\mathbf{Q} _{p},\mathbf{Q} _{p})$, u in $H^{1}(\mathbf{Q} _{p},\mathbf{Q} _{p}(1))$, $v^{-}$ in $V(f)^{-}_{\beta \beta }$ and $v^{+}$ in $V(f)^{+}_{\alpha \alpha }$. Here

$$\begin{aligned} \pi _{fgh}(-1) : V(f)^{+}_{\alpha \alpha }(-1)\otimes _{L}V(f)^{-}_{\beta \beta }{\mathop {\longrightarrow }\limits ^{}}L \end{aligned}$$

is the composition of $\pi _{fgh}\otimes {}\mathbf{Q} _{p}(-1)$ with the evaluation pairing $L(1)\otimes _{L}L(-1){\mathop {\longrightarrow }\limits ^{}}L$.

Recall that we identify $H^{0}(\mathbf{Q} _{p},V^{-})$ with a submodule of $\tilde{H}^{1}_{f}(\mathbf{Q} ,V)$ via the embedding $\jmath$ introduced in Diagram (13). Lemma 3.4 and Eqs. (18) and (20) give

(21)

for each z in $\tilde{H}^{1}_{f}(\mathbf{Q} ,V)$, where $\varvec{\xi }=\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha }$, $u_{o}=2,1,1$ is the centre of $U_{\varvec{\xi }}$, and

$$\begin{aligned} z_{\alpha \alpha }^{+}\in {}H^{1}(\mathbf{Q} _{p},\mathbf{Q} _{p}(1))={\hat{\mathbf{Q }}}_{p}^{*}\otimes _\mathbf{Z _{p}}\mathbf{Q} _{p} \end{aligned}$$

is defined as follows. Let $\text {pr}_{\alpha \alpha }$ denote the projection onto the direct summand $H^{1}(\mathbf{Q} _{p},V(f)^{+}_{\alpha \alpha })$ of the local cohomology group $H^{1}(\mathbf{Q} _{p},V^{+})$, and let $q_{\beta \beta }^{*}$ be the generator of $V(f)^{+}_{\alpha \alpha }(-1)$ dual to $q_{\beta \beta }$ under $\pi _{fgh}(-1)$, namely satisfying

$$\begin{aligned} \pi _{fgh}(-1)(q_{\beta \beta }^{*}\otimes {}q_{\beta \beta })=1. \end{aligned}$$

Then $z_{\alpha \alpha }^{+}$ is defined (via the natural isomorphism (19)) by the identity

$$\begin{aligned} \text {pr}_{\alpha \alpha }(z^{+})=z_{\alpha \alpha }^{+}\otimes {}q_{\beta \beta }^{*}. \end{aligned}$$

(22)

We now determine $z_{\alpha \alpha }^{+}$ for $z=\jmath (q_{\alpha \alpha })$. By definition $\jmath (q_{\alpha \alpha })$ is represented by

$$\begin{aligned} c_{\alpha \alpha }=(0,d\tilde{q}_{\alpha \alpha },\tilde{q}_{\alpha \alpha })\in {}\tilde{\text {C}}^{1}_{f}(\mathbf{Q} ,V), \end{aligned}$$

where $\tilde{q}_{\alpha \alpha }$ in V is a lift of $q_{\alpha \alpha }$ under the the projection $V{\mathop {\longrightarrow }\limits ^{}}V^{-}$, and where

$$\begin{aligned} d\tilde{q}_{\alpha \alpha } : G_\mathbf{Q _{p}}{\mathop {\longrightarrow }\limits ^{}}V^{+} \end{aligned}$$

is its image under the differential in $\mathbf{R} \Gamma _{\text {cont}}(\mathbf{Q} _{p},V)$. By construction $d\tilde{q}_{\alpha \alpha }$ represents the class $q_{\alpha \alpha }^{+}=\jmath (q_{\alpha \alpha })^{+}$ in $H^{1}(\mathbf{Q} _{p},V^{+})$. Since $V(\xi )$ is the direct sum of $V(\xi )_{\alpha }$ and $V(\xi )_{\beta }$ for $\xi =g,h$, we can (and will) choose $\tilde{q}_{\alpha \alpha }$ of the form

$$\begin{aligned} \tilde{q}_{\alpha \alpha }=\sqrt{m_{p}}\cdot {}\tilde{q}(f)\otimes \omega _{g_{\alpha }}\otimes \omega _{h_{\alpha }} \end{aligned}$$

for a lift $\tilde{q}(f)$ of q(f) under the projection $V(f){\mathop {\longrightarrow }\limits ^{}}V(f)^{-}$, so that $d\tilde{q}_{\alpha \alpha }$ represents the image of $q_{\alpha \alpha }$ under the connecting morphism

$$\begin{aligned} \delta _{\alpha \alpha } : V(f)^{-}_{\alpha \alpha }{\mathop {\longrightarrow }\limits ^{}}H^{1}(\mathbf{Q} _{p},V(f)^{+}_{\alpha \alpha }) \end{aligned}$$

arising from the short exact sequence of $G_\mathbf{Q _{p}}$-modules

$$\begin{aligned} 0{\mathop {\longrightarrow }\limits ^{}}V(f)^{+}_{\alpha \alpha }{\mathop {\longrightarrow }\limits ^{}}V(f)_{\alpha \alpha }{\mathop {\longrightarrow }\limits ^{}}V(f)^{-}_{\alpha \alpha }{\mathop {\longrightarrow }\limits ^{}}0, \end{aligned}$$

where $V(f)^{\cdot }_{\alpha \alpha }$ is the $L[G_\mathbf{Q _{p}}]$-direct summand $V(f)^{\cdot }\otimes _\mathbf{Q _{p}}V(g)_{\alpha }\otimes _{L}V(h)_{\alpha }$ of $V^{\cdot }$. Let $q_{A}$ in $p\mathbf{Z} _{p}$ be the Tate period of $A_\mathbf{Q _{p}}$. Tate’s theory gives a rigid analytic isomorphisms between the base change $E_\mathbf{Q _{p}^{2}}$ of the Tate curve $E=\mathbf{G} _{m,\mathbf{Q} _{p}}^{rig}/q_{A}^\mathbf{Z }$ to the quadratic unramified extension $\mathbf{Q} _{p^{2}}$ of $\mathbf{Q} _{p}$ and $A_\mathbf{Q _{p^{2}}}$. Set $V_{p}(E)=H_\mathrm{{\acute{e}t}}^{1}(E_{{\bar{\mathbf{Q }}}_{p}},\mathbf{Q} _{p}(1))$ and let $\wp _{\text {Tate}} : V_{p}(E)\simeq {}V_{p}(A)$ be the isomorphisms of $G_\mathbf{Q _{p^{2}}}$-modules induced by the Tate uniformisation. There is a short exact sequence of $\mathbf{Q} _{p}[G_\mathbf{Q _{p}}]$-modules

$$\begin{aligned} 0{\mathop {\longrightarrow }\limits ^{}}\mathbf{Q} _{p}(1){\mathop {\longrightarrow }\limits ^{a\,}}V_{p}(E){\mathop {\longrightarrow }\limits ^{b\,}}\mathbf{Q} _{p}{\mathop {\longrightarrow }\limits ^{}}0, \end{aligned}$$

(23)

where $a(\zeta _{p^{\infty }})=(\zeta _{p^{n}}\cdot {}q_{A}^\mathbf{Z })_{n\geqslant {}1}$ for each compatible system $\zeta _{p^{\infty }}=(\zeta _{p^{n}})_{n{}\geqslant {}1}$ of $p^{n}$-th roots of unity, and b is the $\mathbf{Q} _{p}$-linear extension of the inverse limit of (canonical) maps

$$\begin{aligned} b_{n} : E({\bar{\mathbf{Q }}}_{p})_{p^{n}}=({\bar{\mathbf{Q }}}_{p}^{*}/q_{A}^\mathbf{Z })_{p^{n}}{\mathop {\longrightarrow }\limits ^{}}\mathbf{Z} /p^{n}\mathbf{Z} \end{aligned}$$

defined by $b_{n}(x\cdot {}q_{A}^\mathbf{Z })=\frac{p^{n}\cdot {}\text {ord}_{p}(x)}{\text {ord}_{p}(q_{A})}+p^{n}\cdot {}\mathbf{Z}$. By definition $q(A)=\wp _{\text {Tate}}^{-}(1)$, where $\wp _{\text {Tate}}^{-}\circ {}b$ is the composition of $\wp _{\text {Tate}}$ and the projection $V_{p}(A){\mathop {\longrightarrow }\limits ^{}}V_{p}(A)^{-}$ onto the maximal $G_\mathbf{Q _{p}}$-unramified quotient, and

$$\begin{aligned} \tilde{q}(f)=\wp _{\infty }^{-1}\circ {}\wp _{\text {Tate}}(\!\root p^{\infty } \of {q_{A}}\,) \end{aligned}$$

is the image of a compatible system $\root p^{\infty } \of {q_{A}}$ of $p^{n}$-th roots of the Tate period $q_{A}$ under the composition of $\wp _{\text {Tate}}$ and the inverse of the isomorphism $\wp _{\infty } : V(f)\simeq {}V_{p}(A)$ induced by the fixed modular parametrisation $\wp _{\infty } : X_{1}(N_{f}){\mathop {\longrightarrow }\limits ^{}}A$. As a consequence 1 in $\mathbf{Q} _{p}$ maps to $q_{A}{\hat{\otimes }}1$ under the connecting map $\mathbf{Q} _{p}{\mathop {\longrightarrow }\limits ^{}}H^{1}(\mathbf{Q} _{p},\mathbf{Q} _{p}(1))=\mathbf{Q} _{p}^{*}{\hat{\otimes }}\mathbf{Q} _{p}$ associated with the short exact sequence (23), hence

$$\begin{aligned} \jmath (q_{\alpha \alpha })^{+}=cl(d\tilde{q}_{\alpha \alpha })=\delta _{\alpha \alpha }(q_{\alpha \alpha })=\sqrt{m_{p}}\cdot {} (\wp _{\infty *}^{-1}\circ {}\wp _{\text {Tate}})_{*}^{+}(q_{A}{\hat{\otimes }}1)\otimes {} \omega _{g_{\alpha }}\otimes \omega _{h_{\alpha }} \end{aligned}$$

(24)

in

$$\begin{aligned} H^{1}(\mathbf{Q} _{p},V(f)^{+}_{\alpha \alpha })=H^{0}\big (\text {Gal}(\mathbf{Q} _{p^{2}}/\mathbf{Q} ),H^{1}(\mathbf{Q} _{p^{2}},V(f)^{+})\otimes _\mathbf{Q _{p}}V(g)_{\alpha }\otimes _{L}V(h)_{\alpha }\big ), \end{aligned}$$

where

$$\begin{aligned} (\wp _{\infty }^{-1}\circ {}\wp _{\text {Tate}})_{*}^{+} : \mathbf{Q} _{p^{2}}^{*}{\hat{\otimes }}\mathbf{Q} _{p}\simeq {}H^{1}(\mathbf{Q} _{p^{2}},V(f)^{+}) \end{aligned}$$

is the map induced in cohomology by the composition of $\wp _{\infty }^{-1}$ and

$$\begin{aligned} \wp _{\text {Tate}}^{+}=\wp _{\text {Tate}}\circ {}a. \end{aligned}$$

If ${\mathcal {A}}$ denotes either A or E, denote by

$$\begin{aligned} \pi _{{\mathcal {A}}} : V_{p}({\mathcal {A}})(-1)\otimes _\mathbf{Q _{p}}V_{p}({\mathcal {A}}){\mathop {\longrightarrow }\limits ^{}}\mathbf{Q} _{p} \end{aligned}$$

the composition of the evaluation pairing $\mathbf{Q} _{p}(1)\otimes _\mathbf{Q _{p}}\mathbf{Q} _{p}(-1){\mathop {\longrightarrow }\limits ^{}}\mathbf{Q} _{p}$ with the base change of the Weil pairing on $V_{p}({\mathcal {A}})$ by $\mathbf{Q} _{p}(-1)$. Set

$$\begin{aligned} q(A)^{*}=\wp _{\text {Tate}}^{+}(\zeta _{p^{\infty }})\otimes {}\zeta _{p^{\infty }}^{*}\in {}V_{p}(A)^{+}(-1), \end{aligned}$$

where $\zeta _{p^{\infty }}$ is a generator of $\mathbf{Q} _{p}(1)$ and $\zeta _{p^{\infty }}^{*}$ in $\mathbf{Q} _{p}(-1)$ is its dual basis, and set

$$\begin{aligned} q(f)^{*}=\deg (\wp _{\infty })\cdot {}\wp _{\infty }^{-1}(q(A)^{*})\in {}V(f)^{+}(-1). \end{aligned}$$

As $\pi _{E}((a(y)\otimes {}z)\otimes {}x)=b(x)\cdot {}z(y)$ for each x in $V_{p}(E)$, y in $\mathbf{Q} _{p}(1)$ and z in $\mathbf{Q} _{p}(-1)$, the functoriality of the Poincaré duality under finite morphisms yields

$$\begin{aligned} \pi _{f}(q(f)^{*}\otimes {}q(f)) =\pi _{A}(q(A)^{*}\otimes {}q(A))=\pi _{E}\big ((a(\zeta _{p^{\infty }})\otimes {}\zeta _{p^{\infty }}^{*})\otimes {}\!\!\root p^{\infty } \of {q_{A}}\,\big )=1, \end{aligned}$$

then (by the definition of the weight-one differentials $\eta _{\xi _{\alpha }}$, cf. Sect. 3.1.1)

$$\begin{aligned} q_{\beta \beta }^{*}=\frac{1}{\sqrt{m_{p}}}\cdot {}q(f)^{*}\otimes {}\omega _{g_{\alpha }}\otimes \omega _{h_{\alpha }}. \end{aligned}$$

Together with Eq. (24) this gives

$$\begin{aligned} \jmath (q_{\alpha \alpha })^{+}=\frac{m_{p}}{\deg (\wp _{\infty })}\cdot {}(q_{A}{\hat{\otimes }}1)\otimes {}q_{\beta \beta }^{*}, \end{aligned}$$

(25)

id est

$$\begin{aligned} \jmath (q_{\alpha \alpha })^{+}_{\alpha \alpha }=\frac{m_{p}}{\deg (\wp _{\infty })}\cdot {}q_{A}{\hat{\otimes }}1. \end{aligned}$$

(26)

According to Theorem 3.18 of [9] ${\mathfrak {L}}_{\varvec{f}}^{\text {an}}=\frac{\log _{p}(q_{A})}{\text {ord}_{p}(q_{A})}$, so that

(27)

by Eqs. (21) and (26).

4.4 Computation of

Assume in this subsection $\alpha _{f}=\beta _{g}\cdot {}\alpha _{h}$, so that $H^{0}(\mathbf{Q} _{p},V^{-})$ is generated by the p-adic periods

$$\begin{aligned} q_{\alpha \beta }=\sqrt{m_{p}}\cdot {}q(f)\otimes \omega _{g_{\alpha }}\otimes \eta _{h_{\alpha }}\ \ \ \text {and}\ \ \ q_{\beta \alpha }=\sqrt{m_{p}}\cdot {}q(f)\otimes \eta _{g_{\alpha }}\otimes \omega _{h_{\alpha }}. \end{aligned}$$

For $\gamma \delta =\alpha \beta ,\beta \alpha$ and $\cdot =\emptyset ,\pm$, define $V(f)^{\cdot }_{\gamma \delta }=V(f)^{\cdot }\otimes _\mathbf{Q _{p}}V(g)_{\gamma } \otimes {}V(h)_{\delta }$. Then

$$\begin{aligned} H^{0}(\mathbf{Q} _{p},V^{-})=V(f)^{-}_{\alpha \beta }\oplus {}V(f)^{-}_{\beta \alpha }, \end{aligned}$$

$G_\mathbf{Q _{p}}$ acts on $V(f)^{+}_{\alpha \beta }$ and $V(f)^{+}_{\beta \alpha }$ via the p-adic cyclotomic character, and the local Tate pairing $\left<{}\cdot ,\cdot \right>_{\text {Tate}}$ introduced in Sect. 3.2.2 induces a perfect duality (denoted by the same symbol) between $H^{1}(\mathbf{Q} _{p},V(f)^{-}_{\alpha \beta })$ and $H^{1}(\mathbf{Q} _{p},V(f)^{+}_{\beta \alpha })$. The argument of the proof of Eq. (25) shows that

$$\begin{aligned} \jmath (q_{\beta \alpha })^{+}=\frac{m_{p}}{\deg (\wp _{\infty })}\cdot {}(q_{A}{\hat{\otimes }}1)\otimes {}q_{\alpha \beta }^{*} \end{aligned}$$

(28)

in the direct summand $H^{1}(\mathbf{Q} _{p},V(f)^{+}_{\beta \alpha })=\mathbf{Q} _{p}^{*}{\hat{\otimes }}V(f)^{+}_{\beta \alpha }(-1)$ of $H^{1}(\mathbf{Q} _{p},V^{+})$, where

$$\begin{aligned} q_{\alpha \beta }^{*}=\frac{1}{\sqrt{m_{p}}}\cdot {}q(f)^{*}\otimes \eta _{g_{\alpha }}\otimes \omega _{h_{\alpha }}\ \ \text {satisfies}\ \ \pi _{fgh}(-1)(q_{\alpha \beta }^{*}\otimes {}q_{\alpha \beta })=1. \end{aligned}$$

(29)

Let $\text {pr}_{\alpha \beta } : H^{1}(\mathbf{Q} _{p},V^{-}){\mathop {\longrightarrow }\limits ^{}}H^{1}(\mathbf{Q} _{p},\mathbf{Q} _{p})\otimes _\mathbf{Q _{p}}V(f)^{-}_{\alpha \beta }$ denote the projection, and write

$$\begin{aligned} \text {pr}_{\alpha \beta }\otimes {}\mathscr {I}/\mathscr {I}^{2}\circ {}\beta ^{-}_{\varvec{f}\varvec{g}_{\alpha }\varvec{h}_{\alpha }}(q_{\alpha \beta })= \sum _{\varvec{u}}\gamma _{\varvec{u}}\otimes {}q_{\alpha \beta }\cdot {}(\varvec{u}-u_{o}) \end{aligned}$$

(30)

with $\gamma _{\varvec{u}}$ in $H^{1}(\mathbf{Q} _{p},\mathbf{Q} _{p})=\text {Hom}_{}(\mathbf{Q} _{p}^{*},\mathbf{Q} _{p})$ for $\varvec{u}=\varvec{k},l,\varvec{m}$, where (with the notations introduced in Sect. 3.2.1) $\beta _{\varvec{f}\varvec{g}_{\alpha }\varvec{h}_{\alpha }}^{-}$ is a shorthand for

$$\begin{aligned} \beta _\mathbf{R \Gamma _{\text {cont}}(\mathbf{Q} _{p},\varvec{V}^{-})/\mathbf{R} \Gamma _{\text {cont}}(\mathbf{Q} _{p},V^{-})} : H^{0}(\mathbf{Q} _{p},V^{-}){\mathop {\longrightarrow }\limits ^{}}H^{1}(\mathbf{Q} _{p},V^{-})\otimes _{L}\mathscr {I}/\mathscr {I}^{2}, \end{aligned}$$

and $u_{o}=2$ if $\varvec{u}=\varvec{k}$ and $u_{o}=1$ if $\varvec{u}=l,\varvec{m}$. Then (cf. Eq. (21))

(31)

where the last equality follows from Eq. (29) and the analogue of Eq. (20) obtained by replacing $\alpha \alpha$ and $\beta \beta$ with $\beta \alpha$ and $\alpha \beta$ respectively. It then remains to compute $\gamma _{\varvec{u}}$ for $\varvec{u}$ equal to $\varvec{k}$, $l$ and $\varvec{m}$.

For $\varvec{\xi }=\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha }$, fix $\mathscr {O}_{\varvec{\xi }}$-bases $b_{\varvec{\xi }}^{\pm }$ of $V(\varvec{\xi })^{\pm }$. After identifying $V(\varvec{\xi })$ with $\mathscr {O}_{\varvec{\xi }}\oplus {}\mathscr {O}_{\varvec{\xi }}$ via the $\mathscr {O}_{\varvec{\xi }}$-basis $(b_{\varvec{\xi }}^{+},b_{\varvec{\xi }}^{-})$, the action of $G_\mathbf{Q _{p}}$ on $V(\varvec{\xi })$ is given by (cf. Eq. (9))

for a continuous map $c_{\varvec{\xi }} : G_\mathbf{Q _{p}}{\mathop {\longrightarrow }\limits ^{}}\mathscr {O}_{\varvec{\xi }}$. Without loss of generality, assume that

$$\begin{aligned} \varvec{q}_{\alpha \beta }=b_{\varvec{f}}^{-}{\hat{\otimes }}b_{\varvec{g}_{\alpha }}^{-}{\hat{\otimes }}b_{\varvec{h}_{\alpha }}^{+}\otimes {}1 \end{aligned}$$

in $\varvec{V}^{-}=V(\varvec{f})^{-}{\hat{\otimes }}_{L}V(\varvec{g}_{\alpha }){\hat{\otimes }}_{L}V(\varvec{h}_{\alpha })\otimes _{\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}}\Xi _{\varvec{f}\varvec{g}\varvec{h}}$ maps to

$$\begin{aligned} q_{\alpha \beta }\in {}V(f)^{-}_{\alpha \beta }=V(f)^{-}\otimes _\mathbf{Q _{p}}V(g)_{\alpha }\otimes _{L}V(h)_{\beta } \end{aligned}$$

under $\rho _{w} : \varvec{V}^{-}{\mathop {\longrightarrow }\limits ^{}}V^{-}$. (Recall that $V(\xi )=V(\varvec{\xi }_{\alpha })\otimes _{1}L$ is the direct sum of the modules $V(\xi )_{\alpha }=V(\varvec{\xi }_{\alpha })^{-}\otimes _{1}L$ and $V(\xi )_{\beta }=V(\varvec{\xi }_{\alpha })^{+}\otimes _{1}L$ for $\xi =g,h$, cf. Sect. 3.1.1.) Then

$$\begin{aligned} d\varvec{q}_{\alpha \beta }=\Gamma \cdot {}\varvec{q}_{\alpha \beta }+\Delta \cdot {}\varvec{q}_{\beta \beta }, \end{aligned}$$

(32)

where $\varvec{q}_{\beta \beta }=b_{\varvec{f}}^{-}{\hat{\otimes }}b_{\varvec{g}_{\alpha }}^{+}{\hat{\otimes }}b_{\varvec{h}_{\alpha }}^{+}\otimes {}1$, where

$$\begin{aligned} \Gamma =\frac{{\check{a}}_{p}(\varvec{f})\cdot {}{\check{a}}_{p}(\varvec{g}_{\alpha })}{{\check{a}}_{p}(\varvec{h}_{\alpha })}\cdot {}\chi _{h}\cdot {}\chi _{\text {cyc}}^{(\varvec{m}-\varvec{k}-l+2)/2}-1 \end{aligned}$$

and where

$$\begin{aligned} \Delta ={\check{a}}_{p}(\varvec{f})\cdot {}{\check{a}}_{p}(\varvec{h}_{\alpha })^{-1}\cdot {}\chi _{h}\cdot {}\chi _{\text {cyc}}^{(\varvec{m}-\varvec{k}-l+2)/2}\cdot {}c_{\varvec{g}_{\alpha }}. \end{aligned}$$

The exceptional zero condition $\alpha _{f}=\beta _{g}\cdot {}\alpha _{h}$ and the self duality condition $\chi _{g}\cdot {}\chi _{h}=1$ imply that $\Gamma$ takes values in $\mathscr {I}$. Moreover, since the $G_\mathbf{Q _{p}}$-module $V(g)=V(\varvec{g}_{\alpha })\otimes _{1}L$ splits as the direct sum of $V(g)_{\beta }=V(\varvec{g}_{\alpha })^{+}\otimes _{1}L$ and $V(g)_{\alpha }=V(\varvec{g}_{\alpha })^{-}\otimes _{1}L$, the map $c_{\varvec{g}_{\alpha }}$ takes values in $(l-1)\cdot {}\mathscr {O}_{\varvec{g}}$, hence $\Delta$ takes values in $\mathscr {I}$. Because by construction $\varvec{q}_{\beta \beta }$ maps to an element of $V(f)^{-}_{\beta \beta }$ under the specialisation map $\rho _{w_{o}} : \varvec{V}^{-}{\mathop {\longrightarrow }\limits ^{}}V^{-}$, Lemma 3.2 and Eqs. (30) and (32) yield the identities

$$\begin{aligned} \gamma _{\varvec{u}}=-\frac{\partial }{\partial \varvec{u}}\Gamma (\cdot )(w_{o}), \end{aligned}$$

hence (as in the previous subsection) a direct computation gives

$$\begin{aligned} \gamma _{\varvec{k}}=\frac{1}{2}\cdot {}\log _{\varvec{f}},\ \ \gamma _{l}=\frac{1}{2}\cdot {}\log _{\varvec{g}_{\alpha }}\ \ \text {and} \ \ \gamma _{\varvec{m}}=-\frac{1}{2}\cdot {}\log _{\varvec{h}_{\alpha }}. \end{aligned}$$

(33)

Recalling that $\log _{\varvec{f}}(q_{A})=0$ by [9, Theorem 3.18], Eq. (31) finally proves

(34)

4.5 Proof of equation (8)

Assume in this subsection that $(A,\varrho )$ is exceptional at p, and fix a Selmer class x in $\text {Sel}(\mathbf{Q} ,V(f,g,h))$. Let

$$\begin{aligned} \tilde{x}=\imath _{\text {ur}}(x)\in {}\tilde{H}^{1}_{f}(\mathbf{Q} ,V(f,g,h)) \end{aligned}$$

be the corresponding extended Selmer class (cf. Sect. 3.1.2). By construction $\tilde{x}^{+}$ belongs to the finite subspace of $H^{1}(\mathbf{Q} _{p},V^{+})$, and its image under the natural map $i^{+} : H^{1}_{\text {fin}}(\mathbf{Q} _{p},V^{+}){\mathop {\longrightarrow }\limits ^{}}H^{1}_{\text {fin}}(\mathbf{Q} _{p},V)$ equals the restriction of x at p:

$$\begin{aligned} \text {res}_{p}(x)=i^{+}(\tilde{x}^{+}). \end{aligned}$$

(35)

The Galois group $G_\mathbf{Q _{p}}$ acts on $V(f)^{+}_{\natural }$ via the p-adic cyclotomic character, hence

$$\begin{aligned} H^{1}_{\text {fin}}(\mathbf{Q} _{p},V(f)^{+}_{\natural })=\mathbf{Z} _{p}^{*}\otimes _\mathbf{Z _{p}}V(f)^{+}_{\natural }(-1) \end{aligned}$$

by Kummer theory. If $q_{\flat }^{*}$ in $V(f)^{+}_{\natural }$ denotes (as in the previous subsections) the dual basis of $q_{\flat }$ in $V(f)_{\natural }^{-}$ under the pairing $\pi _{fgh}$, and if one writes

$$\begin{aligned} \text {pr}_{\natural }(\tilde{x}^{+})=\tilde{x}^{+}_{\natural }\otimes {}q_{\flat }^{*}\in {}H^{1}_{\text {fin}}(\mathbf{Q} _{p},V(f)^{+}_{\natural }) \end{aligned}$$

for some $\tilde{x}^{+}_{\natural }$ in $\mathbf{Z} _{p}^{*}\otimes _\mathbf{Z _{p}}L$, then Eq. (35) yields the equality

$$\begin{aligned} \log _{\natural }(\text {res}_{p}(x))=\langle \log _{p}^{+}(\tilde{x}^{+}),q_{\flat }\rangle _{fgh}= \langle \log _{p}(\tilde{x}^{+}_{\natural })\otimes {}q_{\flat }^{*},q_{\flat }\rangle _{fgh}=\log _{p}(\tilde{x}^{+}_{\natural }), \end{aligned}$$

(36)

where $\log _{p}^{+} : H^{1}_{\text {fin}}(\mathbf{Q} _{p},V^{+})\simeq {}D_{\text {dR}}(V^{+})$ is the Bloch–Kato logarithm and (with a slight abuse of notation) we denote again by $\log _{p} : \mathbf{Z} _{p}^{*}\otimes _\mathbf{Z _{p}}L{\mathop {\longrightarrow }\limits ^{}}L$ the L-linear extension of the p-adic logarithm. In the previous equation we used the functoriality of the Bloch–Kato logarithm and the fact that (by construction) the linear form $\left<{}\cdot ,q_{\flat }\right>_{fgh}$ on $D_{\text {dR}}(V^{+})$ factors through the projection onto $D_{\text {dR}}(V(f)^{+}_{\natural })=V(f)^{+}_{\natural }(-1)$.

Assume ($\alpha _{f}=\alpha _{g}\cdot {}\alpha _{h}$ and) $q_{\flat }=q_{\beta \beta }$. According to Eqs. (21) and (36)

(37)

thus proving Eq. (8) in this case.

Assume $q_{\flat }=q_{\alpha \beta }$. Since (with the notations of Section 3.4) $\Delta$ takes values in $(l-1)\cdot {}\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}$, it follows from Lemma 3.2 and Eqs. (32) and (33) that

$$\begin{aligned} 2\cdot {}\beta _{\varvec{f}\varvec{g}_{\alpha }\varvec{h}_{\alpha }}^{-}(q_{\alpha \beta })=\sum _{\varvec{\xi }}\varepsilon _{\varvec{\xi }}\cdot {}\log _{\varvec{\xi }}\otimes {}q_{\alpha \beta }\cdot {}(\varvec{u}-u_{o})+\vartheta \cdot {}(l-1) \end{aligned}$$

(38)

for some cohomology class $\vartheta$ in $H^{1}(\mathbf{Q} _{p},V(f)^{-}_{\beta \beta })$, where $\varepsilon _{\varvec{h}_{\alpha }}=-1$ and $\varepsilon _{\varvec{\xi }}=+1$ for $\varvec{\xi }=\varvec{f},\varvec{g}_{\alpha }$. One has then

(39)

thus proving Eq. (8) when $q_{\flat }=q_{\alpha \beta }$. Switching the roles of the Hida families $\varvec{g}_{\alpha }$ and $\varvec{h}_{\alpha }$, this also proves Eq. (8) when $q_{\flat }=q_{\beta \alpha }$.

Assume finally $q_{\flat }=q_{\alpha \alpha }$. With the notations of Sect. 3.4, let $(b_{\varvec{\xi }}^{+},b_{\varvec{\xi }}^{-})$ be $\mathscr {O}_{\varvec{\xi }}$-bases of $V(\varvec{\xi })$ such that $\varvec{q}_{\alpha \alpha }=b_{\varvec{f}}^{-}{\hat{\otimes }}b_{\varvec{g}_{\alpha }}^{-}{\hat{\otimes }}b_{\varvec{h}_{\alpha }}^{-}\otimes {}1$ is a lift of $q_{\alpha \alpha }$ under the specialisation map $\rho _{w_{o}} : \varvec{V}^{-}{\mathop {\longrightarrow }\limits ^{}}V^{-}$. Since $c_{\varvec{\xi }}$ takes values in $(\varvec{u}-u_{o})\cdot {}\mathscr {O}_{\varvec{\xi }}$ for $\varvec{\xi }=\varvec{g}_{\alpha },\varvec{h}_{\alpha }$, one has

$$\begin{aligned} d\varvec{q}_{\alpha \alpha }\equiv {} \Big (\chi _{\text {cyc}}^{(4-\varvec{k}-l-\varvec{m})/2}\cdot {}\prod _{\varvec{\xi }}{\check{a}}_{p}(\varvec{\xi })-1\Big ) \cdot {}\varvec{q}_{\alpha \alpha }\ \big (\text {mod}\ (l-1,\varvec{m}-1)\cdot {}\text {C}_{\text {cont}}^{1}(\mathbf{Q} _{p},\varvec{V}^{-})\big ), \end{aligned}$$

hence Lemma 3.2 and a direct computation give

$$\begin{aligned} 2\cdot {}\beta ^{-}_{\varvec{f}\varvec{g}_{\alpha }\varvec{h}_{\alpha }}(q_{\alpha \alpha })= \log _{\varvec{f}}\otimes {}q_{\alpha \alpha }\cdot {}(\varvec{k}-2)+\vartheta \cdot {}(l-1)+\vartheta ^{\prime }\cdot {}(\varvec{m}-1) \end{aligned}$$

(40)

for some local cohomology classes $\vartheta$ and $\vartheta ^{\prime }$ in $H^{1}(\mathbf{Q} _{p},V^{-})$. As in (39) one deduces Eq. (8) for $q_{\flat }=q_{\alpha \alpha }$ from Lemma 3.4 and Eqs. (36) and (40).

5 Proof of theorem 2.1

Let ${\Pi }_{f}$, ${\Pi }_{g}$ and ${\Pi }_{h}$ be the improving planes in $U_{\!\varvec{f}}\times {}U_{\varvec{g}}\times {}U_{\varvec{h}}$ defined respectively by the equations $\varvec{k}=l+\varvec{m}$, $\varvec{k}=l-\varvec{m}+2$ and $\varvec{k}=\varvec{m}-l+2$. For $\xi =f,g,h$ define

$$\begin{aligned} {\mathcal {E}}_{\xi }=1-{\bar{\chi }}_{\xi }(p)\cdot {}\frac{a_{p}(\varvec{\xi })}{a_{p}(\varvec{\xi }^{\prime })\cdot {}a_{p}(\varvec{\xi }^{\prime \!\prime })} \end{aligned}$$

in $\mathscr {O}_{\!\varvec{f}\varvec{g}\varvec{h}}$, where $\{\varvec{\xi },\varvec{\xi }^{\prime },\varvec{\xi }^{\prime \!\prime }\}=\{\varvec{f},\varvec{g}_{\alpha },\varvec{h}_{\alpha }\}$. Lemma 9.8 of [7] implies that

$$\begin{aligned} \mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )|_{{\Pi }_{\xi }}={\mathcal {E}}_{\xi }|_{{\Pi }_{\xi }}\cdot {}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )_{\xi }^{\star } \end{aligned}$$

(41)

for an improved p-adic L-function $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )_{\xi }^{\star }$ in ${\mathcal {O}}({\Pi }_{\xi })$. Indeed loc. cit. (together with its analogue obtained by switching the roles of g and h) proves that the meromorphic function $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )_{\xi }^{\star }$ on ${\Pi }_{\xi }$ defined by the previous equation is (bounded, hence) regular at $w_{o}$. Shrinking the discs $U_{\varvec{\xi }}$ if necessary, we then conclude that the improved p-adic L-function $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )_{\xi }^{\star }$ is analytic on ${\Pi }_{\xi }$, as claimed.

Assume first $\alpha _{f}=\alpha _{g}\cdot {}\alpha _{h}$, so that

$$\begin{aligned} 2\cdot {}{\mathcal {E}}_{f}\!\!\!\!\pmod {\mathscr {I}^{2}}={\mathfrak {L}}_{\varvec{f}}^{\text {an}}\cdot {}(\varvec{k}-2)-{\mathfrak {L}}_{\varvec{g}_{\alpha }}^{\text {an}}\cdot {}(l-1) -{\mathfrak {L}}_{\varvec{h}_{\alpha }}^{\text {an}}\cdot {}(\varvec{m}-1). \end{aligned}$$

(42)

According to Theorem A and Proposition 9.3 of [7], the partial derivative of $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )$ with respect to $\varvec{k}$ vanishes at $w_{o}$, hence

$$\begin{aligned} 2\cdot {}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )\!\!\!\pmod {\mathscr {I}^{2}} \end{aligned}$$

is equal to

$$\begin{aligned} \big (({\mathfrak {L}}_{\varvec{f}}^{\text {an}} -{\mathfrak {L}}^{\text {an}}_{g_{\alpha }})\cdot {}(l-1)+ ({\mathfrak {L}}^{\text {an}}_{\varvec{f}}-{\mathfrak {L}}^{\text {an}}_{\varvec{h}_{\alpha }})\cdot {}(\varvec{m}-1)\big )\cdot {}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )_{f}^{\star }(w_{o}) \end{aligned}$$

by Eqs. (41) and (42). Moreover, with the notations introduced before the statement of Theorem 2.1, one has $\mathsf {L}={\Pi }_{f}\cap {}{\Pi }_{g}$ and $\mathscr {E}_{f}={\mathcal {E}}_{f}|_{\mathsf {L}}$, thus

$$\begin{aligned} \mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )_{f}^{\star }(w_{o})=\mathscr {E}_{g}(w_{o})\cdot {}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )^{\star }(w_{o}). \end{aligned}$$

Noting that $\mathscr {E}_{g}(w_{o})=1-\beta _{h}/\alpha _{h}$ (when $\alpha _{f}=\alpha _{g}\cdot {}\alpha _{h}$), the previous discussion and Eq. (27) conclude the proof of Theorem 2.1 when $\alpha _{f}=\alpha _{g}\cdot {}\alpha _{h}$.

Assume now $\alpha _{f}=\beta _{g}\cdot {}\alpha _{h}$. In this case, for $\xi =g,h$, one has

$$\begin{aligned} 2\cdot {}{\mathcal {E}}_{\xi }\!\!\!\pmod {\mathscr {I}^{2}}={\mathfrak {L}}^{\text {an}}_{\varvec{\xi }_{\alpha }}\cdot {}(\varvec{u}-1)-{\mathfrak {L}}^{\text {an}}_{\varvec{f}}\cdot {}(\varvec{k}-2) -{\mathfrak {L}}^{\text {an}}_{\varvec{\xi }^{\prime }_{\alpha }}\cdot {}(\varvec{u}^{\prime }-1), \end{aligned}$$

(43)

where $\{(\varvec{\xi }_{\alpha },\varvec{u}),(\varvec{\xi }_{\alpha }^{\prime },\varvec{u}^{\prime })\}=\{(\varvec{g}_{\alpha },l),(\varvec{h}_{\alpha },\varvec{m})\}$, and

$$\begin{aligned} -\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )_{h}^{\star }(w_{o})=\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )_{g}^{\star }(w_{o}) =\mathscr {E}_{f}(w_{o})\cdot {}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )^{\star }(w_{o}). \end{aligned}$$

(44)

The second equality in the previous equation follows as above from the definitions, according to which $\mathsf {L}={\Pi }_{f}\cap {}{\Pi }_{g}$ and $\mathscr {E}_{g}={\mathcal {E}}_{g}|_{\mathsf {L}}$. The first equality follows by noting that the restrictions of ${\mathcal {E}}_{g}$ and ${\mathcal {E}}_{h}$ to the line ${\Pi }_{g}\cap {}{\Pi }_{h}$ satisfy

$$\begin{aligned} {\mathcal {E}}_{g}|_{{\Pi }_{g}\cap {}{\Pi }_{h}}=-\left. \frac{{\bar{\chi }}_{g}(p)\cdot {}a_{p}(\varvec{g}_{\alpha })}{a_{p}(\varvec{f})\cdot {}a_{p}(\varvec{h}_{\alpha })}\right| _{{\Pi }_{g}\cap {}{\Pi }_{h}}\cdot {} {\mathcal {E}}_{h}|_{{\Pi }_{g}\cap {}{\Pi }_{h}} \end{aligned}$$

(as $a_{p}(\varvec{f})|_{{\Pi }_{g}\cap {}{\Pi }_{h}}=\alpha _{f}=\alpha _{f}^{-1}$ and $\chi _{g}\cdot {}\chi _{h}=1$ by Assumption 1.1.1) with

$$\begin{aligned} -\frac{{\bar{\chi }}_{g}(p)\cdot {}a_{p}(\varvec{g}_{\alpha })}{a_{p}(\varvec{f})\cdot {}a_{p}(\varvec{h}_{\alpha })}(w_{o})=-1. \end{aligned}$$

(In other words ${\mathcal {E}}_{g}|_{{\Pi }_{g}\cap {}{\Pi }_{h}}$ and $-{\mathcal {E}}_{h}|_{{\Pi }_{g}\cap {}{\Pi }_{h}}$ have the same leading term at $w_{o}$, which together with the equality ${\mathcal {E}}_{g}\cdot {}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )_{g}^{\star }|_{{\Pi }_{g}\cap {}{\Pi }_{h}}= {\mathcal {E}}_{h}\cdot {}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )_{h}^{\star }|_{{\Pi }_{g}\cap {}{\Pi }_{h}}$ implies the first identity in Eq. (44).) Write

$$\begin{aligned} 2\cdot {}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )\!\!\!\pmod {\mathscr {I}^{2}}=a\cdot {}(\varvec{k}-2)+b\cdot {}(l-1)+c\cdot {}(\varvec{m}-1) \end{aligned}$$

with a, b and c in L. Equations (41) and (43) with $\xi =g$ and Eq. (44) give

$$\begin{aligned} a+b=\mathscr {E}_{f}(w_{o})\cdot {}\big ({\mathfrak {L}}_{\varvec{g}_{\alpha }}^{\text {an}}-{\mathfrak {L}}_{\varvec{f}}^{\text {an}}\big )\cdot {} \mathscr {L}_{p}^{\star }(w_{o})\ \ \ \text {and}\ \ \ c-a=\mathscr {E}_{f}(w_{o})\cdot {}\big ({\mathfrak {L}}_{\varvec{f}}^{\text {an}}-{\mathfrak {L}}_{\varvec{h}_{\alpha }}^{\text {an}}\big )\cdot {}\mathscr {L}_{p}^{\star }(w_{o}), \end{aligned}$$

where $\mathscr {L}_{p}^{\star }$ is a shorthand for $\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )^{\star }$. Similarly

$$\begin{aligned} b-a=\mathscr {E}_{f}(w_{o})\cdot {}\big ({\mathfrak {L}}_{\varvec{g}_{\alpha }}^{\text {an}}-{\mathfrak {L}}_{\varvec{f}}^{\text {an}}\big )\cdot {}\mathscr {L}_{p}^{\star }(w_{o}) \ \ \text {and}\ \ a+c=\mathscr {E}_{f}(w_{o})\cdot {}\big ({\mathfrak {L}}_{\varvec{f}}^{\text {an}}-{\mathfrak {L}}_{\varvec{h}_{\alpha }}^{\text {an}}\big )\cdot {}\mathscr {L}_{p}^{\star }(w_{o}) \end{aligned}$$

by Eqs. (41) and (43) with $\xi =h$ and Eq. (44). As a consequence

$$\begin{aligned} -2\cdot {}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )\!\!\!\pmod {\mathscr {I}^{2}} \end{aligned}$$

equals

$$\begin{aligned} \mathscr {E}_{f}(w_{o})\cdot {}\big ( ({\mathfrak {L}}_{\varvec{f}}^{\text {an}}-{\mathfrak {L}}_{\varvec{g}_{\alpha }}^{\text {an}})\cdot {}(l-1) -({\mathfrak {L}}_{\varvec{f}}^{\text {an}}-{\mathfrak {L}}_{\varvec{h}_{\alpha }}^{\text {an}})\cdot {}(\varvec{m}-1)\big )\cdot {}\mathscr {L}_{p}^{\alpha \alpha }(A,\varrho )^{\star }(w_{o}). \end{aligned}$$

Noting that $\mathscr {E}_{f}(w_{o})=1-\frac{\beta _{h}}{\alpha _{h}}$ (when $\alpha _{f}=\beta _{g}\cdot {}\alpha _{h}$), the previous discussion and Eq. (34) prove Theorem  2.1 when $\alpha _{f}=\beta _{g}\cdot {}\alpha _{h}$.