2·1. Rankin–Selberg L-functions

Let F be a totally real field of degree d and discriminant $D_F$ ; we will write $\mathfrak{d}$ for the different ideal of its ring of integers $\mathcal{O}_F$ . If $\hat{h}_F=\#\hat{\mathrm{Cl}}_F$ is the narrow class number of F, then choose finite ideles $t_1,\dots,t_{\hat{h}_F}$ such that the associated $\mathcal{O}_F$ -ideals $\tilde{t}_1,\dots,\tilde{t}_{\hat{h}_F}$ form a set of representatives for $\hat{\mathrm{Cl}}_F$ . We shall write $\mathcal{M}_k(\mathfrak{c},\psi)$ for the $\mathbb{C}$ -vector space of Hilbert modular forms (HMFs) of parallel weight $k>0$ , level $\mathfrak{c}\lhd \mathcal{O}_F$ and character $\psi$ . There is an isomorphism

\begin{align*}\mathcal{M}_k(\mathfrak{c},\psi)\,\buildrel \sim\over\longrightarrow\, \bigoplus_{\lambda=1}^{\hat{h}_F} \,\mathcal{M}_k\big(\Gamma_{\lambda}(\mathfrak{c}),\psi\big)\end{align*}

of complex vector spaces, which sends each $\mathbf{h}\in \mathcal{M}_k(\mathfrak{c},\psi)$ to an $\hat{h}_F$ -tuple of HMFs $\big(\mathbf{h}_{\lambda}\big)_{\lambda=1,\dots,\hat{h}_F}$ defined on the congruence modular groups

\begin{align*}\Gamma_{\lambda}(\mathfrak{c}) \,{:\!=}\,\left\{ \left(\begin{array}{c@{\kern+12pt}c}a & b \\c & d\end{array}\right)\in\mathrm{GL}_2^+\,\bigg|\,b\in(\tilde{t}_{\lambda}\mathfrak{d})^{-1},\,c\in\tilde{t}_{\lambda}\mathfrak{d}\mathfrak{c},\,a,d\in\mathcal{O}_F,\,ad-bc\in\mathcal{O}_F^{\times}\right\} .\end{align*}

If $\mathcal{S}_k(\mathfrak{c},\psi)\subset \mathcal{M}_k(\mathfrak{c},\psi)$ denotes the subspace of cusp forms, the above isomorphism identifies this subspace with $\bigoplus_{\lambda=1}^{\hat{h}_F} \mathcal{S}_k\big(\Gamma_{\lambda}(\mathfrak{c}),\psi\big)$ .

Throughout we assume there is a unique prime ideal $\mathfrak{p}\lhd \mathcal{O}_F$ lying over p. Suppose $\mathcal{F}$ is a $\mathfrak{p}$ -stabilised ordinary cuspidal eigenform of parallel weight two, trivial nebentypus, and conductor $\mathfrak{c}(\mathcal{F})=\mathfrak{p}\cdot\mathfrak{c}(\mathcal{F})^{\mathrm{tame}}$ satisfying $\mathfrak{p}+\mathfrak{c}(\mathcal{F})^{\mathrm{tame}}=\mathcal{O}_F$ . The complex L-series associated to $\mathcal{F}$ is given by the summation

\begin{align*}L(\mathcal{F},s) \,=\, \sum_{\mathfrak{m}\lhd\mathcal{O}_F} C(\mathcal{F},\mathfrak{m})\cdot\mathcal{N}(\mathfrak{m})^{-s}\end{align*}

and converges for Re(s) sufficiently large, with an analytic continuation to all $\mathbb{C}$ (here we have denoted by $C(\mathcal{F},\mathfrak{m})$ the Fourier coefficient for $\mathcal{F}$ at each ideal $\mathfrak{m}$ , and $\mathcal{N}=\mathcal{N}_{F/\mathbb{Q}}$ indicates the norm mapping defined on the fractional ideals of $\mathcal{O}_F$ ). Since $\mathcal{F}$ is $\mathfrak{p}$ -stabilised, for Re $(s)\gg 0$ its L-series has an Euler product expansion

\begin{align*}L(\mathcal{F},s)\,=\,\prod_{\mathfrak{q}\mid\mathfrak{p}\mathfrak{c}(\mathcal{F})}\!\!\!\left(1-C(\mathcal{F},\mathfrak{q})\mathcal{N}(\mathfrak{q})^{-s}\right)\,\times\!\prod_{\mathfrak{q}\nmid\mathfrak{p}\mathfrak{c}(\mathcal{F})}\!\!\!\left(1-C(\mathcal{F},\mathfrak{q})\mathcal{N}(\mathfrak{q})^{-s}+\mathcal{N}(\mathfrak{q})^{1-2s}\right) .\end{align*}

The local Euler factor for $\mathcal{F}$ at each $\mathfrak{q}\in\mathrm{Spec}(\mathcal{O}_F)$ with $\mathfrak{q}\nmid \mathfrak{c}(\mathcal{F})$ factorises into

\begin{align*}1-C(\mathcal{F},\mathfrak{q})\mathcal{N}(\mathfrak{q})^{-s}+\mathcal{N}(\mathfrak{q})^{1-2s}\,=\,\big(1-\alpha_{\mathfrak{q}}\mathcal{N}(\mathfrak{q})^{-s}\big)\big(1-\alpha'_{\mathfrak{q}}\mathcal{N}(\mathfrak{q})^{-s}\big)\end{align*}

and if $\mathfrak{q}$ divides the conductor $\mathfrak{c}(\mathcal{F})$ , we will always choose $\alpha_{\mathfrak{q}}=C(\mathcal{F},\mathfrak{q})$ and $\alpha_{\mathfrak{q}}'=0$ . We will extend $\alpha$ multiplicatively to the group of non-zero fractional ideals of $\mathcal{O}_F$ , so that under the decomposition $\mathfrak{a}=\prod_i \mathfrak{q}_i^{n_i}$ one defines $\alpha_{\mathfrak{a}}\,{:\!=}\,\prod_i \alpha_{\mathfrak{q}_i}^{n_i}$ .

Recall that Hecke operators act on the vector space of Hilbert modular forms. If $\mathbf{h}$ is a HMF and $\mathfrak{a}\lhd \mathcal{O}_F$ , then both ‘ $\mathbf{h}\big|U_{\mathfrak{a}}$ ’ and ‘ $\mathbf{h}\big|\mathfrak{a}$ ’ can be described in terms of their effect on the Fourier coefficients: for every $\mathcal{O}_F$ -ideal $\mathfrak{m}$ ,

\begin{align*}C\big(\mathbf{h}\big|U_{\mathfrak{a}},\mathfrak{m}\big)=C(\mathbf{h},\mathfrak{ma})\quad\mathrm{and}\quad C \big(\mathbf{h}\big|\mathfrak{a},\mathfrak{m}\big)=C\big(\mathbf{h},\mathfrak{ma}^{-1}\big),\end{align*}

where it is understood that $C\big(\mathbf{h},\mathfrak{ma}^{-1}\big)=0$ whenever $\mathfrak{ma}^{-1}$ is not an integral ideal. Now suppose that we are given another Hilbert eigenform $\mathbf{g}$ of parallel weight one, level $\mathfrak{c}(\mathbf{g})$ and character $\varpi$ . The local Euler factor at a prime $\mathfrak{q}$ similarly splits into

\begin{align*}1-C(\mathbf{g},\mathfrak{q})\mathcal{N}(\mathfrak{q})^{-s}+\varpi(\mathfrak{q})\mathcal{N}(\mathfrak{q})^{-2s}\,=\,\big(1-\beta_{\mathfrak{q}}\mathcal{N}(\mathfrak{q})^{-s}\big)\big(1-\beta'_{\mathfrak{q}}\mathcal{N}(\mathfrak{q})^{-s}\big),\end{align*}

where $\beta_{\mathfrak{q}},\beta_{\mathfrak{q}}'$ are algebraic integers, and we again extend both $\beta,\beta'$ multiplicatively. Choosing a finite set $\Sigma_F$ of places of F such that $\mathfrak{p}\not\in \Sigma_F$ and $\mathrm{supp}\big(\mathfrak{c}(\mathbf{g})\big)\subset \Sigma_F\cup\{\mathfrak{p}\}$ , one factorises the conductor of $\mathcal{F}\in \mathcal{S}_2\big(\mathfrak{c}(\mathcal{F}),\mathbf{1}_F\big)$ into

\begin{align*}\mathfrak{c}(\mathcal{F}) \,=\, \mathfrak{p}\cdot \mathfrak{c}(\mathcal{F})^{\mathrm{tame}}\,=\,\, \mathfrak{p}\cdot \mathfrak{c}(\mathcal{F})^{\sharp}\cdot\mathfrak{c}(\mathcal{F})^{\flat}\end{align*}

with $\mathrm{supp}\big(\mathfrak{c}(\mathcal{F})^{\sharp}\big) \subset \Sigma_F$ and $\mathrm{supp}\big(\mathfrak{c}(\mathcal{F})^{\flat}\big) \cap \Sigma_F=\emptyset$ .

We now explain how the theory of Hilbert modular forms relates to double product L-functions, i.e. for the tensor product of two forms of different weights. If $\mathbf{F}$ denotes a cusp form of parallel weight two and $\mathbf{G}$ another form of weight one (with nebentypes $\mathbf{1}_F$ and $\varpi$ respectively), then for Re $(s)\gg 0$ one defines

\begin{align*}\Psi(\mathbf{F},\mathbf{G},s) \,{:\!=}\,\frac{\Gamma(s)^{2d}}{(2\pi)^{2ds}} \times \zeta_{\mathfrak{c}(\mathbf{F})\mathfrak{c}(\mathbf{G})}\big(2s-1,\varpi\big)\times\sum_{\mathfrak{a}\lhd\mathcal{O}_F}C(\mathbf{F},\mathfrak{a})C(\mathbf{G},\mathfrak{a})\cdot\mathcal{N}(\mathfrak{a})^{-s}.\end{align*}

This L-function can be holomorphically continued to the whole complex plane $\mathbb{C}$ , and satisfies a functional equation relating its value at s with the value at $2-s$ . Shimura proved [ Reference Shimura29 , equation (4·32)] that if the levels of $\mathbf{F},\mathbf{G}$ both divide $\mathfrak{c}$ , thenFootnote ¹

\begin{align*}\Psi(\mathbf{F},\mathbf{G},s) \,=\,\frac{\sqrt{D_F}\cdot \Gamma(s)^d}{\pi^{ds}}\times\Big\langle \mathbf{F}^{\#}, \mathbf{G}\cdot K_{1}^0(s-1,\varpi^{-1};\,\mathfrak{c})\Big\rangle_{\mathfrak{c}} \, .\end{align*}

Here $K_{m}^q(s,\eta;\,\mathfrak{c})$ equals the Klingen-Eisenstein series described in [ Reference Panchishkin26 , section 4·1], namely

\begin{align*}K_{m}^q(s,\eta;\,\mathfrak{c})\,=\,\sum_{a,b}\,\mathrm{sgn}\mathcal{N}(b)^m \cdot\eta^*(b) \times\mathcal{N}(az+b)^{-m}\cdot\big|\mathcal{N}(az+b)\big|^{-2s}\end{align*}

and $\mathbf{F}^{\#}$ is the HMF whose Fourier coefficients are complex conjugate to the $C(\mathbf{F},\mathfrak{m})$ .

Notations

1. (i) Given any Hilbert modular form $\mathbf{h}$ of level $\mathfrak{c}$ and character $\psi$ , the J-operator is defined by the rule

   \begin{align*}\big(\mathbf{h}\big|J_{\mathfrak{c}}\big)(x) \,=\, \psi\big(\mathrm{det}(x)^{-1}\big)\cdot\mathbf{h}\left(x\left(\begin{array}{c@{\kern+8pt}c}0 & 1 \\c_0 & 0\end{array}\right)\right)\quad {\rm for\,c_0\in\mathbb{A}_{\mathbb{Q}}^{\mathrm{fin}}\,with \,\tilde{c}_0=\mathfrak{cd}^2.}\end{align*}

2. (ii) Following [ Reference Panchishkin26 , section 4·5], the normalised Eisenstein series $\mathbf{E}_{m}(s,\varpi^{-1};\,\mathfrak{c}\big)$ is given by

   \begin{align*}\frac{2^{-d}\cdot\sqrt{D_F}\cdot\Gamma(m+s)^d}{({-}4\pi)^{-sd}\cdot({-}2\pi i)^{d(m+2s)}}\times ({-}1)^{dm}\mathcal{N}\big(\mathfrak{cd}^2\big)^{s+\frac{m}{2}}\cdot K_{m}^0(s,\varpi^{-1};\,\mathfrak{c}\big) \Big|J_{\mathfrak{c}} .\end{align*}

To construct a p-adic measure, one first makes a shrewd choice of $\mathbf{F}$ and $\mathbf{G}$ . Let us therefore carefully choose

Henceforth we set the $\mathcal{O}_F$ -ideal $\mathfrak{m}_1\,{:\!=}\,\mathrm{lcm}\big(\mathfrak{c}(\mathcal{F})^{\sharp},\mathfrak{c}(\mathbf{g})^{\mathrm{tame}},\mathfrak{p}\mathfrak{m}_{\Sigma_F}^2\big)$ so that $\mathfrak{m}_1 \big| \mathfrak{m}'$ . In particular, from Definition 2·1(c) the quotient ideal ${\mathfrak{m}'}/{\mathfrak{m}_1}=\mathfrak{p}^{\mathbf{e}-1}$ belongs to $\mathfrak{p}^{\mathbb{N}}$ . Unfortunately, to obtain a non-trivial algebraic relation between firstly $\Psi\big(\mathbf{F},\mathbf{G},1\big)$ and secondly the inner product $\big\langle \mathbf{F}^{\#},\mathbf{G}\cdot\mathbf{E}_{1}(0,\varpi^{-1})\big|J\big\rangle$ , we must introduce further assumptions.

Hypotheses 2·2.

1. (i) the coefficient $C\big(\mathcal{F},\mathfrak{c}(\mathcal{F})^{\flat}\big)\neq 0$ ;

2. (ii) $\,\mathbf{g}$ is a primitive Hecke eigenform;

3. (iii) $\,$ the conductor $\mathfrak{c}(\mathbf{g})\subset \mathfrak{p}^2$ ;

4. (iv) $\,\mathrm{lcm}\big(\mathfrak{c}(\mathbf{g})^{\mathrm{tame}},\mathfrak{m}_{\Sigma_F}\big)\subset \mathfrak{c}(\mathcal{F})^{\sharp}$ .

Note that if $\mathbf{g}$ is a primitive eigenform of conductor $\mathfrak{c}(\mathbf{g})$ , then $\mathbf{g}\big|J_{\mathfrak{c}(\mathbf{g})}=\upsilon(\mathbf{g})\mathbf{g}^{\#}$ where the pseudo-eigenvalue of the J-operator, $\upsilon(\mathbf{g})$ , lies on the complex unit circle. Moreover $\mathbf{g}^{\#}$ is also a primitive form, and its local L-factor at $\mathfrak{q}$ factorises into

\begin{align*}1-C(\mathbf{g}^{\#},\mathfrak{q})\mathcal{N}(\mathfrak{q})^{-s}+\varpi^{-1}(\mathfrak{q})\mathcal{N}(\mathfrak{q})^{-2s}\,=\,\big(1-\hat{\beta}_{\mathfrak{q}}\mathcal{N}(\mathfrak{q})^{-s}\big)\big(1-\hat{\beta}'_{\mathfrak{q}}\mathcal{N}(\mathfrak{q})^{-s}\big).\end{align*}

The main example we will consider is when $\mathbf{g}$ is a theta-lift of a Hecke character.

The following statement is just a special case of [ Reference Delbourgo7 , corollary 3·7] which was developed for HMFs of parallel weight k at least two, but which we shall only state for parallel weight $k=2$ . Assuming that Hypotheses 2·2(i)–(iv) hold, if $s=1$ then

\begin{eqnarray*}\frac{\upsilon(\mathbf{g})\!\cdot\! \mathcal{N}\big(\mathfrak{c}(\mathbf{g}) \mathfrak{d}^2\big)^{s-\frac{1}{2}}}{\alpha_{\mathfrak{p}}^{\mathrm{ord}_{\mathfrak{p}}\mathfrak{c}(\mathbf{g})}}\! \cdot \mathrm{Eul}_{\mathcal{F},\Sigma_F}\big(\mathbf{g}^{\#},s\big)\cdot \Psi\big(\mathcal{F},\mathbf{g}^{\#},s\big) \,=\,\frac{({-}4i)^d}{C\big(\mathcal{F},\mathfrak{c}(\mathcal{F})^{\flat}\big) \cdot \alpha_{\frac{\mathfrak{m}'\quad\,\,}{\mathfrak{c}(\mathbf{g})^{\mathrm{tame}}}}} & & \\\,\times\, \Big\langle \mathcal{F}_{\Sigma_F}^{\#}\, ,\,\big(\mathbf{g}_{\mathfrak{m}_0}\cdot \mathbf{E}_{1}(s-1,\varpi^{-1};\,\mathfrak{m}'\mathfrak{c}(\mathcal{F})^{\flat})\big)\Big| U_{\frac{\mathfrak{m}'}{\mathfrak{m}_1}} \circ J_{\mathfrak{m}_1\cdot \mathfrak{c}(\mathcal{F})^{\flat}}\Big\rangle_{\!\mathfrak{m}_1\mathfrak{c}(\mathcal{F})^{\flat}}, & &\end{eqnarray*}

where the Euler factor is given by the finite product

\begin{eqnarray*}\mathrm{Eul}_{\mathcal{F},\Sigma_F}\big(\mathbf{g}^{\#},s\big) & = & \,\,\,\prod_{\mathfrak{q}\in\Sigma_F^{\mathrm{ps}}} \,\, \big(1-\alpha_{\mathfrak{q}}'\hat{\beta}_{\mathfrak{q}}\mathcal{N}(\mathfrak{q})^{-s}\big)\big(1-\alpha_{\mathfrak{q}}'\hat{\beta}'_{\mathfrak{q}}\mathcal{N}(\mathfrak{q})^{-s}\big) \\& \times & \!\!\!\!\!\prod_{\mathfrak{q}\in\Sigma_F^{\mathrm{ps}}\cup\Sigma_F^{\mathrm{spe}}\cup\{\mathfrak{p}\}}\!\!\big(1-\alpha_{\mathfrak{q}}^{-1}\beta_{\mathfrak{q}}\mathcal{N}(\mathfrak{q})^{s-1}\big)\big(1-\alpha_{\mathfrak{q}}^{-1}\beta'_{\mathfrak{q}}\mathcal{N}(\mathfrak{q})^{s-1}\big)\end{eqnarray*}

with $\Sigma_F^{\mathrm{ps}}$ consisting of the subset of places $\mathfrak{q}\in\Sigma_F$ at which $\mathcal{F}$ is principal series, and $\Sigma_F^{\mathrm{spe}}$ consisting of the subset of places $\mathfrak{q}\in\Sigma_F-\Sigma_F^{\mathrm{ps}}$ at which $C\big(\mathcal{F},\mathfrak{q}\big)\neq 0$ .

The reader will notice the presence of these unwanted factors, $\mathrm{Eul}_{\mathcal{F},\Sigma_F}({-})$ , which have the potential to lengthen our calculation of the $\lambda$ -invariant of $\mathcal{F}\otimes \mathbf{g}$ since they will have to be removed at some point. To simplify matters considerably, we shall now introduce the $\Sigma_F$ -depletion of $\mathcal{F}$ by taking

(2·1) \begin{equation}\mathcal{F}_{\Sigma_F}^{\mathrm{dep}}\,{:\!=}\,\sum_{\mathfrak{a}\mid \mathfrak{m}_{\Sigma_F}} \mu_{F}(\mathfrak{a})\cdot\mathcal{F}\big|U_{\mathfrak{a}}\circ\mathfrak{a}\,\,\in\, \mathcal{S}_2\Big(\mathrm{lcm}\big(\mathfrak{c}(\mathcal{F}),\mathfrak{m}_{\Sigma_F}^2\big),\mathbf{1}_F\Big) .\end{equation}

For reasons that will become apparent in the next section, we shall need to work at a smaller level than $\mathfrak{m}_1\mathfrak{c}(\mathcal{F})^{\flat}$ in general; we will thus choose $\mathfrak{m}_2\,{:\!=}\,\mathrm{lcm}\big(\mathfrak{c}(\mathcal{F})^{\sharp},\mathfrak{p}\mathfrak{m}_{\Sigma_F}^2\big)$ . Recall that the trace map $\mathrm{Tr}_{\mathfrak{m}_2}^{\mathfrak{m}_1}\,{:}\,\mathcal{M}_2\big(\mathfrak{m}_1\mathfrak{c}(\mathcal{F})^{\flat},\mathbf{1}_F\big)\rightarrow \mathcal{M}_2\big(\mathfrak{m}_2\mathfrak{c}(\mathcal{F})^{\flat},\mathbf{1}_F\big)$ satisfies

\begin{align*}\big\langle \mathbf{F}_1,\mathbf{F}_2\big\rangle_{\mathfrak{m}_1\mathfrak{c}(\mathcal{F})^{\flat}}\,=\,\big\langle \mathbf{F}_1,\mathrm{Tr}_{\mathfrak{m}_2}^{\mathfrak{m}_1}\mathbf{F}_2\big\rangle_{\mathfrak{m}_2\mathfrak{c}(\mathcal{F})^{\flat}}\end{align*}

for any pair of modular forms $\mathbf{F}_1\in\mathcal{S}_2\big(\mathfrak{m}_2\mathfrak{c}(\mathcal{F})^{\flat},\mathbf{1}_F\big)$ and $\mathbf{F}_2\in\mathcal{M}_2\big(\mathfrak{m}_1\mathfrak{c}(\mathcal{F})^{\flat},\mathbf{1}_F\big)$ . Repeating the Rankin integral calculation with $\mathcal{F}_{\Sigma_F}^{\mathrm{dep}}$ in place of $\mathcal{F}_{\Sigma_F}$ , one obtains:

Proposition 2·3. If the Hypotheses 2·2(i)–(iv) hold, then one has an equality

\begin{eqnarray*}\frac{\upsilon(\mathbf{g})\!\cdot\! \mathcal{N}\big(\mathfrak{c}(\mathbf{g}) \mathfrak{d}^2\big)^{\frac{1}{2}}}{\alpha_{\mathfrak{p}}^{\mathrm{ord}_{\mathfrak{p}}\mathfrak{c}(\mathbf{g})}}\! \cdot \Psi_{\Sigma_F}\big(\mathcal{F},\mathbf{g}^{\#},1\big) \,=\,\frac{({-}4i)^d}{C\big(\mathcal{F},\mathfrak{c}(\mathcal{F})^{\flat}\big) \cdot \alpha_{\frac{\mathfrak{m}'\quad\,\,}{\mathfrak{c}(\mathbf{g})^{\mathrm{tame}}}}}\qquad\qquad\qquad\qquad\quad & & \\\,\times\, \Big\langle (\mathcal{F}_{\Sigma_F}^{\mathrm{dep}})^{\#} ,\,\tilde{T}_{\mathbf{g}}\Big(\mathbf{g}_{\mathfrak{m}_0}\cdot \mathbf{E}_{1}(0,\varpi^{-1};\,\mathfrak{m}'\mathfrak{c}(\mathcal{F})^{\flat})\Big| U_{\frac{\mathfrak{m}'}{\mathfrak{m}_1}}\Big)\Big|J_{\mathfrak{m}_2\cdot \mathfrak{c}(\mathcal{F})^{\flat}}\Big\rangle_{\!\mathfrak{m}_2\mathfrak{c}(\mathcal{F})^{\flat}} & &\end{eqnarray*}

where the notation $\Psi_{\Sigma_F}\big(\mathcal{F},\mathbf{g}^{\#},s\big)$ means the Euler factors at primes dividing $\Sigma_F$ have been stripped away completely from the complex L-function, and the twisted trace mapping $\tilde{T}_{\mathbf{g}}\,{:}\,\mathcal{M}_2\big(\mathfrak{m}_1\mathfrak{c}(\mathcal{F})^{\flat},\mathbf{1}_F\big)\rightarrow \mathcal{M}_2\big(\mathfrak{m}_2\mathfrak{c}(\mathcal{F})^{\flat},\mathbf{1}_F\big)$ is defined by

\begin{align*}\tilde{T}_{\mathbf{g}}({-})\,{:\!=}\,\mathrm{Tr}_{\mathfrak{m}_2}^{\mathfrak{m}_1}\big({-}\big|J_{\mathfrak{m}_1\cdot \mathfrak{c}(\mathcal{F})^{\flat}}\big)\big|J_{\mathfrak{m}_2\cdot \mathfrak{c}(\mathcal{F})^{\flat}}^{-1}.\end{align*}

The left-hand side is independent of the choice of ideals $\mathfrak{m}_1$ , $\mathfrak{m}_2$ and $\mathfrak{m}'$ , hence the right-hand side is too. Note the pseudo-eigenvalue $\upsilon(\mathbf{g})\neq 0$ since $\big|\upsilon(\mathbf{g})\big|_{\infty}=1$ while $C\big(\mathcal{F},\mathfrak{c}(\mathcal{F})^{\flat}\big)\neq 0$ under Hypothesis 2·2(i), so the above is a non-trivial identity.

Proof. From the special case of [ Reference Delbourgo7 , corollary 3·7] given earlier, one deduces

\begin{eqnarray*}\frac{\upsilon(\mathbf{g})\!\cdot\! \mathcal{N}\big(\mathfrak{c}(\mathbf{g}) \mathfrak{d}^2\big)^{\frac{1}{2}}}{\alpha_{\mathfrak{p}}^{\mathrm{ord}_{\mathfrak{p}}\mathfrak{c}(\mathbf{g})}}\cdot \Psi_{\Sigma_F}\big(\mathcal{F},\mathbf{g}^{\#},1\big) \,=\,\frac{({-}4i)^d}{C\big(\mathcal{F},\mathfrak{c}(\mathcal{F})^{\flat}\big) \cdot \alpha_{\frac{\mathfrak{m}'\quad\,\,}{\mathfrak{c}(\mathbf{g})^{\mathrm{tame}}}}}\qquad\qquad\qquad\qquad & & \\\,\times\, \Big\langle (\mathcal{F}_{\Sigma_F}^{\mathrm{dep}})^{\#} ,\big(\mathbf{g}_{\mathfrak{m}_0}\cdot \mathbf{E}_{1}(0,\varpi^{-1};\,\mathfrak{m}'\mathfrak{c}(\mathcal{F})^{\flat})\big)\Big| U_{\frac{\mathfrak{m}'}{\mathfrak{m}_1}} \circ J_{\mathfrak{m}_1\cdot \mathfrak{c}(\mathcal{F})^{\flat}}\Big\rangle_{\!\mathfrak{m}_1\mathfrak{c}(\mathcal{F})^{\flat}} & &\end{eqnarray*}

and then tracing down via $\mathrm{Tr}_{\mathfrak{m}_2}^{\mathfrak{m}_1}$ and cleaning up J-operators, the result follows.

2·2. The Ray–Sujatha–Vatsal pairing

In order to obtain a congruence modulo p, one needs to work in a setting with an underlying integral structure. Following work of Ray, Sujatha and Vatsal in [ Reference Ray, Sujatha and Vatsal27 , section 2] for elliptic modular forms, we now extend their algebraic pairings to include HMFs of parallel weight $k\geqslant 2$ and arbitrary central character $\psi_F$ .

For each ideal $\mathfrak{n}\lhd \mathcal{O}_F$ and a fixed integral extension $R/\mathbb{Z}_p$ , let $\mathbb{T}_{\mathfrak{n}}(R)$ denote the algebra generated by the Hecke operators $T_{\mathfrak{q}}$ , $U_{\mathfrak{q}}$ and $U_{\mathfrak{p}}$ acting upon $\mathcal{S}_k\big(\mathfrak{n};\,R\big)$ . If R contains the Fourier coefficients of some Hecke eigenform $\mathbf{F}\in\mathcal{S}_k^{\mathrm{ord}}\big(\mathfrak{n};\,R\big)$ then one has a ring homomorphism $\lambda_{\mathbf{F}}\,{:}\,\mathbb{T}_{\mathfrak{n}}(R)\rightarrow R$ , and there is a unique maximal ideal $\mathcal{M}$ of $\mathbb{T}_{\mathfrak{n}}(R)$ containing both the kernel of $\lambda_{\mathbf{F}}$ and also the ideal $\mathfrak{m}_R\cdot \mathbb{T}_{\mathfrak{n}}(R)$ . Localising at this maximal ideal, one obtains a perfect pairing

\begin{align*}\mathcal{S}_k(\mathfrak{n};\,R)_{\mathcal{M}}\times \mathbb{T}_{\mathfrak{n}}(R)_{\mathcal{M}}\longrightarrow\, R,\qquad (\mathbf{h},T)\mapsto C(\mathbf{h}|T,\mathcal{O}_F)\end{align*}

which then induces a canonical ring isomorphism $\mathbb{T}_{\mathfrak{n}}(R)_{\mathcal{M}}\cong\mathrm{Hom}_R\big(\mathcal{S}_k(\mathfrak{n};\,R)_{\mathcal{M}},R\big)$ . (In the sequel, we will choose $\mathfrak{n}$ to be a minimal common level for both $\mathcal{F}_1$ and $\mathcal{F}_2$ , we always take the parallel weight to be two, and choose $\mathbf{F}_i=\mathcal{F}_{i,\Sigma_F}^{\mathrm{dep}}$ as in section 2·1.)

Hypotheses 2·4.

1. (i) p is inert in the extension $F/\mathbb{Q}$ , so $p\cdot\mathcal{O}_F=\mathfrak{p}$ and $p\nmid D_F$ ;

2. (ii) the Galois representation $\rho_{\mathcal{F}}\,{:}\,G_F\rightarrow\mathrm{GL}_2(R)$ attached to the eigenform $\mathcal{F}$ is absolutely irreducible, $\overline{\rho}_{\mathcal{F}}\big|_{G_{F_{\mathfrak{p}}}}$ is $\mathfrak{p}$ -ordinary, and lastly $\overline{\rho}_{\mathcal{F}}^{\mathrm{ss}}\big|_{G_{F_{\mathfrak{p}}}}$ is non-scalar;

3. (iii) at each prime ideal $\mathfrak{q}\in\Sigma_F$ , the oldform $\mathcal{F}_{\Sigma_F}^{\mathrm{dep}}\big|\mathfrak{q}$ does not occur at level $\mathfrak{n}$ .

The motivation for imposing these assumptions is identical to that in op. cit. i.e. without them there is no way to ensure that the p-adic L-function is integral. Conditions (i) and (ii) imply $\mathbb{T}_{\mathfrak{n}}(R)_{\mathcal{M}}$ is a Gorenstein ring using [ Reference Dimitrov11 , theorem 6·7], in which case one finds that $\mathbb{T}_{\mathfrak{n}}(R)_{\mathcal{M}}\cong \mathrm{Hom}\big(\mathbb{T}_{\mathfrak{n}}(R)_{\mathcal{M}},R\big)$ as left Hecke modules. The right-hand module is isomorphic to $\mathcal{S}_k(\mathfrak{n};\,R)_{\mathcal{M}}$ from the above pairing, and the left-action of $\mathbb{T}_{\mathfrak{n}}(R)_{\mathcal{M}}$ on $\mathrm{Hom}\big(\mathbb{T}_{\mathfrak{n}}(R)_{\mathcal{M}},R\big)\cong \mathcal{S}_k(\mathfrak{n};\,R)_{\mathcal{M}}$ coincides with the standard right-action of the Hecke algebra. We therefore obtain a duality

(2·2) \begin{equation}({-},-)_{\mathfrak{n}} \,{:}\,\mathcal{S}_k(\mathfrak{n};\,R)_{\mathcal{M}}\times\, \mathcal{S}_k(\mathfrak{n};\,R)_{\mathcal{M}} \longrightarrow R\end{equation}

which is a Hecke-equivariant pairing of localised modules (cf. Hida’s pairing [ Reference Hida18 ]).

Statements (a) and (b) were proved in [ Reference Ray, Sujatha and Vatsal27 , lemma 2·14 and corollary 2·15] respectively, but only over the rational numbers. However their inductive argument extends seamlesslyFootnote ² to totally real fields $F\neq \mathbb{Q}$ , under these hypotheses of course. To illustrate how their method works in practice, at every $\mathfrak{q}\in\Sigma_F$ one considers the subspaces $\mathcal{S}({\mathfrak{q}})\subset \mathcal{S}_k(\mathfrak{n},\mathbf{1}_F)$ defined as follows:

1. (i) if $\mathfrak{q}=\mathfrak{p}$ and $\mathfrak{p}\nmid\mathfrak{c}(\mathcal{F}^{\dagger})$ where $\mathcal{F}^{\dagger}$ indicates the newform associated to $\mathcal{F}$ , then the space $\mathcal{S}({\mathfrak{p}})$ is generated by $\mathcal{F}^{\dagger}$ and $\mathcal{F}^{\dagger}\big|\mathfrak{p}$ ;

2. (ii) if $\mathfrak{q}=\mathfrak{p}$ and $\mathfrak{p}\big|\!\big|\mathfrak{c}(\mathcal{F}^{\dagger})$ , then $\mathcal{S}({\mathfrak{p}})$ is a line generated by $\mathcal{F}=\mathcal{F}^{\dagger}$ ;

3. (iii) if $\mathfrak{q}\neq\mathfrak{p}$ and $\mathfrak{q}\nmid\mathfrak{c}(\mathcal{F})^{\sharp}$ , then $\mathcal{S}({\mathfrak{q}})$ is generated by $\mathcal{F}^{\dagger}$ , $\mathcal{F}^{\dagger}\big|\mathfrak{q}$ and $\mathcal{F}^{\dagger}\big|\mathfrak{q}^2$ ;

4. (iv) if $\mathfrak{q}\neq\mathfrak{p}$ and $\mathfrak{q}\big|\!\big|\mathfrak{c}(\mathcal{F})^{\sharp}$ , then $\mathcal{S}({\mathfrak{q}})$ is generated by $\mathcal{F}^{\dagger}$ and $\mathcal{F}^{\dagger}\big|\mathfrak{q}$ ;

5. (v) if $\mathfrak{q}\neq\mathfrak{p}$ and $\mathfrak{q}^2\big|\!\big|\mathfrak{c}(\mathcal{F})^{\sharp}$ , then $\mathcal{S}({\mathfrak{q}})$ is a line generated by $\mathcal{F}^{\dagger}$ .

For all $\mathfrak{q}\in\Sigma_F$ , the spaces $\mathcal{S}({\mathfrak{q}})$ are stable under the full Hecke algebra at level $\mathfrak{n}$ . Furthermore, a result of Shemanske and Walling in [ Reference Shemanske and Walling28 , theorem 3·3] implies that the eigenvalues of the $U_{\mathfrak{q}}$ -operator on $\mathcal{S}({\mathfrak{q}})$ are distinct. Under Hypothesis 2·4(iii), we easily see that $\mathcal{F}_{\Sigma_F}^{\mathrm{dep}}$ is the unique Hecke eigenform of level $\mathfrak{n}$ (in the $\mathcal{F}^{\dagger}$ -isotypic component) which has $U_{\mathfrak{q}}$ -eigenvalues equal to zero at all the prime ideals $\mathfrak{q}\in\Sigma_F$ . It follows that each localised component $\mathcal{S}({\mathfrak{q}})_{\mathcal{M}}$ as above must be one-dimensional, which provides the necessary induction step in [ Reference Ray, Sujatha and Vatsal27 , lemma 2·14].

The pairing $({-},-)_{\mathfrak{n}}$ in (2·2) is self-adjoint with respect to the action of $\mathbb{T}_{\mathfrak{n}}(R)$ . Similarly, the twisted Petersson inner product $\big\langle \mathbf{F}_1,\mathbf{F}_2\big\rangle_{\mathfrak{n}}'\,{:\!=}\,\big\langle \mathbf{F}_1^{\#},\mathbf{F}_2\big|J_{\mathfrak{n}}\big\rangle_{\mathfrak{n}}$ defined for all $\mathbf{F}_1,\mathbf{F}_2\in\mathcal{S}_k(\mathfrak{n},\psi_F;\,\mathbb{C})$ is self-adjoint under the right-action of the ring $\mathbb{T}_{\mathfrak{n}}(\mathbb{C})$ . In fact an analogous argument involving the subspaces $\mathcal{S}({\mathfrak{q}})$ , but applied to the twisted $\mathbb{C}$ -linear pairing $\big\langle -,-\big\rangle_{\mathfrak{n}}'$ instead, establishes that $\big\langle\mathcal{F}_{\Sigma_F}^{\mathrm{dep}},\mathcal{F}_{\Sigma_F}^{\mathrm{dep}}\big\rangle_{\mathfrak{n}}'\neq 0$ as well. If we identify $\mathbb{C}\cong\mathbb{C}_p$ as fields, then one obtains two $\mathbb{C}_p$ -valued linear functionals

\begin{align*}\phi_{\mathfrak{n},\mathbf{F}}^{\mathrm{alg}}({-})=\big(\mathcal{F}_{\Sigma_F}^{\mathrm{dep}},-\big)_{\mathfrak{n}}\,,\,\phi_{\mathfrak{n},\mathbf{F}}^{\mathrm{Pet}}({-})=\big\langle\mathcal{F}_{\Sigma_F}^{\mathrm{dep}},-\big\rangle_{\mathfrak{n}}'\,:\, \mathcal{S}_k(\mathfrak{n},\psi_F;\,\mathbb{C}_p)_{\mathcal{M}}\longrightarrow \mathbb{C}_p\end{align*}

which are both non-zero on the $\Sigma_F$ -depleted Hilbert modular eigenform $\mathbf{F}=\mathcal{F}_{\Sigma_F}^{\mathrm{dep}}$ .

Clearly one can redo the construction of these pairings by working at level $\mathfrak{c}(\mathcal{F})$ and without $\Sigma_F$ -depleting $\mathcal{F}$ first, in which case one obtains a period $\Omega_{\mathfrak{c}(\mathcal{F}),\emptyset}(\mathcal{F})$ . We conjecture that if p is unramified in F, then $\Omega_{\mathfrak{n},\Sigma}(\mathcal{F})$ coincides with $\Omega_{\mathfrak{c}(\mathcal{F}),\emptyset}(\mathcal{F})$ up to a p-adic unit (and some other simple factors). Indeed if $F=\mathbb{Q}$ then Ray, Sujatha and Vatsal [ Reference Ray, Sujatha and Vatsal27 , lemma 2·20] prove this statement using an appropriate version of Ihara’s lemma – we will return to this question again in Section 2.5.

2·3. Congruences between algebraic L-values

We shall now combine together the results in the previous two sections to obtain our p-adic congruences. Suppose we are given a pair of $\mathfrak{p}$ -stabilised ordinary eigenforms $\mathcal{F}_1\in \mathcal{S}_2\big(\mathfrak{c}(\mathcal{F}_1),\mathbf{1}_F;\,R\big)$ and $\mathcal{F}_2\in \mathcal{S}_2\big(\mathfrak{c}(\mathcal{F}_2),\mathbf{1}_F;\,R\big)$ , where R is a finite extension of $\mathbb{Z}_p$ with uniformiser $\wp$ . Henceforth, one assumes that for some fixed positive integer r:

(2·3) \begin{equation}C\big((\mathcal{F}_1)_{\Sigma_F}^{\mathrm{dep}},\mathfrak{m}\big) \equiv C\big((\mathcal{F}_2)_{\Sigma_F}^{\mathrm{dep}},\mathfrak{m}\big)\!\!\mod\wp^r\cdot R\quad\textit{at every ideal}\ \mathfrak{m}.\end{equation}

In the next section, we will discuss in greater detail how this situation arises for a pair of congruent modular elliptic curves $E_1$ and $E_2$ which are defined over F.

We again fix a primitive Hecke eigenform $\mathbf{g}$ of parallel weight one which satisfies Hypotheses 2·2 for both $\mathcal{F}_1$ and $\mathcal{F}_2$ , where the finite set $\Sigma_F\subset\mathrm{Spec}(\mathcal{O}_F)-\{\mathfrak{p}\}$ consists of the prime $\mathcal{O}_F$ -ideals dividing $\mathfrak{c}(\mathcal{F}_1)^{\sharp}\cdot\mathfrak{c}(\mathcal{F}_2)^{\sharp}\cdot\mathfrak{c}(\mathbf{g})$ but not $\mathfrak{c}(\mathcal{F}_1)^{\flat}\cdot\mathfrak{c}(\mathcal{F}_2)^{\flat}$ . In order to express the algebraic L-values for $\mathcal{F}_j\otimes \mathbf{g}^{\#}$ at each $j\in\{1,2\}$ using a common inner product, the calculations are done at a level with stable $\mathfrak{p}$ -part, i.e.

(2·4) \begin{equation}\mathfrak{n} \,{:\!=}\, \mathfrak{m}_2\cdot\mathrm{lcm}\big(\mathfrak{c}(\mathcal{F}_1)^{\flat},\mathfrak{c}(\mathcal{F}_2)^{\flat}\big) .\end{equation}

Here one now takes $\mathfrak{m}_2=\mathrm{lcm}\big(\mathfrak{c}(\mathcal{F}_1)^{\sharp},\mathfrak{c}(\mathcal{F}_2)^{\sharp},\mathfrak{p}\mathfrak{m}_{\Sigma_F}^2\big)$ , which still divides $\mathfrak{m}'$ of course.

In the last three cases $\mathrm{ord}_{\mathfrak{q}}(\mathfrak{n})=\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\overline{\rho})\big)+1=2$ , hence the $\mathfrak{q}$ -level of both $\Sigma_F$ -depleted Hilbert modular forms $\mathcal{F}_{1,\Sigma_F}^{\mathrm{dep}}\big|\mathfrak{q}$ and $\mathcal{F}_{2,\Sigma_F}^{\mathrm{dep}}\big|\mathfrak{q}$ exceeds $\mathfrak{n}_{\mathfrak{q}}=\mathfrak{q}^2$ again.

Proof. Beginning with assertion (a), we note that ${\mathfrak{m}_1}/{\mathfrak{m}_2}=\prod_{\mathfrak{q}\mid \mathfrak{c}(\mathbf{g})^{\mathrm{tame}}}\mathfrak{q}^{e(\mathfrak{q})}$ say with each exponent $e(\mathfrak{q})\geqslant 0$ . If we define an auxiliary level $\mathfrak{m}_2^{\dagger}\,{:\!=}\,\prod_{e(\mathfrak{q})\geqslant 2}\mathfrak{q}^{e(\mathfrak{q})-1}$ then the twisted trace operator $\tilde{T}_{\mathbf{g}}({-})$ can be factorised into $\tilde{T}_{\mathbf{g}}= \tilde{T}_{\mathbf{g}}^{\dagger\!\dagger}\circ \tilde{T}_{\mathbf{g}}^{\dagger}$ , where

\begin{align*}\tilde{T}_{\mathbf{g}}^{\dagger}({-})\,{:\!=}\,\mathrm{Tr}_{\mathfrak{m}_2^{\dagger}}^{\mathfrak{m}_1}\big({-}\big|J_{\mathfrak{m}_1\cdot \mathfrak{c}(\mathcal{F})^{\flat}}\big)\big|J_{\mathfrak{m}_2^{\dagger}\cdot \mathfrak{c}(\mathcal{F})^{\flat}}^{-1} \\\,\,\mathrm{and} \,\, \tilde{T}_{\mathbf{g}}^{\dagger\!\dagger}({-})\,{:\!=}\,\mathrm{Tr}^{\mathfrak{m}_2^{\dagger}}_{\mathfrak{m}_2}\big({-}\big|J_{\mathfrak{m}_2^{\dagger}\cdot \mathfrak{c}(\mathcal{F})^{\flat}}\big)\big|J_{\mathfrak{m}_2\cdot \mathfrak{c}(\mathcal{F})^{\flat}}^{-1} .\end{align*}

Let us make the important observation that $\mathrm{supp}(\mathfrak{m}_2^{\dagger})=\mathrm{supp}(\mathfrak{m}_1)$ , whilst tracing down from level $\mathfrak{m}_2^{\dagger}$ to level $\mathfrak{m}_2$ consists exclusively of removing primes with $e(\mathfrak{q})\geqslant 1$ . Let $\mathfrak{a}$ be an $\mathcal{O}_F$ -ideal. We employ the following two claims about the trace mapping:

1. (I) if $\mathfrak{q}\mid \mathfrak{a}$ then for each Hilbert modular form $\mathbf{H}$ of level $\mathfrak{aq}$ and weight $k\geqslant 2$ ,

   \begin{align*}\mathrm{Tr}_{\mathfrak{a}}^{{\mathfrak{aq}}}\big(\mathbf{H}\big|J_{\mathfrak{aq}}\big)=\,\mathcal{N}(\mathfrak{q})^{1-k/2}\cdot \mathbf{H}\big|U_{\mathfrak{q}}\circ J_{\mathfrak{a}} \, ;\end{align*}

2. (II) if $\mathfrak{q}\nmid \mathfrak{a}$ then for each Hilbert modular form $\mathbf{H}$ of level $\mathfrak{aq}$ and weight $k\geqslant 2$ ,

   \begin{align*}\quad\mathrm{Tr}_{\mathfrak{a}}^{{\mathfrak{aq}}}\big(\mathbf{H}\big|J_{\mathfrak{aq}}\big)=\,\mathcal{N}(\mathfrak{q})^{1-k/2}\cdot \mathbf{H}\big|\big(U_{\mathfrak{q}}+\mathcal{N}(\mathfrak{q})^{k-1}\!\cdot \langle\mathfrak{q}\rangle\big) \circ J_{\mathfrak{a}}\, .\end{align*}

Applying a simple induction argument, from statement (I) one readily deduces that

\begin{align*}\mathbf{H}^{\dagger}\,{:\!=}\,\tilde{T}_{\mathbf{g}}^{\dagger}(\mathbf{H}) \,=\, \mathcal{N}\big(\mathfrak{m}_1\big/\mathfrak{m}_2^{\dagger}\big)^{1-k/2}\cdot \mathbf{H}\,\Big|\prod_{e(\mathfrak{q})\geqslant 2}U_{\mathfrak{q}}^{e(\mathfrak{q})-1} .\end{align*}

Similarly, applying induction and using statement (II) instead yields the identity

\begin{align*}\tilde{T}_{\mathbf{g}}(\mathbf{H})=\tilde{T}_{\mathbf{g}}^{\dagger\!\dagger}(\mathbf{H}^{\dagger}) \,=\, \mathcal{N}\big(\mathfrak{m}_2^{\dagger}\big/\mathfrak{m}_2\big)^{1-k/2}\cdot \mathbf{H}^{\dagger}\Big|\prod_{e(\mathfrak{q})\geqslant 1}\big(U_{\mathfrak{q}}+\mathcal{N}(\mathfrak{q})^{k-1}\!\cdot \langle\mathfrak{q}\rangle\circ \mathfrak{q}\big) .\end{align*}

At weight $k=2$ one has $\tilde{T}_{\mathbf{g}}(\mathbf{H})=\mathbf{H}\Big|\prod_{e(\mathfrak{q})\geqslant 2}U_{\mathfrak{q}}^{e(\mathfrak{q})-1}\Big|\prod_{e(\mathfrak{q})\geqslant 1}\big(U_{\mathfrak{q}}+\mathcal{N}(\mathfrak{q})\cdot \langle\mathfrak{q}\rangle\circ \mathfrak{q}\big)$ ; because $U_{\mathfrak{q}}$ and the diamond operators $\langle\mathfrak{q}\rangle$ preserve $p$ -integrality, part (a) follows.

In order to justify these claims made above, statement (I) is equation (4·11) in [ Reference Panchishkin26 , chapter IV] (note that there is a factor of $({-}1)^k$ missing in the latter formula). For claim (II) we may study the effect of these operators on the Fourier coefficients:

\begin{align*}C\Big(\mathbf{H}\Big|\big(U_{\mathfrak{q}}+\mathcal{N}(\mathfrak{q})^{k-1}\!\cdot \langle\mathfrak{q}\rangle\circ \mathfrak{q}\big) , \mathfrak{m}\Big)= \mathcal{N}(\mathfrak{q})^{1-k/2}\cdot\!\sum_{\mathfrak{q}+\mathfrak{m}\subset \mathfrak{b}} \psi^*(\mathfrak{b})\cdot\mathcal{N}\big(\mathfrak{b}^{k-1}\big)\cdot C \big(\mathbf{H} , \mathfrak{b}^2\mathfrak{mq}\big)\end{align*}

which coincides with $\mathcal{N}(\mathfrak{q})^{k/2-1}\cdot\mathrm{Tr}_{\mathfrak{a}}^{{\mathfrak{aq}}}\big({-}\big|J_{\mathfrak{aq}}\big)\big|J_{\mathfrak{a}}^{-1}$ acting on the $\mathfrak{m}$ -th coefficient – we refer the reader to Appendix A for the full (but quite tedious) calculation.

To prove that part (b) is true, clearly the primitive form $\mathbf{g}$ has an algebraic integer Fourier expansion and so does $\mathbf{g}_{\mathfrak{m}_0}$ , while by the formula in [ Reference Panchishkin26 , proposition 4·2]

\begin{align*}\mathbf{E}_{1}(0,\varpi^{-1};\,\mathfrak{m}'\mathfrak{c}(\mathcal{F})^{\flat})_{\lambda}\,=\, \mathcal{N}(\tilde{t}_{\lambda})^{-3/2}\cdot\sum_{0\ll \xi\in\tilde{t}_{\lambda}}\,\sum_{\tilde{\xi}=\tilde{b}\tilde{c}}\varpi^*(\tilde{c})^{-1}\cdot\exp\big(2\pi i \cdot \mathrm{Tr}(\xi z)\big) .\end{align*}

Therefore $\mathbf{g}_{\mathfrak{m}_0}\cdot \mathbf{E}_{1}(0,\varpi^{-1};\,\mathfrak{m}'\mathfrak{c}(\mathcal{F})^{\flat})$ also has an algebraic integer Fourier expansion, and the $U_{\frac{\mathfrak{m}'}{\mathfrak{m}_1}}$ -operator will preserve integrality (on holomorphic HMFs at least).

The following result establishes that once they have been sufficiently depleted, the special values of the L-functions attached to $\mathcal{F}_1\otimes\mathbf{g}^{\#}$ and $\mathcal{F}_2\otimes\mathbf{g}^{\#}$ are congruent.

Theorem 2·10. Assume that the Hypotheses 2·4(i)-(iii) hold for $\mathcal{F}_1$ and $\mathcal{F}_2$ . Then in the notation of Section 2·1,

\begin{eqnarray*}& & \!\!\!\!\!\!\!\!\quad \,\,\mathfrak{u}_{\mathfrak{m}'}(\mathcal{F}_1)\times \upsilon(\mathbf{g})\cdot \frac{ \mathcal{N}\big(\mathfrak{c}(\mathbf{g}) \mathfrak{d}^2\big)^{\frac{1}{2}}}{\alpha_{\mathfrak{p}}(\mathcal{F}_1)^{\mathrm{ord}_{\mathfrak{p}}\mathfrak{c}(\mathbf{g})}}\! \cdot \frac{\Psi_{\Sigma_F}\big(\mathcal{F}_1,\mathbf{g}^{\#},1\big)}{\Omega_{\mathfrak{n},\Sigma}(\mathcal{F}_1)} \\& & \!\!\!\!\!\!\!\! \equiv\,\,\mathfrak{u}_{\mathfrak{m}'}(\mathcal{F}_2)\times \upsilon(\mathbf{g})\cdot\frac{ \mathcal{N}\big(\mathfrak{c}(\mathbf{g}) \mathfrak{d}^2\big)^{\frac{1}{2}}}{\alpha_{\mathfrak{p}}(\mathcal{F}_2)^{\mathrm{ord}_{\mathfrak{p}}\mathfrak{c}(\mathbf{g})}}\! \cdot \frac{\Psi_{\Sigma_F}\big(\mathcal{F}_2,\mathbf{g}^{\#},1\big)}{\Omega_{\mathfrak{n},\Sigma}(\mathcal{F}_2)}\mod \wp^r\cdot\mathcal{O}_{\mathbb{C}_p}, \qquad\end{eqnarray*}

where the scalar terms $\mathfrak{u}_{\mathfrak{m}'}(\mathcal{F}_j)\,{:\!=}\,C\big(\mathcal{F}_j,\mathfrak{c}(\mathcal{F}_j)^{\flat}\big) \cdot \alpha_{\frac{\mathfrak{m}'\quad\,\,}{\mathfrak{c}(\mathbf{g})^{\mathrm{tame}}}}\!(\mathcal{F}_j)\cdot ({-}4i)^{-d}\in\mathcal{O}_{\mathbb{C}_p}^{\times}$ .

Proof. We first observe that by Proposition 2·9(a) and (b),

\begin{align*}\Phi\big(\mathbf{g}\cdot \mathbf{E}_{1}\big)\,{:\!=}\,\, \tilde{T}_{\mathbf{g}}\Big(\mathbf{g}_{\mathfrak{m}_0}\cdot \mathbf{E}_{1}(0,\varpi^{-1};\,\mathfrak{p}^{\mathbf{e}-1}\mathfrak{n})\Big| U_{\frac{\mathfrak{m}'}{\mathfrak{m}_1}}\Big)\end{align*}

is holomorphic of level $\mathfrak{n}$ and character $\mathbf{1}_F$ , and furthermore $\Phi\big(\mathbf{g}\cdot \mathbf{E}_{1}\big)\in\mathcal{O}_{\mathbb{C}_p}[\![q]\!]$ : enlarging R if necessary, one may therefore assume that $\Phi\big(\mathbf{g}\cdot \mathbf{E}_{1}\big)\in\mathcal{M}_2\big(\mathfrak{n},\mathbf{1}_F;\,R\big)$ . If we recall from [ Reference Hida18 ] that Hida’s ordinary idempotent equals $\mathfrak{e}^{\mathrm{ord}}\,{:\!=}\,\lim_{n\rightarrow\infty} U_{\mathfrak{p}}^{n!}$ , then one may conclude that $\Phi\big(\mathbf{g}\cdot \mathbf{E}_{1}\big)^{\mathrm{ord}}\!\,{:\!=}\,\Phi\big(\mathbf{g}\cdot \mathbf{E}_{1}\big)\Big|\mathfrak{e}^{\mathrm{ord}}$ clearly also has p-integral Fourier coefficients, although it might not end up being a cusp form in general.

Writing $\mathbf{H}_{\mathfrak{n}}^{\mathrm{ord}}(R)$ for the algebra generated by the Hecke operators over F acting on $\mathcal{M}_2^{\mathrm{ord}}\big(\mathfrak{n},\mathbf{1}_F;\,R\big)$ , there is a surjection of R-algebras $\vartheta:\mathbf{H}_{\mathfrak{n}}^{\mathrm{ord}}(R)\twoheadrightarrow\mathbb{T}_{\mathfrak{n}}^{\mathrm{ord}}(R)$ . Moreover $\mathbb{T}_{\mathfrak{n}}^{\mathrm{ord}}(R)\buildrel \sim\over\rightarrow\bigoplus_i \mathbb{T}_{\mathfrak{n}}(R)_{\mathcal{M}_i}$ where the direct sum ranges over a finite set of maximal ideals $\mathcal{M}_i\lhd \mathbb{T}_{\mathfrak{n}}(R)$ , namely the associated primes of the Hecke algebra. The congruence in Equation (2·3) between the weight two HMFs ensures that the specialisation maps $\lambda_{\mathbf{F}}\,{:}\,\mathbb{T}_{\mathfrak{n}}(R)\twoheadrightarrow\mathbb{T}_{\mathfrak{n}}^{\mathrm{ord}}(R)\rightarrow R$ for both of $\mathbf{F}=(\mathcal{F}_1)_{\Sigma_F}^{\mathrm{dep}}$ and $\mathbf{F}=(\mathcal{F}_2)_{\Sigma_F}^{\mathrm{dep}}$ are congruent mod $\wp^r$ , hence they share the same maximal ideal $\mathcal{M}$ .

We define $\mathcal{M}^{\dagger}\,{:\!=}\,\vartheta^{-1}(\mathcal{M})$ , which is a maximal ideal of $\mathbf{H}_{\mathfrak{n}}^{\mathrm{ord}}(R)$ . Consequently one may write $\mathfrak{e}_{\mathcal{M}}^{\dagger}\in \mathbf{H}_{\mathfrak{n}}^{\mathrm{ord}}(R)$ and $\mathfrak{e}_{\mathcal{M}}\in \mathbb{T}_{\mathfrak{n}}^{\mathrm{ord}}(R)$ for the corresponding projectors. If we choose $\mathbf{F}=(\mathcal{F}_1)_{\Sigma_F}^{\mathrm{dep}}$ or $\mathbf{F}=(\mathcal{F}_2)_{\Sigma_F}^{\mathrm{dep}}$ , a simple series of calculations reveals

\begin{eqnarray*}\phi_{\mathfrak{n},\mathbf{F}}^{\mathrm{Pet}}\big(\Phi\big(\mathbf{g}\cdot \mathbf{E}_{1}\big)\big)&=&\big\langle\mathbf{F},\Phi(\mathbf{g}\cdot \mathbf{E}_{1})\big\rangle_{\mathfrak{n}}'=\big\langle\mathbf{F},\Phi(\mathbf{g}\cdot \mathbf{E}_{1})^{\mathrm{ord}}\big\rangle_{\mathfrak{n}}' \\&=&\big\langle\mathbf{F}\big|\mathfrak{e}_{\mathcal{M}},\Phi(\mathbf{g}\cdot \mathbf{E}_{1})^{\mathrm{ord}}\big\rangle_{\mathfrak{n}}'=\big\langle\mathbf{F},\Phi(\mathbf{g}\cdot \mathbf{E}_{1})^{\mathrm{ord}}\big|\mathfrak{e}^{\dagger}_{\mathcal{M}}\big\rangle_{\mathfrak{n}}'.\end{eqnarray*}

Using the period from Definition 2·6, if $\mathbf{F}=(\mathcal{F}_j)_{\Sigma_F}^{\mathrm{dep}}$ then one has a compatibility

\begin{align*}\phi_{\mathfrak{n},\mathbf{F}}^{\mathrm{Pet}}\big(\Phi\big(\mathbf{g}\cdot \mathbf{E}_{1}\big)^{\mathrm{ord}}\big|\mathfrak{e}^{\dagger}_{\mathcal{M}}\big)\,=\,\phi_{\mathfrak{n},\mathbf{F}}^{\mathrm{alg}}\big(\Phi\big(\mathbf{g}\cdot \mathbf{E}_{1}\big)^{\mathrm{ord}}\big|\mathfrak{e}^{\dagger}_{\mathcal{M}}\big)\times \Omega_{\mathfrak{n},\Sigma}(\mathcal{F}_j) .\end{align*}

Putting $\mathbf{h}=\Phi\big(\mathbf{g}\cdot \mathbf{E}_{1}\big)^{\mathrm{ord}}\big|\mathfrak{e}^{\dagger}_{\mathcal{M}}\in \mathcal{S}_2(\mathfrak{n},\mathbf{1}_F;\,R)_{\mathcal{M}}$ , one immediately deduces that

\begin{align*}\phi_{\mathfrak{n},\mathbf{F}}^{\mathrm{alg}}(\mathbf{h})\,=\, \Omega_{\mathfrak{n},\Sigma}(\mathcal{F}_j)^{-1}\!\times \Big\langle(\mathcal{F}_j)_{\Sigma_F}^{\mathrm{dep}}\,,\Phi(\mathbf{g}\cdot \mathbf{E}_{1})\Big\rangle_{\mathfrak{n}}'\,=\,\frac{\big\langle(\mathcal{F}_j)_{\Sigma_F}^{\mathrm{dep},\#},\Phi(\mathbf{g}\cdot \mathbf{E}_{1})\big|J_{\mathfrak{n}}\big\rangle_{\mathfrak{n}}}{\Omega_{\mathfrak{n},\Sigma}(\mathcal{F}_j)}.\end{align*}

The numerator can be readily evaluated using Proposition 2·3, in which case

\begin{align*}\phi_{\mathfrak{n},\mathbf{F}}^{\mathrm{alg}}(\mathbf{h})\,=\,\,\mathfrak{u}_{\mathfrak{m}'}(\mathcal{F}_j)\cdot\frac{\upsilon(\mathbf{g})\!\cdot\! \mathcal{N}\big(\mathfrak{c}(\mathbf{g}) \mathfrak{d}^2\big)^{\frac{1}{2}}}{\alpha_{\mathfrak{p}}(\mathcal{F}_j)^{\mathrm{ord}_{\mathfrak{p}}\mathfrak{c}(\mathbf{g})}}\! \cdot \frac{\Psi_{\Sigma_F}\big(\mathcal{F}_j,\mathbf{g}^{\#},1\big)}{\Omega_{\mathfrak{n},\Sigma}(\mathcal{F}_j)} \, .\end{align*}

By assumption, $C\big((\mathcal{F}_1)_{\Sigma_F}^{\mathrm{dep}},\mathfrak{m}\big) \equiv C\big((\mathcal{F}_2)_{\Sigma_F}^{\mathrm{dep}},\mathfrak{m}\big)\!\!\mod\wp^r\cdot R$ at every $\mathcal{O}_F$ -ideal $\mathfrak{m}$ . Thus the algebraic pairing $\big((\mathcal{F}_j)_{\Sigma_F}^{\mathrm{dep}},\,-\,\big)_{\mathfrak{n}}$ must preserve congruences modulo $\wp^r$ when we switch between indices $j=1$ and $j=2$ , and the result follows easily.

2·4. Elliptic curves over totally real fields

Let $E_1$ and $E_2$ be two modular elliptic curves defined over the totally real field F, and assume they are $\mathfrak{p}$ -congruent. Again one supposes that the prime $p>2$ is inert in F, so that $p\cdot\mathcal{O}_F=\mathfrak{p}$ say. Lastly, we ensure that both elliptic curves $E_1$ and $E_2$ either have good ordinary or bad multiplicative reduction over $F_{\mathfrak{p}}$ , thence $a_{\mathfrak{p}}(E_1)$ and $a_{\mathfrak{p}}(E_2)$ are p-adic units. We shall write $\mathcal{F}_1^{\dagger}$ and $\mathcal{F}_2^{\dagger}$ to denote the Hilbert modular cusp forms of parallel weight 2 and trivial nebentypus, associated to the curves $E_1$ and $E_2$ respectively: to be consistent, their $\mathfrak{p}$ -stabilisations will be denoted by $\mathcal{F}_1$ and $\mathcal{F}_2$ as before.

We also suppose $\tau\,{:}\,G_{F}\rightarrow \mathrm{GL}_2(\overline{\mathbb{Q}})$ is a two-dimensional Artin representation induced from a Hecke character over K, where $K/F$ is a CM-extension of fields. One can consider the weight one theta-series, $\mathcal{G}_{\tau}$ , arising from $\tau$ in [ Reference Shimura29 , section 5], so that

\begin{align*}L\big(\mathcal{G}_{\tau},s\big) \,=\!\prod_{\mathfrak{q}\in\mathrm{Spec}(\mathcal{O}_F)} \,\frac{1}{\mathrm{det}\big(1-\tau(\mathrm{Frob}_{\mathfrak{q}})\cdot\mathcal{N}(\mathfrak{q})^{-s}\big)}\quad\text{for all}\, s\in\mathbb{C} \,\text{with}\, \mathrm{Re(s)\gg 0}.\end{align*}

In particular, one has $\mathcal{G}_{\tau}\in\mathcal{M}_1\big(\mathfrak{c}(\mathcal{G}_{\tau}),\varpi\big)$ where $\varpi=\mathrm{det}(\tau)$ and $\mathfrak{c}(\mathcal{G}_{\tau})=\mathrm{cond}(\tau)$ . To make any further progress, we must restrict the allowable levels of the forms $\mathcal{F}_1$ , $\mathcal{F}_2$ and $\mathcal{G}_{\tau}$ appropriately, in order to apply our preceding congruence formulae.

Hypothesis 2·11. For every prime ideal $\mathfrak{q}\in\Sigma^{\mathrm{bad}}$ , either:

\begin{eqnarray*}\textrm{(i)} & & \mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(E_1)\big)=\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(E_2)\big)=\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\overline{\rho})\big); \quad\mathrm{or} \\\textrm{(ii)} & & \mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\overline{\rho})\big)=0,\, \mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(E_1)\big)\leqslant 2\,\,\mathrm{and} \,\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(E_2)\big)\leqslant 2; \quad\mathrm{or} \\\textrm{(iii)} & & 1=\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\overline{\rho})\big)\leqslant \mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(E_j)\big)\leqslant \mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\overline{\rho})\big)+1 \,\mathrm{at each $j\in\{1,2\}$.}\end{eqnarray*}

Moreover, if $\mathfrak{q}^2\big|\mathfrak{c}(\mathcal{F}_1)$ or $\mathfrak{q}^2\big|\mathfrak{c}(\mathcal{F}_2)$ then $\mathrm{ord}_{\mathfrak{q}}\big(\mathfrak{c}(\mathcal{F}_j)\big)\leqslant \mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\tau)\big)$ for $j\in\{1,2\}$ .

The first three conditions are symmetric in $E_1$ and $E_2$ , and imply that the levels of the eigenforms $\mathcal{F}_j$ are close to each other, as well as to the Artin conductor of $\overline{\rho}$ . The fourth condition will be satisfied whenever the Hecke character $\vartheta_K$ that induces $\tau$ has its conductor sufficiently divisible by the primes dividing $\mathfrak{c}(\mathcal{F}_1)\cdot \mathfrak{c}(\mathcal{F}_2)\cdot\mathcal{O}_K$ . Henceforth we will assume these conditions in Hypothesis 2·11 hold for the modular elliptic curves $E_1$ and $E_2$ , and the Artin representation $\tau\,{:}\,G_{F}\rightarrow \mathrm{GL}_2(\overline{\mathbb{Q}})$ . The following technical result guarantees that Theorem 2·10 is applicable here.

Lemma 2·12. If $\theta\neq\mathbf{1}_F$ is a finite order character factoring through $F(\mu_{p^{\infty}})/F$ and if we adopt the following choice of finite places

\begin{align*}\Sigma_F \,{:\!=}\, \Big\{\mathfrak{q}\in\mathrm{Spec}(\mathcal{O}_F)\,\mathrm{such that}\,\,\mathfrak{q}\big|\mathfrak{c}(\mathcal{F}_1)\cdot \mathfrak{c}(\mathcal{F}_2)\cdot \mathfrak{c}(\mathcal{G}_{\tau})\,\mathrm{and}\, \mathfrak{q}\neq\mathfrak{p} \Big\} ,\end{align*}

then Hypotheses 2·2(i)-(iv) and 2·4(i)-(iii) hold for $(\mathcal{F}_1,\mathcal{G}_{\tau}\otimes\theta)$ and $(\mathcal{F}_2,\mathcal{G}_{\tau}\otimes\theta)$ .

All seven required conditions have now been verified, hence the proof is finished.

Theorem 2·13. Under Hypothesis 2·11 and for the same choice of $\Sigma_F$ ,

\begin{align*}\mathbf{L}_{p,\Sigma_F}^{\mathrm{an},\Omega_{\mathfrak{n}}} \big(E_1,\tau\otimes\omega^t\big)\,\equiv\, \mathbf{L}_{p,\Sigma_F}^{\mathrm{an},\Omega_{\mathfrak{n}}} \big(E_2,\tau\otimes\omega^t\big) \mod\mathfrak{p}\end{align*}

at each separate branch $t\in\{0,\dots,p-2\}$ .

Proof. We begin by choosing the integer $r\geqslant 1$ such that $\mathfrak{p}\cdot\mathcal{O}_{\mathbb{C}_p}=\wp^r\cdot\mathcal{O}_{\mathbb{C}_p}$ , which means that $C\big((\mathcal{F}_1)_{\Sigma_F}^{\mathrm{dep}},\mathfrak{m}\big) \equiv C\big((\mathcal{F}_2)_{\Sigma_F}^{\mathrm{dep}},\mathfrak{m}\big)\!\!\mod\wp^r$ for every ideal $\mathfrak{m}\lhd \mathcal{O}_F$ . Now at any non-trivialFootnote ³ Hecke character $\theta$ of $\mathfrak{p}$ -power conductor with $\theta\big|_{\mathbb{F}_p^{\times}}=\omega^t$ ,

\begin{eqnarray*}\theta\left(\mathbf{L}_{p,\Sigma_F}^{\mathrm{an},\Omega_{\mathfrak{n}}} \big(E_j,\tau\otimes\omega^t\big)\right)\,=\,\frac{\epsilon_F(\tau\otimes\theta,0)}{\alpha_{\mathfrak{p}}^{\mathrm{ord}_{\mathfrak{p}}(\mathrm{cond}(\tau\otimes\theta))}}\cdot\mathcal{E}_{\mathfrak{p}}\big(E_j,\tau\otimes\theta\big)\cdot \frac{\Psi_{\Sigma_F}\big(E_j,\tau^*\otimes\theta^{-1},1\big)}{\Omega_{\mathfrak{n},\Sigma}(\mathcal{F}_j)} & & \\\,=\, i^{[F\,{:}\,\mathbb{Q}]} \cdot\upsilon(\mathcal{G}_{\tau}\otimes\theta)\cdot \frac{ \mathcal{N}\big(\mathfrak{c}(\mathcal{G}_{\tau}\otimes\theta)\cdot \mathfrak{d}^2\big)^{\frac{1}{2}}}{\alpha_{\mathfrak{p}}(\mathcal{F}_j)^{\mathrm{ord}_{\mathfrak{p}}\mathfrak{c}(\mathcal{G}_{\tau}\otimes\theta)}}\! \cdot \frac{\Psi_{\Sigma_F}\big(\mathcal{F}_j,(\mathcal{G}_{\tau}\otimes\theta)^{\#},1\big)}{\Omega_{\mathfrak{n},\Sigma}(\mathcal{F}_j)} & &\end{eqnarray*}

where the modified factor at $\mathfrak{p}$ is $\mathcal{E}_{\mathfrak{p}}\big(E_j,\tau\otimes\theta\big)\!=\!\Big(1-\frac{\theta^{-1}(\mathfrak{p})\cdot\beta_{\mathfrak{p}}}{\alpha_{\mathfrak{p}}(\mathcal{F}_j)}\Big)\Big(1-\frac{\theta^{-1}(\mathfrak{p})\cdot\beta_{\mathfrak{p}}'}{\alpha_{\mathfrak{p}}(\mathcal{F}_j)}\Big)$ . Here we have combined from [ Reference Delbourgo6 , lemma B·1] the standard epsilon-factor identity

\begin{align*}\epsilon_F(\tau\otimes\theta,s)\Big|_{s=0} \,=\,\,i^{[F\,{:}\,\mathbb{Q}]}\cdot D_F\cdot \sqrt{\mathcal{N}_{F/\mathbb{Q}}\big(\mathfrak{c}(\mathcal{G}_{\tau}\otimes\theta)\big)}\cdot \upsilon(\mathcal{G}_{\tau}\otimes\theta)\end{align*}

together with the property that $\mathcal{E}_{\mathfrak{p}}\big(E_j,\tau\otimes\theta\big)=1$ whenever one twists by $\theta\neq\mathbf{1}_F$ .

Lemma 2·12 confirms the congruence in Theorem 2·10 is valid for the family of weight one twisted HMFs $\mathcal{G}_{\tau}\otimes\theta$ with $\theta\neq\mathbf{1}_F$ , and it follows immediately that

\begin{align*}\mathfrak{u}_{\mathfrak{m}'}(\mathcal{F}_1)\cdot\theta\left(\mathbf{L}_{p,\Sigma_F}^{\mathrm{an},\Omega_{\mathfrak{n}}} \big(E_1,\tau\otimes\omega^t\big)\right)\equiv\mathfrak{u}_{\mathfrak{m}'}(\mathcal{F}_2)\cdot\theta\left(\mathbf{L}_{p,\Sigma_F}^{\mathrm{an},\Omega_{\mathfrak{n}}} \big(E_2,\tau\otimes\omega^t\big)\right)\!\!\mod\wp^r\cdot\mathcal{O}_{\mathbb{C}_p} .\end{align*}

Of course, we have been forced to omit the twist by the trivial character $\theta=\mathbf{1}_F$ . Fortunately the (punctured) set of finite order characters

\begin{align*}\mathrm{Hom}_{\mathrm{cont}}\Big(\mathrm{Gal}\big(F(\mu_{p^{\infty}})/F\big),\overline{\mathbb{Q}}_p^{\times}\Big)_{\mathrm{tors}}\!-\big\{\mathbf{1}_F\big\}\end{align*}

is p-adically dense inside the space parameterising the cyclotomic deformations of the double product $\mathcal{F}_j\otimes(\mathcal{G}_{\tau}\otimes\theta)^{\#}$ . Using properties of continuity, one can therefore extend the mod $\wp^r$ congruence above to include the p-adic L-functions themselves.

Lastly, the scalar terms from Theorem 2·10 satisfy $\mathfrak{u}_{\mathfrak{m}'}(\mathcal{F}_1)\equiv\mathfrak{u}_{\mathfrak{m}'}(\mathcal{F}_2)\!\mod\mathfrak{p}$ because of the congruence $\alpha_{\mathfrak{p}}(\mathcal{F}_1)\equiv \alpha_{\mathfrak{p}}(\mathcal{F}_2)\!\mod\mathfrak{p}$ , and the result follows.

2·5. Canonical periods and Ihara’s lemma

The reader may have noticed that the periods $\Omega_{\mathfrak{n},\Sigma}(\mathcal{F}_j)$ differ from the automorphic periods associated to $E_j\,\!_{/F}$ . Let us briefly recall from Definition 2·6 that

\begin{align*}\Omega_{\mathfrak{c}(\mathcal{F}_j),\emptyset}(\mathcal{F}_j)\,=\,\frac{\langle\mathcal{F}_j,\mathcal{F}_j\rangle_{\mathfrak{c}(\mathcal{F}_j)}'}{(\mathcal{F}_j,\mathcal{F}_j)_{\mathfrak{c}(\mathcal{F}_j)}} \,=\,\frac{\langle\mathcal{F}_j,\mathcal{F}_j|J_{\mathfrak{c}(\mathcal{F}_j)}\rangle_{\mathfrak{c}(\mathcal{F}_j)}}{(\mathcal{F}_j,\mathcal{F}_j)_{\mathfrak{c}(\mathcal{F}_j)}} \, .\end{align*}

The numerator is the Petersson inner product of $\mathcal{F}_j^{\dagger}$ with itself (at weight two), and the denominator is the congruence number $\mathfrak{h}_{E_j}=(\mathcal{F}_j,\mathcal{F}_j)_{\mathfrak{c}(\mathcal{F}_j)}$ for the HMF. Therefore, if we divide through by this value, the p-adic L-function for $E_j\otimes\tau$ has $\mathfrak{h}_{E_j}$ appearing in its numerator, and also $\Omega_{E_j/F}^{\mathrm{Pet}}\,{:\!=}\,\langle\mathcal{F}_j^{\dagger},\mathcal{F}_j^{\dagger}\rangle_{\mathfrak{c}(\mathcal{F}_j)}$ in its denominator.

To work in more generality, suppose that $\mathbf{F}$ is a Hilbert modular eigenform of parallel weight $k\geqslant 2$ , character $\psi_{\mathbf{F}}$ and level $\mathfrak{c}$ , with maximal ideal $\mathcal{M}\lhd\mathbb{T}_{\mathfrak{c}}$ . Applying the Eichler–Shimura–Harder isomorphism, an R-valued Hecke eigenform $\mathbf{F}$ corresponds to a d-cocycle in $\delta(\mathbf{F})\in H_{\mathrm{cusp}}^d\big(Y_1(\mathfrak{c})^{\mathrm{an}},V_{k-2}(R)\big)$ where $Y_1(\mathfrak{c})^{\mathrm{an}}$ is the (analytic) Hilbert modular variety of level $K_1(\mathfrak{c})$ , secondly $V_{k-2}(R)$ denotes the sheaf of homogeneous polynomials of degree $k-2$ , and lastly $d=[F\,{:}\,\mathbb{Q}]$ as before. In fact, if $k>2$ then $H_{\mathrm{cusp}}^d\big(Y_1(\mathfrak{c})^{\mathrm{an}},V_{k-2}(R)\big)\cong H_{!}^d\big(Y_1(\mathfrak{c})^{\mathrm{an}},V_{k-2}(R)\big)$ where the latter R-module denotes the image of the cuspidal cohomology inside the full $H^d$ .

Hida [ Reference Hida19 ] proved for $k>2$ that $\delta({-})$ induces an isomorphismFootnote ⁴ of $\mathbb{T}_{\mathfrak{c}}(R)$ -modules

\begin{align*}\bigoplus_{\mathfrak{c}'\mathfrak{c}''|\mathfrak{c}}\,\bigoplus_{\psi} \,\mathcal{S}_k^{\mathrm{new}}\big(\mathfrak{c}',\psi;\,R\big)\big|\mathfrak{c}''\,\buildrel \sim\over\longrightarrow\, H_{!}^d\big(Y_1(\mathfrak{c})^{\mathrm{an}},V_{k-2}(R)\big)\end{align*}

where $\psi$ runs over primitive finite order Hecke characters of conductor dividing $\mathfrak{c}$ . Note also that the twisted Poincaré pairing (e.g. see [ Reference Dimitrov11 , equation (6)] for details)

\begin{align*}[-,-]_{\mathfrak{c}}\,{:}\,H_{!}^d\big(Y_1(\mathfrak{c})^{\mathrm{an}},V_{k-2}(R)\big)\times\, H_{!}^d\big(Y_1(\mathfrak{c})^{\mathrm{an}},V_{k-2}(R)\big) \longrightarrow R\end{align*}

is compatible with the algebraic pairing $({-},-)_{\mathfrak{c}}\,{:}\,\mathcal{S}_k(\mathfrak{c},R)_{\mathcal{M}}^{\times 2}\rightarrow R$ under $\delta\times \delta$ . Further, $H_{!}^d\big(Y_1(\mathfrak{c})^{\mathrm{an}},V_{k-2}(R)\big)_{\mathcal{M}}$ is free of rank $2^d$ over $\mathbb{T}_{\mathfrak{c}}(R)_{\mathcal{M}}$ by [ Reference Dimitrov11 , theorem 6·7].

Using exactly the same argument as described in [ Reference Ray, Sujatha and Vatsal27 , section 2·8] over $F=\mathbb{Q}$ , one can deduce that both periods $\Omega_{\mathfrak{n},\Sigma}(\mathbf{F})$ and $\Omega_{\mathfrak{c}(\mathcal{F}_j),\emptyset}(\mathbf{F})$ coincide (up to a unit) if the following two conditions hold:

1. (i) for the dual lattices, $\mathrm{dep}_{\mathfrak{q}}^*\big(L(\mathfrak{c},R)^*\big)\subset L(\mathfrak{c}\mathfrak{q}^{\mathrm{ord}_{\mathfrak{q}}(\mathfrak{n})},R)^*$ at all $\mathfrak{q}\in\Sigma_F$ ;

2. (ii) the quotient module $\frac{L(\mathfrak{c}\mathfrak{q}^{\mathrm{ord}_{\mathfrak{q}}(\mathfrak{n})},R)^*}{\mathrm{dep}_{\mathfrak{q}}^*(L(\mathfrak{c},R)^*)}$ is R-torsion free at every $\mathfrak{q}\in\Sigma_F$ .

Whilst the inclusion of $\mathrm{dep}_{\mathfrak{q}}^*\big(L(\mathfrak{c},R)^*\big)$ inside $L(\mathfrak{c}\mathfrak{q}^{\mathrm{ord}_{\mathfrak{q}}(\mathfrak{n})},R)^*$ can be established by using the Hecke-equivariance of the twisted pairings $[-,-]$ at levels $\mathfrak{c}$ and $\mathfrak{c}\mathfrak{q}^{\mathrm{ord}_{\mathfrak{q}}(\mathfrak{n})}$ , the second statement above requires an appropriate version of Ihara’s lemma.

If $\mathfrak{q}\big|\!\big|\mathfrak{n}$ and $k>2$ , a result of Dimitrov [ Reference Dimitrov12 , theorem 3·1] implies that the map

\begin{align*}\mathrm{id}^*\!+\!\big({-}\big|\mathfrak{q}\big)^*\,{:}\, H^d\big(Y_1(\mathfrak{c})^{\mathrm{an}},V_{k-2}(R)\big)^{\oplus 2}\rightarrow H^d\big(Y_1(\mathfrak{c}\mathfrak{q})^{\mathrm{an}},V_{k-2}(R)\big)\end{align*}

is injective with flat cokernel, so after localisation one obtains the second statement. Alternatively, if $\mathfrak{q}\big|\!\big|\mathfrak{n}$ and $k=2$ then one can likely deduce the second statement from a mod p version established by Manning and Shotton in [ Reference Manning and Shotton25 , theorem 6·8].

However if $\mathfrak{q}^2\big|\mathfrak{n}$ then it is still open as to whether the second statement is true, to the best of our knowledge. Fortunately, some important new work of Maletto (thesis in preparation) should treat the remaining cases, thereby removing all the Ihara’s lemma hypotheses from the theorems in this paper and from [ Reference Ray, Sujatha and Vatsal27 ] as well.

3·1. $\Lambda$ -structure of the cyclotomic Selmer group

Fix an odd prime p. If $E=E_1$ or $E=E_2$ is one of our elliptic curves (over the totally real field F) then the p-adic Tate module is given by the projective limit $\mathrm{Ta}_p(E)=\varprojlim_{n}E[p^n]$ . We relax our assumption that p is inert in F and no longer insist $p\nmid D_F$ . Provided E has good ordinary or bad multiplicative reduction at each place $\mathfrak{p}\big|p$ , there exists a short exact sequence of discrete $\mathrm{Gal}\big(\overline{F}_{\mathfrak{p}}/F_{\mathfrak{p}}\big)$ -modules

\begin{align*}0\rightarrow \widehat{E}_{/\mathcal{O}_{F_{\mathfrak{p}}}}[p^{\infty}] \rightarrow E[p^{\infty}] \rightarrow \mathcal{C}_{\mathfrak{p}}^{\acute{\mathrm{e}}\mathrm{t}}\rightarrow 0,\end{align*}

where $ \widehat{E}_{/\mathcal{O}_{F_{\mathfrak{p}}}}$ is the height one formal group, and $\mathcal{C}_{\mathfrak{p}}^{\acute{\mathrm{e}}\mathrm{t}}$ is an étale p-divisible group. For example, if E has good ordinary reduction at $\mathfrak{p}$ then $\mathcal{C}_{\mathfrak{p}}^{\acute{\mathrm{e}}\mathrm{t}}\cong \widetilde{E}[p^{\infty}]$ where $\widetilde{E}_{/\mathbb{F}}$ denotes the reduced (non-singular) curve defined over $\mathbb{F}\,{:\!=}\,\mathcal{O}_F\big/\mathfrak{p}$ .

Recall that $\tau\,{:}\,G_F\rightarrow\mathrm{GL}_2(\overline{\mathbb{Q}})$ was the two-dimensional Artin representation induced from a Hecke character $\vartheta_K$ over the CM-field K, so that $\tau=\mathrm{Ind}_F^K(\vartheta_K)$ . By enlarging the finite field $\mathbb{F}$ if necessary, we may assume that $\mathrm{Im}(\overline{\tau})\subset\mathrm{GL}_2(\mathbb{F})$ . Taking Teichmüller twists by $\omega^t$ with $t\in\{0,\dots,p-2\}$ , the Galois representation

\begin{align*}\rho_{E}\otimes\tau\otimes\omega^t\,{:}\,G_F\rightarrow \mathrm{GL}_4(\overline{\mathbb{Q}}_p)\end{align*}

has an associated free rank four R-module, $\mathbf{T}_{\omega^t}^{(\tau)}(E)$ , equipped with a $G_F$ -action:

\begin{align*}\mathbf{T}_{\omega^t}^{(\tau)}(E)\,{:\!=}\,\mathrm{Ta}_p(E)\otimes \mathrm{Ind}_F^K(\vartheta_K)\otimes\omega^t .\end{align*}

We write $\mathcal{E}_{\omega^t}^{(\tau)} =\mathbb{Q}_p/\mathbb{Z}_p\otimes_{\mathbb{Z}_p} \mathbf{T}_{\omega^t}^{(\tau)}(E)$ to denote the corresponding p-primary object. Viewed as a local $G_{F_{\mathfrak{p}}}$ -module, at each place $\mathfrak{p}\big|p$ one sets

\begin{align*}\mathrm{Fil}_{\mathfrak{p}}^+\mathbf{T}_{\omega^t}^{(\tau)}(E) \,{:\!=}\, \mathrm{Ta}_p\big(\widehat{E}_{/\mathcal{O}_{F_{\mathfrak{p}}}}\big)\otimes \mathrm{Ind}_F^K(\vartheta_K)\otimes\omega^t\quad\mathrm{and}\quad\mathrm{Fil}_{\mathfrak{p}}^-\mathbf{T}_{\omega^t}^{(\tau)}(E) \,{:\!=}\, \frac{\mathbf{T}_{\omega^t}^{(\tau)}(E)}{\mathrm{Fil}_{\mathfrak{p}}^+\mathbf{T}_{\omega^t}^{(\tau)}(E)}\end{align*}

with analogous notation for the p-divisible versions, namely $\mathrm{Fil}_{\mathfrak{p}}^{+}\mathcal{E}_{\omega^t}^{(\tau)}$ and $\mathrm{Fil}_{\mathfrak{p}}^{-}\mathcal{E}_{\omega^t}^{(\tau)}$ .

Definition 3·1. For each finite subset $\Sigma=\Sigma_F\subset\mathrm{Spec}(\mathcal{O}_F)$ not containing the places above p, we define $\mathrm{Sel}_{\omega^t}^{\Sigma}\big(E,\tau\big)$ to be the kernel of the restriction maps

\begin{align*}H^1\big(\overline{F}/F^{\mathrm{cyc}},\,\mathcal{E}_{\omega^t}^{(\tau)}\big) \,\buildrel {\prod\mathrm{res}_{\nu}}\over\longrightarrow\!\prod_{\nu\not\in\Sigma^{\mathrm{cyc}}\cup\{\mathfrak{p}\mid p\}} \!\!\!\! H^1\big(F_{\nu}^{\mathrm{cyc}},\,\mathcal{E}_{\omega^t}^{(\tau)}\big)\times\,\prod_{\mathfrak{p}\mid p} H^1\big(I_{F_{\mathfrak{p}}^{\mathrm{cyc}}},\,\mathrm{Fil}_{\mathfrak{p}}^{-}\mathcal{E}_{\omega^t}^{(\tau)}\big)\end{align*}

which is then a module over the Iwasawa algebra $\Lambda=R\big[\!\big[\mathrm{Gal}(F^{\mathrm{cyc}}\!/F)\big]\!\big]\cong R[\![X]\!]$ . Here we have written $\Sigma^{\mathrm{cyc}}$ to denote the (finite) set of places of $F^{\mathrm{cyc}}$ lying over $\Sigma$ .

The Pontrjagin dual $\mathfrak{X}_{\omega^t}^{\Sigma}\big(E,\tau\big)\,{:\!=}\,\mathrm{Hom}_{\mathrm{cont}}\big( \mathrm{Sel}_{\omega^t}^{\Sigma}\big(E,\tau\big), \mathbb{Q}_p/\mathbb{Z}_p\big)$ is compact since the $\tau$ -twisted cyclotomic Selmer group above is discrete. One may therefore apply the structure theory of such Iwasawa modules to carefully analyse $\mathfrak{X}_{\omega^t}^{\Sigma}\big(E,\tau\big)$ . In particular, we will study the irreducible factors in its characteristic power series. Let us again recall that we fixed a uniformiser $\wp$ of the discrete valuation ring R. The following technical lemma provides a summary of the results we need later.

Proof. Statements (i)-(iii) follow from [ Reference Delbourgo9 , proposition 4·3], which is itself a generalisation of Greenberg’s argument in [ Reference Greenberg16 , proposition 4·14] for elliptic curves. We should point out that in (iii) the vanishing condition $H^0\big(F,\mathbf{T}_{\omega^t}^{(\tau)}(E)\otimes\mathbb{Q}\big)=\{0\}$ is required in order to be able to apply [ Reference Delbourgo9 , 4·3(c)]; in fact if $K'=\overline{\mathbb{Q}}^{\mathrm{Ker}(\vartheta_K)}\!$ then

\begin{align*}H^0\big(F,\mathbf{T}_{\omega^t}^{(\tau)}(E)\otimes\mathbb{Q}\big)\,\hookrightarrow\,H^0\big(K'(\mu_p),\mathbf{T}_{\omega^t}^{(\tau)}(E)\otimes\mathbb{Q}\big)\cong H^0\big(K'(\mu_p),V_{\rho_E}\otimes_{\mathbb{Z}_p}R\big)^{\oplus 2}\end{align*}

and the right-hand group is zero, as the residual representation $\overline{\rho}_E\,{:}\,G_{F}\rightarrow\mathrm{GL}_2(\mathbb{F}_p)$ arising from the action on the p-torsion points is absolutely irreducible.

To show (iv), we use the reduction formula in [ Reference Livné22 , section 1] for the Swan conductor:

\begin{align*}\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\rho_E\otimes\tau)\big)+\mathrm{dim}\big(V_{\rho_E\otimes\tau}^{I_{F_{\mathfrak{q}}}}\big) =\,\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\overline{\rho}_E\otimes\overline{\tau})\big)+ \mathrm{dim}_{\mathbb{F}}\big(\mathcal{E}_{\omega^0}^{(\tau)}[\wp]^{I_{F_{\mathfrak{q}}}}\big) .\end{align*}

The stated condition in (iv) instantly implies that $\mathrm{dim}_{\mathbb{F}}\big(\mathcal{E}_{\omega^0}^{(\tau)}[\wp]^{I_{F_{\mathfrak{q}}}}\big)=\mathrm{dim}\big(V_{\rho_E\otimes\tau}^{I_{F_{\mathfrak{q}}}}\big)$ and as a consequence, the inertia invariants $H^0\big(I_{F_{\mathfrak{q}}},\mathcal{E}_{\omega^0}^{(\tau)}\big)$ are $\wp$ -divisible.

Clearly $H^0\big(I_{F_{\mathfrak{q}}},\mathcal{E}_{\omega^0}^{(\tau)}\big)\cong H^0\big(I_{F_{\mathfrak{q}}},\mathcal{E}_{\omega^t}^{(\tau)}\big)$ for all of the $\omega^t$ -twists because $\omega\big|_{G_F}$ is unramified at $\mathfrak{q}$ . Secondly, as $\mathfrak{q}$ does not ramify in the cyclotomic $\mathbb{Z}_p$ -extension, $H^0\big(I_{F^{\mathrm{cyc}}_{\nu}},\mathcal{E}_{\omega^t}^{(\tau)}\big)=H^0\big(I_{F_{\mathfrak{q}}},\mathcal{E}_{\omega^t}^{(\tau)}\big)$ must be $\wp$ -divisible. Lastly we may conclude that

\begin{align*}H^0\big(F^{\mathrm{cyc}}_{\nu},\mathcal{E}_{\omega^t}^{(\tau)}\big)\,=\,H^0\big(I_{F^{\mathrm{cyc}}_{\nu}},\mathcal{E}_{\omega^t}^{(\tau)}\big)^{\mathrm{Frob}_{\mathfrak{q}}=1}\end{align*}

is also a $\wp$ –divisible group, since taking $\mathrm{Frob}_{\mathfrak{q}}$ -invariants preserves this property.

This condition (iv) on the conductors plays a pivotal role later in our work. Indeed the existence of an elliptic curve congruent to E modulo p and satisfying property (iv) at primes $\mathfrak{q}\in\mathrm{Spec}(\mathcal{O_F})$ not lying above p will prove to be invaluable.

Whilst we have been unable to establish that this $\tau$ -optimal congruent elliptic curve $E^{\mathrm{opt}}_{/F}$ exists over general totally real fields F, if we consider exclusively the situation from the Introduction, then one can prove the existence of such a curve.

Fix a place $\mathfrak{Q}\in\mathrm{Spec}(\mathcal{O}_K)$ of the CM-field which lies above a prime $q\neq p$ say, and let $\mathfrak{q}\in\mathrm{Spec}(\mathcal{O}_F)$ be the unique prime ideal lying under $\mathfrak{Q}$ . If $\mathfrak{q}\nmid \mathrm{cond}(\tau)$ then $\mathfrak{q}\nmid D_K$ and q is unramified in $K/\mathbb{Q}$ . Moreover, it is automatically true [ Reference Livné22 ] that

(3·1) \begin{equation}\mathrm{ord}_{\mathfrak{Q}}\big(\mathrm{cond}\big(\rho_{E^{\mathrm{opt}}}\big|_{G_K}\big)\big) \geqslant\mathrm{ord}_{\mathfrak{Q}}\big(\mathrm{cond}\big(\overline{\rho}_E\big|_{G_K}\big)\big) .\end{equation}

If (3·1) were a strict inequality, one would find $\mathrm{ord}_{q}\big(\mathrm{cond}(\rho_{E^{\mathrm{opt}}})\big)>\mathrm{ord}_{q}\big(\mathrm{cond}(\overline{\rho}_E)\big)$ because $E_{/\mathbb{Q}}^{\mathrm{opt}}$ has the same reduction type (good, multiplicative or additive) at q as $E_{/K}^{\mathrm{opt}}$ does at the place $\mathfrak{Q}$ since the extension of local fields $K_{\mathfrak{Q}}/\mathbb{Q}_q$ is unramified – but this contradicts the minimality property of the $\mathbb{Q}$ -optimal curve $E^{\mathrm{opt}}$ stated in the previous paragraph, hence we deduce that (3·1) is an actual equality.

Let us next recall that $\tau=\mathrm{Ind}_F^K(\vartheta_K)$ where $\vartheta_K$ was a finite order Hecke character. Because $\mathfrak{q}\nmid \mathrm{cond}(\tau)$ it follows that $\vartheta_K$ is unramified at $\mathfrak{Q}$ , whence

as the $\mathfrak{Q}$ -part of the conductors does not changeFootnote ⁵ if we twist by the character $\vartheta_K$ . Taguchi [ Reference Taguchi30 , p·2866] has shown that if one induces from K down to $F=K^+$ , the conductors of both $\rho_{E^{\mathrm{opt}}}\big|_{G_F}\otimes\tau\cong\mathrm{Ind}_F^K\big(\rho_{E^{\mathrm{opt}}}\big|_{G_K}\otimes\vartheta_K\big)$ and its residual version satisfy precisely the same descent formula, so we therefore conclude that $\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\rho_{E^{\mathrm{opt}}}\big|_{G_F}\otimes\tau)\big)=\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\overline{\rho}_E\big|_{G_F}\otimes\overline{\tau})\big)$ at all primes $\mathfrak{q}\nmid p\cdot \mathrm{cond}(\tau)$ .

3·2. $\mathbb{F}$ -rank of the residual Selmer group

We now introduce a residual version of the $\Lambda$ -module ‘ $\mathrm{Sel}_{\omega^t}^{\Sigma}(E,\tau)$ ’ which has the structure of an $\mathbb{F}$ -vector space. Recall that $E=E_1$ , $E=E_2$ , and (if it exists) $E=E_{/F}^{\mathrm{opt}}$ , each share a common mod p representation $\overline{\rho}_E\,{:}\,G_F\rightarrow\mathrm{GL}_2(\mathbb{F}_p)$ . One can then consider a tensor product of Galois representations (in characteristic p),

\begin{align*}\overline{\rho}_{E}\otimes\overline{\tau}\otimes\omega^t\,{:}\,G_F\rightarrow \mathrm{Aut}_{\mathbb{F}}\big(\overline{\mathcal{V}}_{\omega^t}\big)\cong\mathrm{GL}_4(\mathbb{F}),\end{align*}

where $\overline{\mathcal{V}}_{\omega^t}\cong {1}/{p}\cdot\mathbb{Z}_p\otimes_{\mathbb{Z}_p}\mathbf{T}_{\omega^t}^{(\tau)}(E)$ denotes the underlying rank four $\mathbb{F}$ -vector space. N.B. We identify $\overline{\mathcal{V}}_{\omega^t}$ with the $\wp$ -torsion subgroup of $\mathcal{E}_{\omega^t}^{(\tau)} =\mathbb{Q}_p/\mathbb{Z}_p\otimes_{\mathbb{Z}_p} \mathbf{T}_{\omega^t}^{(\tau)}(E)$ .

At each place $\mathfrak{p}\big|p$ , there exists an exact sequence of local $\mathrm{Gal}\big(\overline{F}_{\mathfrak{p}}\big/F_{\mathfrak{p}}\big)$ -modules

\begin{align*}0\,\rightarrow\, \overline{\mathcal{V}}_{\omega^t}^{\mathfrak{p},+} \,\rightarrow\,\overline{\mathcal{V}}_{\omega^t} \,\rightarrow\, \overline{\mathcal{V}}_{\omega^t}^{\mathfrak{p},-}\,\rightarrow\, 0\end{align*}

satisfying $\mathrm{dim}_{\mathbb{F}}\big(\overline{\mathcal{V}}_{\omega^t}^{\mathfrak{p},+}\big)=\mathrm{dim}_{\mathbb{F}}\big(\overline{\mathcal{V}}_{\omega^t}^{\mathfrak{p},-}\big)=2$ where $\overline{\mathcal{V}}_{\omega^t}^{\mathfrak{p},\pm}\,{:\!=}\,\frac{1}{p}\cdot\mathbb{Z}_p\otimes_{\mathbb{Z}_p}\mathrm{Fil}_{\mathfrak{p}}^{\pm}\mathbf{T}_{\omega^t}^{(\tau)}(E)$ .

Definition 3·5. Let $\Sigma=\Sigma_F\subset\mathrm{Spec}(\mathcal{O}_F)$ be a finite subset not containing any of the primes above p, and write $\Sigma^{\mathrm{cyc}}$ for those places of $F^{\mathrm{cyc}}$ lying above $\Sigma$ . For $t\in\{0,\dots,p-2\}$ , we define the residual Selmer group $\mathrm{Sel}_{\omega^t}^{\Sigma}\big(\overline{\rho}_E\otimes\tau\big)$ to equal

\begin{align*}\mathrm{Ker}\bigg(H^1\big(F^{\mathrm{cyc}},\overline{\mathcal{V}}_{\omega^t}\big) \buildrel {\prod\mathrm{res}_{\nu}}\over\longrightarrow\!\!\prod_{\nu\nmid p,\,\nu\not\in\Sigma^{\mathrm{cyc}}} H^1\big(F_{\nu}^{\mathrm{cyc}},\overline{\mathcal{V}}_{\omega^t}\big)\,\times\,\prod_{\mathfrak{p}\mid p} H^1\big(I_{F_{\mathfrak{p}}^{\mathrm{cyc}}},\overline{\mathcal{V}}_{\omega^t}^{\mathfrak{p},-}\big)\bigg)\end{align*}

with the first product ranging over all places $\nu$ of $F^{\mathrm{cyc}}$ not lying above $\Sigma\cup\big\{\mathfrak{p}\big| p\big\}$ .

We can compare the residual Selmer group with the $\wp$ -torsion in the usual one. For each $i\geqslant 0$ and closed normal subgroup $G\lhd G_F$ , there is a short exact sequence

(3·2) \begin{equation}0\rightarrow H^i\big(G,\mathcal{E}_{\omega^t}^{(\tau)}\big)\big/\wp\buildrel \,\partial_i\over\longrightarrow\, H^{i+1}\big(G,\overline{\mathcal{V}}_{\omega^t}\big)\buildrel {\beta_{i+1}}\over\longrightarrow H^{i+1}\big(G,\mathcal{E}_{\omega^t}^{(\tau)}\big)[\wp]\rightarrow 0 .\end{equation}

In particular, one obtains a large commutative diagram with exact columns:

\begin{eqnarray*}& & \qquad\, 0 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad 0 \\& & \!\!\!\!\!\!\!\!\!\!\qquad\qquad \big\downarrow \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\big\downarrow \\& & \!\!\!\!\!\!\!\!\!\!H^0\big(F^{\mathrm{cyc}},\mathcal{E}_{\omega^t}^{(\tau)}\big)\big/\wp \quad\buildrel {\prod\mathrm{loc}_{\mathfrak{p}}^{(0)}}\over\longrightarrow\,\prod_{\nu\notin\Sigma^{\mathrm{cyc}}, \, \nu\nmid p}\! \big\{0\big\}\,\,\times\,\,\prod_{\mathfrak{p}\mid p} H^0\big(I_{F_{\mathfrak{p}}^{\mathrm{cyc}}},\mathrm{Fil}_{\mathfrak{p}}^{-}\mathcal{E}_{\omega^t}^{(\tau)}\big)\big/\wp \\& & \!\!\!\!\!\!\!\!\!\!\qquad\qquad \Big\downarrow\,_{\partial_0} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Big\downarrow\,_{(0,\partial_0^-)} \\& & \!\!\!\!\!\!\!\!\!\!H^1\big(F^{\mathrm{cyc}},\overline{\mathcal{V}}_{\omega^t}\big)\qquad\buildrel{\prod\mathrm{res}_{\nu}}\over\longrightarrow\quad\prod_{\nu\notin\Sigma^{\mathrm{cyc}}, \, \nu\nmid p}\!\!\!\! H^1\big(F_{\nu}^{\mathrm{cyc}},\mathcal{E}_{\omega^t}^{(\tau)}\big)[\wp]\times\prod_{\mathfrak{p}\mid p} H^1\big(I_{F_{\mathfrak{p}}^{\mathrm{cyc}}},\overline{\mathcal{V}}_{\omega^t}^{\mathfrak{p},-}\big) \\& & \!\!\!\!\!\!\!\!\!\!\qquad\qquad \Big\downarrow\,_{\beta_1} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\Big\downarrow\,_{(\mathrm{id},\beta_1^-)} \\& & \!\!\!\!\!\!\!\!\!\!H^1\big(F^{\mathrm{cyc}},\mathcal{E}_{\omega^t}^{(\tau)}\big)[\wp] \,\,\buildrel{\prod\mathrm{res}_{\nu}}\over\longrightarrow\!\!\prod_{\nu\notin\Sigma^{\mathrm{cyc}}, \, \nu\nmid p}\!\!\!\!\!\!\!\! H^1\big(F_{\nu}^{\mathrm{cyc}},\mathcal{E}_{\omega^t}^{(\tau)}\big)[\wp]\times\prod_{\mathfrak{p}\mid p} H^1\big(I_{F_{\mathfrak{p}}^{\mathrm{cyc}}},\mathrm{Fil}_{\mathfrak{p}}^{-}\mathcal{E}_{\omega^t}^{(\tau)}\big)[\wp] \\& & \!\!\!\!\!\!\!\!\!\!\qquad\qquad \big\downarrow \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\big\downarrow \\& & \qquad 0 \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\, 0\end{eqnarray*}

From Lemma 3·2(iv), if $\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\rho_E\otimes\tau)\big)=\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\overline{\rho}_E\otimes\overline{\tau})\big)$ for some $\mathfrak{q}\nmid p$ then $H^0\big(F_{\nu}^{\mathrm{cyc}},\mathcal{E}_{\omega^t}^{(\tau)}\big)\!$ is $\wp$ -divisible at all places $\nu\big| \mathfrak{q}$ , in which case

(3·3) \begin{equation}\beta_1\,{:}\, H^1\big(F_{\nu}^{\mathrm{cyc}},\overline{\mathcal{V}}_{\omega^t}\big)\buildrel \sim\over\longrightarrow H^1\big(F_{\nu}^{\mathrm{cyc}},\mathcal{E}_{\omega^t}^{(\tau)}\big)[\wp]\end{equation}

is an isomorphism (using the exactness of the sequence in (3·2) above at $i=0$ ). For example, if $E=E_{/F}^{\mathrm{opt}}$ is an optimal elliptic curve then this condition on the conductors holds for all $\mathfrak{q}\nmid p\cdot\mathrm{cond}(\tau)$ . More generally, if we choose $\Sigma=\Sigma_F$ ‘sufficiently large’ then $\beta_1$ will be an isomorphism on these local cohomology groups.

Lemma 3·6. If $\Sigma$ contains all $\mathfrak{q}$ with $\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\rho_E\otimes\tau)\big)>\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\overline{\rho}_E\otimes\overline{\tau})\big)$ then the induced mapping

\begin{align*}\beta_{1,*}\,{:}\,\mathrm{Sel}_{\omega^t}^{\Sigma}\big(\overline{\rho}_E\otimes\tau\big)\longrightarrow \mathrm{Sel}_{\omega^t}^{\Sigma}(E,\tau)\big[\wp\big]\end{align*}

yields an isomorphism of $\mathbb{F}$ -vector spaces.

Proof. From the previous discussion, at places $\nu\not\in\Sigma^{\mathrm{cyc}}$ not lying above p one deduces that $H^1\big(F_{\nu}^{\mathrm{cyc}},\overline{\mathcal{V}}_{\omega^t}\big)\cong H^1\big(F_{\nu}^{\mathrm{cyc}},\mathcal{E}_{\omega^t}^{(\tau)}\big)[\wp]$ . Applying the Snake Lemma to our large commutative diagram, one obtains the exact sequence

\begin{align*}0\rightarrow\, \prod_{\mathfrak{p}|p}\mathrm{Ker}\big(\mathrm{loc}_{\mathfrak{p}}^{(0)}\big)\,\buildrel \partial_{0,\star}\over\longrightarrow\, \mathrm{Sel}_{\omega^t}^{\Sigma}\big(\overline{\rho}_E\otimes\tau\big)\,\buildrel \beta_{1,*}\over\longrightarrow\,\mathrm{Sel}_{\omega^t}^{\Sigma}(E,\tau)[\wp]\,\rightarrow\, \prod_{\mathfrak{p}|p}\mathrm{Coker}\big(\mathrm{loc}_{\mathfrak{p}}^{(0)}\big) .\end{align*}

The kernel of $\mathrm{loc}_{\mathfrak{p}}^{(0)}$ vanishes at each place $\mathfrak{p}\big| p$ since $H^0\big(F,\overline{\mathcal{V}}_{\omega^t}\big)$ is identically zero. Furthermore, the vanishing of the cokernel of $\mathrm{loc}_{\mathfrak{p}}^{(0)}$ at each $\mathfrak{p}\big| p$ can be established by using exactly the same argument as in [ Reference Delbourgo9 , proof of 4·7(c)], so we are done.

Henceforth we will always choose our finite set of places $\Sigma\subset\mathrm{Spec}(\mathcal{O}_F)$ to include the primes dividing $\mathrm{cond}(\tau)$ , excluding of course the primes lying above p. One also fixes an elliptic curve $E_{/F}$ , and let us pick any branch $t\in\{0,\dots,p-2\}$ .

Proof. Let us begin by comparing the residual Selmer group with the one for the optimal elliptic curve $E_{/F}^{\mathrm{opt}}$ . Since $\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\rho_{E^{\mathrm{opt}}}\otimes\tau)\big)=\mathrm{ord}_{\mathfrak{q}}\big(\mathrm{cond}(\overline{\rho}_E\otimes\overline{\tau})\big)$ for all $\mathfrak{q}\nmid p\cdot\mathrm{cond}(\tau)$ , clearly Lemma 3·6 implies that there is an isomorphism

\begin{align*}\beta_{1,*}\,{:}\,\mathrm{Sel}_{\omega^t}^{\Sigma}\big(\overline{\rho}_E\otimes\tau\big)\buildrel \sim\over\longrightarrow \mathrm{Sel}_{\omega^t}^{\Sigma}\big(E^{\mathrm{opt}},\tau\big)\big[\wp\big] .\end{align*}

In particular, $\mathrm{Sel}_{\omega^t}^{\Sigma}\big(\overline{\rho}_E\otimes\tau\big)$ will be finite if and only if $\mathrm{Sel}_{\omega^t}^{\Sigma}\big(E^{\mathrm{opt}},\tau\big)\big[\wp\big]$ is finite.

Next, replacing the set $\Sigma$ by a possibly larger set $\Sigma'$ containing all primes $\mathfrak{q}'\neq p$ such that $\mathrm{ord}_{\mathfrak{q}'}\big(\mathrm{cond}(\rho_{E'}\otimes\tau)\big)>\mathrm{ord}_{\mathfrak{q}'}\big(\mathrm{cond}(\overline{\rho}_E\otimes\overline{\tau})\big)$ , one calculates that

The stated result (for the larger set $\Sigma'$ ) now follows directly from Lemma 3·2(ii). Finally to descend from $\Sigma'$ back down to $\Sigma$ , the following quotient module satisfies

\begin{align*}\frac{\mathfrak{X}_{\omega^t}^{\Sigma'}(E',\tau)}{\mathfrak{X}_{\omega^t}^{\Sigma}(E',\tau)}\,\,\cong\, \prod_{\nu|\mathfrak{q}\in\Sigma'-\Sigma} \mathrm{Hom}_{\mathrm{cont}}\Big(H^1 \big(F_{\nu}^{\mathrm{cyc}}, \mathcal{E}^{\prime{(\tau)}}_{\omega^{t}}\big),\mathbb{Q}_p/\mathbb{Z}_p\Big)\end{align*}

and each one of the right-hand groups is $\Lambda$ -torsion with a trivial $\mu$ -invariant.

3·3. Transition formulae for $\lambda$ -invariants

We should remind the reader that $E_1$ and $E_2$ denote a pair of p-congruent elliptic curves defined over the totally real field F, and that $\tau=\mathrm{Ind}_F^K(\vartheta_K)$ was a two-dimensional Artin representation. Throughout this section we shall assume the finiteness of $\mathrm{Sel}_{\omega^t}^{\Sigma}\big(\overline{\rho}_E\otimes\tau\big)$ . As we have already mentioned, this condition is equivalent to the cyclotomic Selmer groups attached to the curves $E_1$ and $E_2$ being both p-divisible and $\Lambda$ -cotorsion. Here our precise goal is to show that the algebraic analogue of Corollary 2·14 holds. Most of the technical results have already been established, so all that is left to do is essentially read off the Iwasawa invariants from the existing dimension formulae.

To establish that statement (b) is also true, an identical argument to that used in the proof of the previous corollary implies that

\begin{align*}\mathrm{dim}_{\mathbb{F}}\Big(\mathrm{Sel}_{\omega^t}^{\Sigma}(E,\tau)\big[\wp\big]\Big)\, =\,\mathrm{dim}_{\mathbb{F}}\Big(\mathrm{Sel}_{\omega^t}^{\emptyset}\big(\overline{\rho}_E\otimes\tau\big)\Big)+\sum_{\nu|\mathfrak{q}\in\Sigma}\mathrm{dim}_{\mathbb{F}}\Big(H^1\big(F_{\nu}^{\mathrm{cyc}},\overline{\mathcal{V}}_{\omega^t}\big)\Big)\end{align*}

for both the elliptic curves $E=E_1$ and $E=E_2$ .

Because the right-hand side is independent of this choice of either elliptic curve, it follows that $\mathrm{Sel}_{\omega^t}^{\Sigma}(E_1,\tau)\big[\wp\big]$ and $\mathrm{Sel}_{\omega^t}^{\Sigma}(E_2,\tau)\big[\wp\big]$ must have the same $\mathbb{F}$ -dimension. Furthermore, Lemma 3·2(iii) informs us that this $\mathbb{F}$ -dimension coincides with the $\lambda$ -invariant of $\mathrm{char}_{\Lambda}\left(\mathfrak{X}_{\omega^t}^{\Sigma}(E,\tau)\right)$ , in other words

(3·4) \begin{equation}\lambda\big(\mathbf{L}_{p,\Sigma}^{\mathrm{alg}} (E_1,\tau\otimes\omega^t)\big) =\,\lambda\big(\mathbf{L}_{p,\Sigma}^{\mathrm{alg}} (E_2,\tau\otimes\omega^t)\big) .\end{equation}

Choosing either $E=E_1$ or $E=E_2$ , a simple calculation reveals that

\begin{align*}\lambda\big(\mathbf{L}_{p,\Sigma}^{\mathrm{alg}} (E,\tau\otimes\omega^t)\big)=\, \lambda\big(\mathbf{L}_p^{\mathrm{alg}} (E,\tau\otimes\omega^t)\big) +\sum_{\nu|\mathfrak{q}\in\Sigma} \lambda\left(\mathrm{char}_{\Lambda}\Big(H^1{\big(F_{\nu}^{\mathrm{cyc}},\,\mathcal{E}^{\prime(\tau)}_{\omega^t}\big)}^{\wedge}\Big)\right) .\end{align*}

However $\lambda\left(\mathrm{char}_{\Lambda}\Big(\bigoplus_{\nu|\mathfrak{q}}H^1\big(F_{\nu}^{\mathrm{cyc}},\,\mathcal{E}^{\prime(\tau)}_{\omega^t}\big)^{\wedge}\Big)\right)$ equals $\mathbf{e}_{\mathfrak{q}}^{(\omega^t)}(E,\tau)$ at each prime $\mathfrak{q}\in\Sigma$ by applying [ Reference Delbourgo8 , lemma 3·5], and as a direct consequence

(3·5) \begin{equation}\lambda\big(\mathbf{L}_{p,\Sigma}^{\mathrm{alg}} (E,\tau\otimes\omega^t)\big)=\, \lambda\big(\mathbf{L}_p^{\mathrm{alg}} (E,\tau\otimes\omega^t)\big) +\sum_{\mathfrak{q}\in\Sigma} \,\mathbf{e}_{\mathfrak{q}}^{(\omega^t)}(E,\tau).\end{equation}

Combining together Equations (3·4) and (3·5), statement (b) is now proved.

3·4 Numerical examples

We begin with a basic but very useful lemma. At every branch $\omega^t$ , the $\Sigma$ -imprimitive p-adic L-function factorises into a product

(3·6) \begin{equation}\mathbf{L}_{p,\Sigma_F}^{\mathrm{an},\Omega} \big(E,\tau\otimes\omega^t\big)\,=\,\mathbf{L}_{p,\emptyset}^{\mathrm{an},\Omega} \big(E,\tau\otimes\omega^t\big)\times \prod_{\mathfrak{q}\in\Sigma_F}\Delta_{\mathfrak{q}}^{(\omega^t)}(E,\tau)\end{equation}

where each factor $\Delta_{\mathfrak{q}}^{(\omega^t)}(E,\tau)\in R[\![\Gamma]\!]\cong R[\![X]\!]$ interpolates $\iota_p\left(L_{\mathfrak{q}}\big(E,\tau^*\otimes\theta^{-1},1\big)\right)$ at characters $\theta$ of $\mathfrak{p}$ -power conductor satisfying $\theta\big|_{\mathbb{F}_p^{\times}}\cong\omega^t$ . In particular, one has

\begin{align*}L_{\mathfrak{q}}\big(E,\tau^*\otimes\theta^{-1},s\big)=\mathcal{P}_{\mathfrak{q}}(E,\tau^*;\,T)\Big|_{T=\theta^{-1}(\mathfrak{q})\cdot\mathcal{N}(\mathfrak{q})^{-s}},\end{align*}

where the above polynomial $\mathcal{P}_{\mathfrak{q}}(E,\tau^*;\,T)$ divides into the $\mathfrak{q}$ -th Euler factor term

\begin{align*}\big(1-\alpha_{\mathfrak{q}}\beta_{\mathfrak{q}} T\big)\big(1-\alpha_{\mathfrak{q}}'\beta_{\mathfrak{q}} T\big)\big(1-\alpha_{\mathfrak{q}}\beta_{\mathfrak{q}}' T\big)\big(1-\alpha_{\mathfrak{q}}'\beta_{\mathfrak{q}}' T\big) .\end{align*}

Individually any one of these four brackets is either 1, or else of the form $(1-w_{\mathfrak{q}}T)$ , where $w_{\mathfrak{q}}$ is a p-adic unit; since $\Delta_{\mathfrak{q}}^{(\omega^t)}(E,\tau)\Big|_{X=\theta(\gamma_0)-1}\!=\,\mathcal{P}_{\mathfrak{q}}(E,\tau^*;\theta^{-1}(\mathfrak{q})\cdot\mathcal{N}(\mathfrak{q})^{-1})$ it is then straightforward to establish that

\begin{align*}\Delta_{\mathfrak{q}}^{(\omega^t)}(E,\tau)\,=\,\prod_{w_{\mathfrak{q}}\in\mathcal{W}_{\mathfrak{q}}} \Big(1-\frac{w_{\mathfrak{q}}\cdot\omega^{-t}(\mathfrak{q})}{\mathcal{N}(\mathfrak{q})}(1+X)^{-\frac{\log_p(\mathcal{N}(\mathfrak{q}))}{\log_p(\gamma_0)}}\Big),\end{align*}

for some subset $\mathcal{W}_{\mathfrak{q}}\subset\big\{\alpha_{\mathfrak{q}}\beta_{\mathfrak{q}},\alpha_{\mathfrak{q}}'\beta_{\mathfrak{q}},\alpha_{\mathfrak{q}}\beta_{\mathfrak{q}}',\alpha_{\mathfrak{q}}'\beta_{\mathfrak{q}}'\big\}$ . Because the $\mu$ -invariant of $1+X$ is trivial it follows that the $\mu$ -invariant of $\Delta_{\mathfrak{q}}^{(\omega^t)}(E,\tau)$ is also trivial as $\mathfrak{q}\neq\mathfrak{p}$ , thence

\begin{align*}\mu\big(\mathbf{L}_{p,\Sigma_F}^{\mathrm{an},\Omega} \big(E,\tau\otimes\omega^t\big)\big)=\mu\big(\mathbf{L}_{p,\emptyset}^{\mathrm{an},\Omega} \big(E,\tau\otimes\omega^t\big)\big)\end{align*}

upon using Equation (3·6) together with the additivity property of the $\mu$ -invariant.

Lemma 3·9. If $p\nmid \mathfrak{h}_E\cdot\epsilon_F(\tau\otimes\omega^t,0)\cdot\mathcal{E}_{\mathfrak{p}}\big(E,\tau\otimes\omega^t\big)\cdot \frac{L(E,\tau^*\otimes\omega^{-t},1)}{(2\pi)^{[K\,{:}\,\mathbb{Q}]}\cdot\Omega_{E/F}^{\mathrm{Pet}}}$ then

\begin{align*}\mu\Big(\mathbf{L}_{p,\Sigma_F}^{\mathrm{an},\Omega} \big(E,\tau\otimes\omega^t\big)\Big)=\mu\Big(\mathbf{L}_{p,\emptyset}^{\mathrm{an},\Omega} \big(E,\tau\otimes\omega^t\big)\Big) =0,\end{align*}

where the $\mathfrak{p}$ -Euler factor is equal to $\mathcal{E}_{\mathfrak{p}}\big(E,\tau\otimes\omega^t\big)=\Big(1-\frac{\omega^{-t}(\mathfrak{p})\cdot\beta_{\mathfrak{p}}}{\alpha_{\mathfrak{p}}(E)}\Big)\Big(1-\frac{\omega^{-t}(\mathfrak{p})\cdot\beta_{\mathfrak{p}}'}{\alpha_{\mathfrak{p}}(E)}\Big)$ .

The above result provides a simple criterion to check the “ $\mu=0$ ” condition holds true for both these analytic objects. Throughout we shall only consider pairs of p-congruent elliptic curve $E_1$ and $E_2$ defined over F such that

\begin{align*}\mu\big(\mathbf{L}_{p,\emptyset}^{\mathrm{an},\Omega} \big(E_1,\tau\otimes\omega^t\big)\big) =\mu\big(\mathbf{L}_{p,\emptyset}^{\mathrm{an},\Omega} \big(E_2,\tau\otimes\omega^t\big)\big)=0 \quad\textit{at the branch}\ \omega^t=\mathbf{1}_F.\end{align*}

However there is one caveat: we replace $(2\pi)^{[K\,{:}\,\mathbb{Q}]}\cdot\Omega_{E/F}^{\mathrm{Pet}}\big/\mathfrak{h}_E$ with $(2\Omega_E^+\Omega_E^-)^{[F\,{:}\,\mathbb{Q}]}$ where $\Omega_E^{\pm}=\int_{E(\mathbb{C})^{\pm}}\varpi_E$ are the real/imaginary Néron periods of the elliptic curve. This swap is fine as long as Ihara’s Lemma holds (e.g. if $F=\mathbb{Q}$ ), but if $F\neq \mathbb{Q}$ the period ratio might not be a unit, although conjecturally(!) this should not happen.

We shall work exclusively over totally real fields F of small degree because the computer calculations are far too slow if we consider larger field extensions $F/\mathbb{Q}$ . The examples are computed via the Dokchitser brothers’ MAGMA routines [ Reference Bosma, Cannon and Playoust1 , Reference Dokchitser and Dokchitser13 ], as well as John Cremona’s LMFDB database of elliptic curves. The Ph.D. thesis of the first named author ([ Reference Maletto24 ]) will contain many similar computations, but mainly for reasons of space we have decided to include only the three examples listed below.

Example 3·10. Consider the pair of elliptic curves $E_1 = \texttt{24a4}$ and $E_2 = \texttt{312e1}$ . Their minimal Weierstrass equations are given by

\begin{align*} E_1&\,{:}\, y^2 = x^3 - x^2 + x\\ E_2&\,{:}\, y^2 = x^3 - x^2 - 651x + 6228.\end{align*}

One can check that these two elliptic curves are congruent modulo $p=3$ , meaning

\[ a_m(\mathcal{F}_{E_1}) \equiv a_m(\mathcal{F}_{E_2})\ (\mathrm{mod}\ 3) \quad \mathrm{for all } m \in \mathbb{N}\; \mathrm{with } \gcd(m, 2 \cdot 13) = 1. \]

Now $E_1$ and $E_2$ both have a non-split multiplicative reduction over $\mathbb{Q}_3$ , therefore both these curves are p-ordinary. Let $F = \mathbb{Q}$ and $K = \mathbb{Q}(\sqrt{-3})$ , and choose $\Delta > 1$ to be a cube-free integer. We consider the two-dimensional Artin representation

\[ \tau^{(\Delta)} \,{:\!=}\, \,\mathrm{Ind}_F^K (\vartheta), \]

where $\vartheta\,{:}\, \mathrm{Gal}(K(\Delta^{1/3})/K) \xrightarrow{\sim} \mu_3$ . Then $\tau^{(\Delta)}\,{:}\, G_{\mathbb{Q}} \to \mathrm{GL}_2(\mathbb{C})$ is realisable over the field of rational numbers. It follows that $\tau^{(\Delta)}$ will be self-dual (that is $\tau = \tau^\ast$ ), thereby providing an alternate way to compute $L(E, \tau^{(\Delta)},s)$ :

\[ L(E, \tau^{(\Delta)}, s) = \frac{L(E/\mathbb{Q}(\Delta^{1/3}), s)}{L(E/\mathbb{Q}, s)}. \]

Note that for any choice of $\Delta$ , the pairs $(E_1,\tau^{(\Delta)})$ and $(E_2,\tau^{(\Delta)})$ may not satisfy condition (iii) of Theorem 1·2. However, if we replace them by twisted pairs $\big(E_1\otimes\eta^{-1},\tau^{(\Delta)}\otimes\eta\big)$ and $\big(E_2\otimes\eta^{-1},\tau^{(\Delta)}\otimes\eta\big)$ where $\eta$ denotes a Hecke character over F whose $\mathfrak{q}$ -conductor is large at the primes $\mathfrak{q}$ dividing the non-square-free part of cond $(E_1)$ and cond $(E_2)$ , then these twisted pairs will satisfy condition (iii) and so one would still expect mod p congruences to hold.

Put $\Delta = 2$ and assign $\Sigma^{\mathrm{bad}} = \{2, 13\}$ . Evaluating $L(E_1, \tau^{(2)}, s)$ at $s = 1$ yields

\[ L(E_1, \tau^{(2)}, 1) \approx 1.399245719600 \]

implying that the algebraic L-value is given by

\[ \mathcal{L}^{\ast}(E_1, \tau^{(2)}) = \left| \frac{L(E_1, \tau^{(2)}, 1) \cdot \sqrt{\mathrm{disc}(\mathbb{Q}(\Delta^{1/3}))}}{(2\Omega^+_{E_1} \Omega^-_{E_1})^{(3-1)/2}} \right| \approx 0.999999999999. \]

Similarly, evaluating $L(E_2, \tau^{(2)}, s)$ at $s = 1$ yields

\[ L(E_2, \tau^{(2)}, 1) \approx 1.645733674248 \]

which we then use to compute

\[ \mathcal{L}^{\ast}(E_2, \tau^{(2)}) = \left| \frac{L(E_2, \tau^{(2)}, 1) \cdot \sqrt{\mathrm{disc}(\mathbb{Q}(\Delta^{1/3}))}}{(2\Omega^+_{E_2} \Omega^-_{E_2})^{(3-1)/2}} \right| \approx 24.9999999999879. \]

Next we will evaluate the $\tau^{(2)}$ -twisted p-adic L-functions of $E=E_1$ and $E=E_2$ at the trivial character, utilising the interpolation formula

\[ \textbf{1}\big(\textbf{L}_{p, \Sigma}(E, \tau)\big) = \iota_3 \circ \iota_\infty^{-1} \left( \frac{\epsilon_F(\tau, 0)_{\mathfrak{p}}}{a_{\mathfrak{p}}(E)^{\mathfrak{f}(\tau, \mathfrak{p})}} \cdot \mathcal{E}_{\mathfrak{p}}(E, \tau) \cdot\!\!\!\prod_{q \in \Sigma^{\mathrm{bad}}}\! L_q(E, \tau^\ast, 1) \cdot \frac{L(E, \tau^\ast, 1)}{(2\Omega_E^+ \Omega_E^-)^{[F: \mathbb{Q}]}} \!\right) \! . \]

Applying the Dokchitsers’ method for computing local $\epsilon$ -factors, we then obtain

\begin{align*}\epsilon_F(\tau^{(2)})_3 \,=\, -1.04520385168448\times 10^{-14} + 5.19615242270663i \,\approx\, 3\sqrt{-3} .\end{align*}

Checking the Fourier expansion of $\mathcal{F}_{E_1}$ and $\mathcal{F}_{E_2}$ shows $a_3(\mathcal{F}_{E_1}) = -1 = a_3(\mathcal{F}_{E_2})$ and as $\mathrm{cond}(\tau^{(2)}) = 108$ , one deduces that

\[ a_3(\mathcal{F}_{E_1})^{\mathfrak{f}(\tau^{(2)}, 3)} = -1 = a_3(\mathcal{F}_{E_2})^{\mathfrak{f}(\tau^{(2)}, 3)}. \]

The Euler factors that we require can be calculated using an in-built routine from the MAGMA package, and thus we determine that

\begin{align*}&L_2(E_1, \tau^{(2)}, s) = 1,\\&L_2(E_2, \tau^{(2)}, s) = 1,\\&L_{13}(E_1, \tau^{(2)}, s) = 169\cdot 13^{-4s} - 26 \cdot 13^{-3s} - 9 \cdot 13^{-2s} - 2 \cdot 13^{-s} + 1,\\&L_{13}(E_2, \tau^{(2)}, s) = 13^{-2s} - 13^{-s} + 1 .\end{align*}

Lastly observe that as we chose $\Delta>1$ so that $\mathrm{ord}_p(\Delta^{p-1} - 1) - 1 = 0$ , one finds

\begin{align*}\mathcal{E}_3(E_1, \tau^{(2)}) = (1 - a_3(E_1) \vartheta(\mathfrak{P})) = 1 \quad\mathrm{and}\quad\mathcal{E}_3(E_2, \tau^{(2)}) = 1.\end{align*}

Plugging in all these factors that we have computed above, we conclude that

\begin{align*} \textbf{1}\big(\textbf{L}_{3, \Sigma}(E_1, \tau^{(2)})\big) &= \iota_3 \circ \iota_\infty^{-1} \Bigg( \frac{\epsilon_F(\tau)_3}{\sqrt{\mathrm{disc}(\mathbb{Q}(\Delta^{1/3}))}} \cdot \frac{2\cdot\mathcal{E}_{3}(E_1, \tau)}{a_{3}(E_1)^{\mathfrak{f}(\tau, 3)}} \cdot \prod_{q \in \Sigma^{\mathrm{bad}}} L_q(E_1, \tau^\ast, 1)\\ &\qquad\qquad\quad\cdot \frac{L(E_1, \tau^\ast, 1) \cdot \sqrt{\mathrm{disc}(\mathbb{Q}(\Delta^{1/3}))}}{(2 \Omega_{E_1}^+ \Omega_{E_1}^-)} \Bigg)\\ &\approx \frac{1}{2} \cdot \frac{2}{-1} \cdot \left(1 \cdot \frac{133}{169}\right) \cdot 1\\ &\approx 1358 \,=\, [2,2,0,2,1,2,1,0,\dots] \,\,\mathrm{written \, as \, a \, 3-adic \, number.}\end{align*}

Repeating the process for $E_2$ , one readily obtains

\begin{align*} \textbf{1}(\textbf{L}_{3, \Sigma}\big(E_2, \tau^{(2)})\big) &\approx \frac{1}{2} \cdot \frac{2}{-1} \cdot \left( 1 \cdot \frac{157}{169} \right) \cdot 25\\ &\approx 365 \,=\, [2,1,1,1,1,1,0,0,\dots] \,\,\mathrm{written \, as \, a 3-adic \, number.}\end{align*}

These values are indeed congruent modulo 3, as was predicted by Theorem 1·2.

Example 3·11. Consider the elliptic curves $E_1 = \texttt{52a1}$ and $E_2 = \texttt{1612a1}$ with minimal Weierstrass equations

\begin{align*} E_1&\,{:}\, y^2 = x^3 + x - 10\\ E_2&\,{:}\, y^2 = x^3 - 272x + 80980.\end{align*}

These curves are congruent modulo $p=3$ . We consider the same two-dimensional Artin representation $\tau^{(\Delta)}$ as above with $\Delta = 2$ , so we have $\Sigma^{\mathrm{bad}} = \{ 2, 13, 31 \}$ . Evaluating $\mathcal{L}^\ast(E_1, \tau^{(2)}, s)$ at $s = 1$ , one obtains

\[ L(E_1, \tau^{(2)}, 1) \approx 1.823895795706 \]

which we then use to calculate the algebraic L-value

\[ \mathcal{L}^\ast(E_1, \tau^{(2)}) = \left| \frac{L(E_1, \tau^{(2)}, 1) \cdot \sqrt{\mathrm{disc}(\mathbb{Q}(\Delta^{1/3}))}}{(2\Omega^+_{E_1} \Omega^-_{E_1})^{(3-1)/2}} \right| \approx 3.999999999998. \]

Repeating the same process for the congruent elliptic curve $E_2$ , we compute that

\[ L(E_2, \tau^{(2)}, 1) \approx 0.363500752332 \]

and as a consequence,

\[ \mathcal{L}^\ast(E_2, \tau^{(2)}) \approx 15.999999999980. \]

We thus make the rational approximations $\mathcal{L}^\ast(E_1, \tau^{(2)}) \approx 4$ and $\mathcal{L}^\ast(E_1, \tau^{(2)}) \approx 16$ .

Both $E_1$ and $E_2$ have supersingular reduction at $p=3$ so $a_3(E_1) = 0 = a_3(E_2)$ ; we will fix the same root $\alpha_3(E_1) = \sqrt{-3} = \alpha_3(E_2)$ of the Hecke polynomial $T^2+3$ . Using the MAGMA package again, one easily determines the following Euler factors:

\begin{align*} &L_2(E_1, \tau^{(2)}, s) = 1,\\ &L_{13}(E_1, \tau^{(2)}, s) = 13^{-2s} - 13^{-s} + 1,\\ &L_{23}(E_1, \tau^{(2)}, s) = 961 \cdot 31^{-4s} - 620 \cdot 31^{-3s} + 162 \cdot 31^{-2s} - 20 \cdot 13^{-s} + 1,\\ &L_2(E_2, \tau^{(2)}, s) = 1,\\ &L_{13}(E_2, \tau^{(2)}, s) = 13^{-2s} - 13^{-s} + 1,\\ &L_{23}(E_2, \tau^{(2)}, s) = 31^{-2s} + 2 \cdot 31^{-2s} + 1.\end{align*}

Abusing notation and writing $\textbf{1}(\textbf{L}_{3,\Sigma}\big(E_1, \tau^{(2)})\big)$ for the 3-adic L-value, one finds

\begin{align*} \textbf{1}(\textbf{L}_{3,\Sigma}\big(E_1, \tau^{(2)})\big) &\approx \iota_3 \circ \iota_\infty^{-1} \left[ \frac{1}{2} \cdot \frac{2}{(\sqrt{-3})^3} \cdot \left( 1 \cdot \frac{157}{169} \cdot \frac{484}{961} \right) \cdot 4 \right]\\ &\approx 1306 \cdot \iota_3 \circ \iota_\infty^{-1} \left( -3^{-3/2} \right).\end{align*}

On the other hand, abusing notation in an identical manner for $E_2$ twisted by $\tau^{(2)}$ :

\begin{align*} \textbf{1}(\textbf{L}_{3,\Sigma}\big(E_2, \tau^{(2)})\big) &\approx \iota_3 \circ \iota_\infty^{-1} \left[ \frac{1}{2} \cdot \frac{2}{(\sqrt{-3})^3} \cdot \left( 1 \cdot \frac{157}{169} \cdot \frac{1024}{961} \right) \cdot 16 \right]\\ &\approx \left( 5359 + \mathrm{O}(3^8) \right) \cdot \iota_3 \circ \iota_\infty^{-1} \left( -3^{-3/2} \right). \end{align*}

Despite both values lying in $\mathbb{Q}_3(\sqrt{-3})$ , they are in fact congruent modulo $p=3$ .

Example 3·12. The elliptic curves $E_1 = \texttt{19a3}$ and $E_2 = \texttt{323a1}$ are given by

\begin{align*} E_1&\,{:}\, y^2 + xy + y = x^3 + 4x - 6\\ E_2&\,{:}\, y^2 + xy = x^3 + 7x - 7 .\end{align*}

By checking the q-expansion of the associated newforms, one sees that these two curves are congruent modulo $p=5$ . Let $K = \mathbb{Q}(\mu_5)$ and write $F = \mathbb{Q}(\mu_5)^+$ for its totally real subfield. We next consider the two-dimensional Artin representation

\[ \tau^{(\Delta)} \,{:\!=}\, \mathrm{Ind}_F^K(\vartheta) \]

induced from a Hecke character $\vartheta\,{:}\, \mathrm{Gal}(K(\Delta^{1/5})/K) \xrightarrow{\sim} \mu_5$ , where $\Delta > 1$ denotes an arbitrary fifth-power free integer. Applying the Artin formalism once more,

\[ L\big(E/F, \tau^{(\Delta)}, s\big) \,=\, L\big(E/\mathbb{Q}, \mathrm{Ind}_\mathbb{Q}^K(\vartheta), s\big)\,=\, \frac{L(E/\mathbb{Q}(\Delta^{1/5}), s)}{L(E/\mathbb{Q}, s)} . \]

To ease notation, we shall further write $\tau_{\mathbb{Q}}^{(\Delta)}$ for the induced representation $\mathrm{Ind}_\mathbb{Q}^K(\vartheta)$ . If we set $\Delta = 2$ and $\Sigma^{\mathrm{bad}} = \{ 2,7,13 \}$ , then using the built-in MAGMA functions

\begin{align*}L\big(E_1, \tau_{\mathbb{Q}}^{(2)}, 1\big) \approx 1.267565010650\end{align*}

so that

\[ \mathcal{L}^{\ast}\big(E_1, \tau_{\mathbb{Q}}^{(2)}\big) = \left| \frac{L\big(E_1, \tau_{\mathbb{Q}}^{(2)}, 1\big) \cdot \sqrt{\mathrm{disc}(\mathbb{Q}(2^{1/5}))}}{(2\Omega^+_{E_1} \Omega^-_{E_1})^{(5-1)/2}} \right|\,\,\approx\,\, 0.999999999999 \,\,\approx\,\, 1. \]

N.B. Because $E_2$ has a point of infinite order over $\mathbb{Q}(2^{1/5})$ clearly $L\big(E_2, \tau_{\mathbb{Q}}^{(2)}, 1\big)= 0$ , in which case it follows that $\textbf{1}\big(\textbf{L}_{5}(E_2, \tau^{(2)})\big) = 0$ via the interpolation formula.

Applying the Dokchitsers’ method for computing local $\epsilon$ -factors, one finds that

\begin{align*}\epsilon_F\big(\tau^{(2)}\big)_5 \,=\, -55.90169945 - 1.647223047i \times 10^{-8} \,\approx\, 25\sqrt{-5} .\end{align*}

The Euler factors we require can be calculated using the MAGMA package, namely

\begin{align*}&L_2(E_1, \tau^{(2)}, s)= 1,\\&L_{17}(E_1, \tau^{(2)}, s) = 83521 \cdot 17^{-8s} - 47 \cdot 17^{-4s} + 1,\\&L_{19}(E_1, \tau^{(2)}, s) = 19^{-4s} - 2 \cdot 19^{-2s} + 1.\end{align*}

One readily computes that $\mathcal{E}_5(E_1, \tau^{(2)}) = 1$ , and plugging in all the factors above:

\begin{align*} \textbf{1}\big(\textbf{L}_{5, \Sigma}(E_1, \tau^{(2)})\big) &= \iota_5 \circ \iota_\infty^{-1} \Bigg( \frac{\epsilon_F(\tau^{(2)})_5}{\sqrt{\mathrm{disc}(\mathbb{Q}(2^{1/5}))}} \cdot \frac{2\cdot\mathcal{E}_{5}(E_1, \tau)}{\alpha_5(E_1)^{\mathfrak{f}(\tau, 5)}} \cdot \prod_{q \in \Sigma^{\mathrm{bad}}} L_q(E_1, \tau^\ast, 1)\\ &\qquad\qquad\quad\cdot \frac{L(E_1, \tau^\ast, 1) \cdot \sqrt{\mathrm{disc}(\mathbb{Q}(\Delta^{1/5}))}}{(2 \Omega_{E_1}^+ \Omega_{E_1}^-)} \Bigg)\\ &\approx \frac{i}{4} \cdot \frac{2}{\alpha_5(E_1)^5} \cdot \left(1 \cdot \frac{83475}{83521} \cdot \frac{129600}{130321}\right) \cdot ({-}1)\\ &\approx 386250 \,=\, [0,0,0,0,3,3,4,4,\dots] \,\,\mathrm{written \, as \, a \, 5-adic \, number.}\end{align*}

In particular, $\textbf{1}\big(\textbf{L}_{5, \Sigma}(E_1, \tau^{(2)})\big)$ and $\textbf{1}\big(\textbf{L}_{5, \Sigma}(E_2, \tau^{(2)})\big)=0$ are congruent modulo 5.

3·5. Tables of mod p congruences

Listed below are some calculations of congruences between p-adic L-functions evaluated at the trivial character; here we take $\tau$ to be the Artin representation $\mathrm{Ind}_\mathbb{Q}^K(\vartheta)$ where $\vartheta\,{:}\, \mathrm{Gal}(K(\Delta^{1/p})/K) \xrightarrow{\sim} \mu_p$ .

Table 1.

Pairs with good ordinary or multiplicative reduction at 3

The numbers above are truncated at $O(3^8)$ with the coefficients of $3^m$ for m negative written in boldface (e.g. $[\boldsymbol{2},1,0,2,\dots]$ means $2\times 3^{-1}+1\times 3^0 + 0\times 3^1+ 2\times 3^2\dots$ ).

Table 2.

Pairs with good supersingular reduction at $p=3$ .

As in Example 3·11, the values in the last two columns of Table 2 belong to the local field $\mathbb{Q}_3(\sqrt{-3})$ . It is also noticeable from these tables that the constant term of the p-adic L-function, namely $\mathbf{1}(\textbf{L}_{p,\Sigma}\big(E, \tau)\big)$ , is frequently highly divisible by p. This is for the most part attributable to the Iwasawa power series interpolating the Euler factors $L_q(E, \tau^\ast, s)$ that we are taking out at primes $q \in \Sigma^{\mathrm{bad}}$ having zeroes very close to the trivial character. Nevertheless, provided it is the case that

\begin{align*}p \nmid\mathfrak{h}_E\cdot \epsilon_F(\tau,0)\cdot \mathcal{E}_{\mathfrak{p}}\big(E,\tau\big) \cdot \frac{L(E,\tau^*,1)}{(2\pi)^{[K\,{:}\,\mathbb{Q}]}\cdot\Omega_{E/F}^{\mathrm{Pet}}}\end{align*}

for either $E=E_1$ or $E_2$ , the analytic $\mu$ -invariant of $E\otimes\tau$ vanishes by Lemma 3·9.