\begin{document}

\title[Non-vanishing mod $p$ of Hecke $L$-values]{Non-vanishing modulo $p$ of Hecke $L$-values over imaginary quadratic fields}

\author[D.~Kundu]{Debanjana Kundu}
\address[Kundu]{Fields Institute \\ University of Toronto \\ Toronto ON, M5T 3J1, Canada}
\email{dkundu@math.toronto.edu}

\author[A.~Lei]{Antonio Lei}
\address[Lei]{Department of Mathematics and Statistics\\University of Ottawa\\
150 Louis-Pasteur Pvt\\
Ottawa, ON\\
Canada K1N 6N5}
\email{antonio.lei@uottawa.ca}

\date{\today}

\keywords{Hecke characters, imaginary quadratic fields, $p$-divisibility of Hecke $L$-values}
\subjclass[2020]{Primary 11S40, 11G15; Secondary 11F67, 11R20, 11R23}

\begin{abstract}
Let $p$ and $q$ be two distinct odd primes.
Let $K$ be an imaginary quadratic field over which $p$ and $q$ are both split.
Let $\Psi$ be a Hecke character over $K$ of infinity type $(k,j)$ with $0\le-j< k$.
Under certain technical hypotheses, we show that for a Zariski dense set of finite-order characters $\kappa$ over $K$ which factor through the $\mathbb{Z}_q^2$-extension of $K$, the $p$-adic valuation of the algebraic part of the $L$-value $L(\overline{\kappa\Psi},k+j)$ is a constant independent of $\kappa$.
In addition, when $j=0$ and certain technical hypothesis holds, this constant is zero.
\end{abstract}

\maketitle

\section{Introduction}

Let $p$ and $q$ be two distinct odd primes.
It is a classical problem to study the divisibility of the algebraic part of (Hecke) $L$-values by a given prime $p$ as one varies the (Hecke) characters of $q$-power conductor.
For Dirichlet $L$-values, such questions were studied by L.~Washington in \cite{Was75, Was78}.
He showed that for almost all Dirichlet characters of $q$-power conductor, the algebraic parts of their $L$-values are coprime to $p$.
As an application, he proved that the $p$-part of the class number stabilizes in cyclotomic $\mathbb Z_q$-extensions of abelian number fields.
Washington's results have been extended to the case of (finite) product of cyclotomic $\mathbb{Z}_{q_i}$-extensions of abelian number fields (for distinct primes $q_i$ with $q_i\ne p$) by E.~Friedman in \cite{friedman}.

In \cite{sinnott}, W.~Sinnott introduced the idea of relating non-vanishing of such $L$-values modulo $p$ to Zariski density (modulo $p$) of special points of the algebraic variety underlying the $L$-values.
Using this machinery, J.~Lamplugh generalized Washington's theorem to \emph{split prime} $\mathbb Z_q$-extensions of imaginary quadratic fields in \cite{Lam15}.
Let $K$ be an imaginary quadratic field such that $q \mathcal O_K= \mathfrak q{\mathfrak q^*}$ with $\mathfrak q \neq \mathfrak q^*$, then the \emph{split prime} $\mathbb{Z}_q$-extension of $K$ is one where only one of $\mathfrak q$ or ${\mathfrak q^*}$ is ramified.

In \cite{hida1,hida2}, H.~Hida studied analogous questions for anticyclotomic characters.
He proved that when $p$ is split in $K$ and the tame conductor of characters is a product of split primes (which excludes the self-dual characters), the algebraic parts of the $L$-values of "almost all" anticyclotomic characters of $q$-power conductor over a CM field are non-zero mod $p$.
Here, "almost all" means "Zariski dense" after identifying the characters with a product of the multiplicative group (see Remark~\ref{rk:Zar}).
This has been generalized by M.-L.~Hsieh to include self-dual characters assuming that $p$ is split in $K$ in \cite{hsieh}  and that the inert part of the conductor is square-free.
The hypothesis on the inert part of the conductor was removed in \cite[Remark~6.4]{hsieh2014mu}.
In the case where the CM field is an imaginary quadratic field, T.~Finis has proved similar results for self-dual characters allowing $p$ to be either inert or ramified in $K$, and has determined precisely the $p$-adic valuations of the algebraic parts of anticyclotomic Hecke characters of $q$-power conductor; see \cite{finis}.
More recently, A.~Burungale showed in \cite{burungale} that these results may be extended to Hida families of anticyclotomic characters under the same hypotheses as those in the works of Hsieh.

We study a generalization of the aforementioned results on anticyclotomic characters to Hecke characters (not necessarily anticyclotomic) of $q$-power conductor over an imaginary quadratic field.
It can be regarded as a 2-variable version of \cite[Theorem~6.9]{Lam15}.

\begin{lthm}\label{thmA}
Let $K$ be an imaginary quadratic field over which $p$ and $q$ are both split.
Suppose that both the prime ideals above $q$ are principal in $K$.
Let $\Psi$ be a Hecke character over $K$ of infinity type $(k,j)$ and conductor $\mathfrak f$, where $0\le -j<k$ and $\mathfrak f$ is coprime to $pq$.
Assume that $q\nmid [\mathscr{R}(\mathfrak f):K]$, where $\mathscr{R}(\mathfrak f)$ denotes the ray class field of $K$ of conductor $\mathfrak f$.
Let $\mathcal F_\infty$ be the $\mathbb{Z}_q^2$-extension of $K$.
There exists a constant $C_\Psi$ such that for a Zariski dense set of finite-order characters $\kappa$ of $\Gal(K_\infty/K)$,
\[
\ord_{p}\left( L^{\alg} \left(\overline{\kappa\Psi}\right)\right)=C_\Psi.
\]
\end{lthm}

Under additional hypotheses, we prove:
\begin{lthm}\label{thmB}
With notation as in the statement of Theorem~\ref{thmA}, if $j=0$ and the character of $\Gal(\mathscr{R}(\mathfrak f)/K)$ induced by $\Psi$ satisfies a technical hypothesis \eqref{eq:condition-rho}, then $C_\Psi=0$.
\end{lthm}

\begin{rem}
If $p\nmid [\mathscr{R}(\mathfrak f):K]$, then it is easy to show (see Remark~\ref{remark: sufficient conditions for technical hypothesis}) that one may multiply $\Psi$ by a character $\rho$ of $\Gal(\mathscr{R}(\mathfrak f)/K)$ such that the technical hypothesis is satisfied.
\end{rem}

\subsection*{Outline of proofs} The proofs of Theorems~\ref{thmA} and \ref{thmB} follow closely the line of argument of \cite[Theorem~6.9]{Lam15}.
It consists of the following ingredients:
\begin{itemize}
 \item[(1)]Establish a theory of Gamma transform of "elliptic function measures" on $\mathbb{Z}_q^2$, which are measures that arise from a rational function on an elliptic curve.
 \item[(2)]Show that the $\pi$-adic valuations of the aforementioned Gamma transforms have the same $p$-adic valuation for almost all finite characters on $\mathbb{Z}_q^2$.
 \item[(3)]Show that by defining an elliptic function measure (see Definition~\ref{def:elliptic-fun-meas}) arising from a rational function on the CM elliptic curve $E$ attached to $\Psi$, the Gamma transforms of this measure is related to the special values of $L$-series that we are interested in.
This proves Theorem~\ref{thmA}.
\item[(4)]To prove Theorem~\ref{thmB}, we show that the $\pi$-adic valuation discussed in (2) is zero under our additional hypotheses.
\end{itemize}
Step (1) is carried out in Section~\ref{section: distribution and measures}.
We follow the strategy of Lamplugh in \cite[Section~3]{Lam15}, where the theory for elliptic function measures on $\mathbb{Z}_q$ was developed.
To execute (2), we use a lemma of Hida on the Zariski density of characters on $\mathbb{Z}_q^d$ from \cite{hida1} to prove a result on the algebraic independence of functions on elliptic curves with positive characteristic.
In particular, we prove Theorem~\ref{thm:Hida-Lam}, which is a two-variable version of \cite[Theorem~4.9]{Lam15}.
Next, we prove Theorem~\ref{thm:valuation-gamma-transform}, which completes step (2) outlined above.
The corresponding 1-variable version of this theorem was proved in \cite[Section~5]{Lam15}.
The construction of the elliptic function measure of step (3) is discussed in Section~\ref{S:elliptic-function}; this is a generalization of the rational function on the CM elliptic curve $E$ utilized in \cite[Section~6.3]{Lam15} and crucially uses the work of E.~de Shalit \cite{dS87}.
The link between the Gamma transforms of this elliptic function measure and the $L$-values of interest is given by Lemma~\ref{thm: Lam15 6.6}.
In step (2), we see that the $\pi$-adic valuation mentioned above is in fact given by the valuation of the rational function (see Definition~\ref{defn:ord}).
Using ideas of the proof of \cite[Lemma~6.7]{Lam15} in the one-variable case, we show in Lemma~\ref{lemma: Lam15 Lemma 6.7} that this valuation is zero; this allows us to conclude step (4).

\begin{rem}
While Theorems~\ref{thmA} and \ref{thmB} are deduced using Lamplugh's techniques developed in \cite{Lam15}, our results are strictly stronger than the one-variable analogue \cite[Theorem~6.9]{Lam15}.
Indeed, after identifying the characters of $\Gal(F_\infty/F)$ with a subset of $\mathbb G_{m/\overline{\mathbb Q}_q}^2$, the Zariski closure of the set of characters given by \emph{loc. cit.} is one copy of $\mathbb G_{m/\overline{\mathbb Q}_q}$.
In particular, it is not Zariski dense in $\mathbb G_{m/\overline{\mathbb Q}_q}^2$.

Furthermore, we consider Hecke characters of much more general infinity type than the ones considered in \cite{Lam15}.
In addition, the class number of $K$ is assumed to be $1$ in \cite{Lam15}, whereas Theorem~\ref{thmA} assumes that $q$ does not divide {$ [\mathscr{R}(\mathfrak f):K]$ instead.}
\end{rem}

\begin{rem}
Using an argument similar to the one presented in \cite[Section~7]{Lam15}, we expect that {Theorem~\ref{thmB}} combined with the Iwasawa main conjecture (proved by K.~Rubin) should show that the $p$-part of the class groups over a $\mathbb{Z}_q^2$-tower is "generically zero".
However, it does not seem to be enough to give a generalization of \cite[Theorem~7.10]{Lam15} in our setting, unless we replace "almost all" by "all but finitely many".
\end{rem}

\noindent We conclude by discussing some follow-up questions.
\begin{itemize}
\item In \cite{KL2}, we study the growth of the $p$-part of the class groups in the anticyclotomic $\mathbb{Z}_q$-extension making use of the aforementioned result of Hida.

\item Similar to how we build on Lamplugh's results to obtain {our results}, it may be possible to prove a similar result for Hecke characters of $q$-power conductor over general CM fields, relying on results of Hida and Hsieh on anticyclotomic characters.
\item It may also be interesting to generalize {our results} to the setting of Hida families, utilizing ideas of Burungale developed in \cite{burungale}.
\end{itemize}

\section{Basic Notions}
\label{section: preliminaries}

Let $K$ be a fixed imaginary quadratic field of discriminant $d_K$ and $H$ denote its Hilbert class field.
Throughout, we assume that $q$ is coprime to the class number of $K$ {and we fix a Hecke character $\Psi$ given as in the statement of Theorem~\ref{thmA}.
The character $\overline{\Psi}\circ N^{-j}$ (where $N$ is the norm map on $K$) is of infinity-type $(0,k-j)$.
There exists a character $\chi_0$ of $\Gal(\mathscr{R}(\mathfrak f)/K)$ and an elliptic curve $E$ defined over $\mathscr{R}(\mathfrak f)$ with complex multiplication by $\mathcal O_K$, i.e., $\mathcal O_K \simeq \End(E)$, such that 
\[
\overline{\Psi} N^{-j}=\overline{\varphi^{k-j}}\chi_0,
\]
where $\varphi$ is a Hecke character of infinity type $(1,0)$ satisfying
\[
\psi=\varphi\circ N_{\mathscr{R}(\mathfrak f)/K}
\]
with $\psi$ being the Hecke character over $\mathscr{R}(\mathfrak f)$ attached to $E$.
Furthermore, $\mathscr{R}(\mathfrak f)(E_{\mathrm{tor}})$ is an abelian extension of $K$.
(See \cite[Chapter II, proofs of Theorems~4.12 and 4.14]{dS87} where the existence of $E$ and $\chi_0$ is discussed.)
} 
Let $q\ge 5$ be a prime number that splits in $K$, i.e., 
\[q\mathcal O_K = \mathfrak q {\mathfrak q^*} \textrm{ with } \mathfrak q\neq {\mathfrak q^*}.\] 

For any integral ideal $\mathfrak a $ in $\mathcal O_K$, we write $E_{\mathfrak a}$ to denote
\[\ker\left( \mathfrak a: E \rightarrow E\right).\]
We write $\mu_K$ to denote the set of roots of unity in $K$ and $w_K$ to denote the size of this set.

We fix a different prime $p$ {such that $p \mathcal O_K= \mathfrak p {\mathfrak p^*}$ in $K$ with $\mathfrak p \neq \mathfrak p^*$ and $\gcd(p, 6\mathfrak f q)=1$.
Note in particular that $E$ has good reduction at all primes above $pq$.}

\section{Distributions and measures on \texorpdfstring{$\mathbb{Z}_q^2$}{}}
\label{section: distribution and measures}

The goal of this section is to generalize the notion of Gamma transform from \cite{sinnott} and elliptic function measures studied in \cite[Section~3.2]{Lam15} to the two-variable setting.

Let $E$ be {the elliptic curve given in \S\ref{section: preliminaries}} and $k/\mathbb Q_p$ be a finite unramified extension containing $\mathbb Q_p(E_{\mathfrak{f}q})$.
Set $J=k(\mu_{q^\infty})$.
This is the unramified $\mathbb{Z}_q$-extension of $k$ (since $\mu_{q}\subset k$ by assumption).
Let $\mathfrak{O}$ denote the ring of integers of $J$.
Fix a uniformizer $\pi$ of $k$ and let $\ord_{\pi}$ denote the normalized valuation map
\[
\ord_{\pi}: J \rightarrow \mathbb Z \cup \{ \infty\}.
\]

\begin{Defi}
\label{def:dist}
Let $\alpha$ be a \emph{$J$-valued distribution} on $\mathbb{Z}_q^2$, i.e., $\alpha$ is a finitely additive function on the set of compact open subsets of $\mathbb Z_q^2$ with values in $J$.
\begin{itemize}
 \item[(i)] Given any $c=(c_1,c_2)\in(\mathbb{Z}_q^\times)^2$, define $\alpha\circ c$ to be the distribution given by $\alpha\circ c(X)=\alpha(c X)$ for all open compact subsets $X$ of $\mathbb{Z}_q^2$.
 \item[(ii)] The \emph{Fourier transform} of $\alpha$ is defined to be
 \begin{align*}
 \hat{\alpha}:\mu_{q^\infty}^2&\rightarrow J\\
 (\zeta_1,\zeta_2)&\mapsto \int_{(x,y)\in\mathbb{Z}_q^2}\zeta_1^x\zeta_2^yd\alpha(x,y).
\end{align*}
 \item[(iii)] Given a finite character $\chi $ on $(\mathbb{Z}_q^\times)^2$ with values in $J$, we define \emph{Leopoldt's $\Gamma$-transform} as
 \[
 \Gamma_\alpha(\chi)=\int_{\mathbb{Z}_q^2}\chi d\alpha,
 \]
 where we extend $\chi$ to $\mathbb{Z}_q^2$ by sending all elements not inside $(\mathbb{Z}_q^\times)^2$ to zero.
 \item[(iv)]We call $\alpha$ a \emph{measure} on $\mathbb Z_q^2$ if the image of $\alpha$ has bounded values with respect to $\ord_\pi$.
\end{itemize}
\end{Defi}

\begin{Lemma}
\label{lem:Gamma-Fourier}
Suppose that $\chi$ is a finite-order character on $(\mathbb Z_q^\times)^2$ factoring through $(\mathbb Z/q^m)^\times \times (\mathbb Z/q^{n})^\times$, then
\[
\Gamma_\alpha(\chi)=\tau(\chi)\sum_{\underline x\in \mathbb Z/q^m\times\mathbb Z/q^n}\chi^{-1}(\underline x)\hat\alpha(\underline\zeta^{\underline x}),
\]
where $\underline\zeta=(\zeta_m,\zeta_n)$ with $\zeta_m$ and $\zeta_n$ being primitive $p^m$-th and $p^n$-th roots of unity respectively, and $\tau(\chi)$ is the \emph{Gauss sum} of $\chi$ defined by
\[
\tau(\chi)=\frac{1}{q^{m+n}}\sum_{ (x_1,x_2)\in \mathbb Z/q^m\times \mathbb Z/q^n}\chi( x_1,x_2)\zeta_m^{-x_1}\zeta_n^{-x_2}.
\]
\end{Lemma}

\begin{proof}
See \cite[proof of Proposition~2.2, equation (2.6)]{sinnott} (or \cite[proof of Lemma~2.2.3]{Lam-thesis}).
\end{proof}

\begin{Lemma}
\label{lem:dist-fourier}
A distribution $\alpha$ on $\mathbb{Z}_q^2$ is uniquely determined by its Fourier transform $\hat\alpha$.
\end{Lemma}

\begin{proof}
The characteristic function on the open subset $U_{a,b} := (a+q^m\mathbb{Z}_q)\times (b+q^n\mathbb{Z}_q)$ of $\mathbb{Z}_q^2$ satisfies
\[
\mathbf{1}|_{U_{a,b}}=\frac{1}{q^{m+n}}\sum_{(\zeta_1,\zeta_2)\in\mu_{q^m }\times\mu_{q^n}}\zeta_1^{-a}\zeta_2^{-b}\chi_{(\zeta_1,\zeta_2)},
\]
where $\chi_{(\zeta_1,\zeta_2)}:\mathbb{Z}_q^2\mapsto J$ is the character sending $(x,y)$ to $\zeta_1^{x}\zeta_2^{y}$.
In particular, we see that $\alpha\left(U_{a,b}\right)$ is a linear combination of $\hat\alpha(\zeta_1,\zeta_2)$.
Since the subsets $U_{a,b}$ form a basis of open compact sets of $\mathbb{Z}_q^2$, $\alpha$ is uniquely determined by $\hat\alpha$.
\end{proof}

For the rest of the article, we fix an isomorphism of groups $\delta:(\mu_{q^\infty})^2\stackrel{\sim}{\longrightarrow} E_{q^\infty}$.

\begin{Defi}\label{defn:ellipticmeasure}
A $J$-valued distribution $\alpha$ on $\mathbb{Z}_q^2$ is an \emph{elliptic function measure} for our fixed elliptic curve $E$ (with respect to $\delta$) if there exists a rational function $R\in J(E)$ such that for almost all $\underline\zeta\in(\mu_{q^\infty})^2$, we have
\[
\hat\alpha(\underline\zeta)=R(\delta(\underline\zeta)).
\]
\end{Defi}

\begin{Lemma}
\label{lem:integral-elliptic-function}
Let $f\in \mathfrak{O}[x,y]$ such that the image of $f$ in $J(E)$ is non-zero.
Then, there exists a unique integer $n\ge0$ such that 
\[
\ord_\pi(f(Q))\ge n\ \forall Q\in E_{q^\infty}\setminus\{0\}
\]
with equality holding for almost all $Q\in E_{q^\infty}$.
\end{Lemma}

\begin{proof}
The proof of \cite[Lemma~3.2]{Lam15} goes through in verbatim on replacing $E_{\mathfrak q^\infty}$ by $E_{q^\infty}$.
\end{proof}

This lemma allows us to define a valuation on $J(E)$.
\begin{Defi}\label{defn:ord}
Given an $R\in J(E)$.
If $R\ne 0$, we define $\ord_\pi(R)$ to be the integer $n $ such that $\ord_\pi(R(Q))=n$ for almost all $Q\in E_{q^\infty}$.
If $R=0$, we set $\ord_\pi(R)=\infty$.
\end{Defi}

By Lemma~\ref{lem:integral-elliptic-function}, if $\alpha$ is an elliptic function measure, then it is in fact a measure (not just a distribution) since the values of $\alpha$ are linear combinations of $\hat\alpha=R\circ \delta$ as we have seen in the proof of Lemma~\ref{lem:dist-fourier} and $\frac{1}{q^{m+n}}\in \mathfrak{O}^\times$ (as $p\ne q$).

Note that for any given rational function $R\in J(E)$, we can define a $J$-valued measure attached to $R$ as given by the following lemma:

\begin{Lemma}
\label{lem:EFM}
Let $R\in J(E)$ be a rational function.
There exists a unique measure $\alpha$ on $\mathbb{Z}_q^2$ such that the Fourier transform $\hat\alpha$ coincides with $R\circ \delta$.
In other words, $\alpha$ is an elliptic function measure associated to $R$ in the sense of Definition~\ref{defn:ellipticmeasure}.
\end{Lemma}

\begin{proof}
By the proof of Lemma~\ref{lem:dist-fourier}, we may define a measure $\alpha$ satisfying
\[
\alpha\left((a+q^m\mathbb{Z}_q)\times (b+q^n\mathbb{Z}_q)\right)=\frac{1}{q^{m+n}}\sum_{(\zeta_1,\zeta_2)\in\mu_{q^m }\times\mu_{q^n}}\zeta_1^{-a}\zeta_2^{-b}R\circ \delta (\zeta_1,\zeta_2).
\]
It follows from direct calculations that $\alpha$ is additive and that $\hat\alpha =R\circ \delta$.
\end{proof}

\begin{comment}
\begin{Prop}
If $\alpha$ is a $J$-valued elliptic function measure for $E$ on $\mathbb{Z}_q^2$, then so is $[\phi]\alpha$ for any locally constant function $\phi: \mathbb{Z}_q^2 \rightarrow J$.
Furthermore, if $\phi$ takes values in $\mathfrak{O}$, then
\[
\ord_{\pi}\left( [\phi]\alpha\right)\geq \ord_{\pi}\left( \alpha\right).
\]
\end{Prop}

\begin{proof}
See \cite[Proposition~2.2.7]{Lam-thesis}.
The proof generalizes verbatim to our situation.
\end{proof}
\end{comment}

We now show how Gamma transforms behave under Galois actions.
This will be utilized in subsequent sections.
Let us define the following homomorphisms of groups
\begin{align*}
 \chi_\mu&:\Gal(J/k)\hookrightarrow \Aut(\mu_{q^\infty})^2\simeq (\mathbb{Z}_q^\times)^2,\\
 \chi_E&:\Gal(J/k)\hookrightarrow \Aut(E_{\mathfrak q^\infty})\times\Aut(E_{\overline\mathfrak q^\infty})\simeq (\mathbb{Z}_q^\times)^2.
\end{align*}
Note that $\chi_\mu=\chi_{\cyc}\times\chi_{\cyc}$, where $\chi_{\cyc}$ is the cyclotomic character.

\begin{Defi}\label{def:elliptic-fun-meas}
An \textbf{elliptic function measure} $\alpha$ for $E$ is said to be \emph{defined over $k$}, if $\hat\alpha=R\circ\delta$ for a rational function $R\in k(E)$.
\end{Defi}

\begin{Lemma}
\label{lem:Gamma-sigma}
Suppose that $\alpha$ is an elliptic function measure defined over $k$.
Then, for almost all finite-order characters $\kappa$ of $(\mathbb{Z}_q^\times)^2$ and for all $\sigma\in\Gal(J/k)$, we have
\[
\Gamma_\alpha(\kappa)^ \sigma=\frac{\kappa^\sigma(\chi_E(\sigma))}{\kappa^\sigma(\chi_\mu(\sigma))}\Gamma_\alpha(\kappa^\sigma).
\]
\end{Lemma}

\begin{proof}
It follows from Lemma~\ref{lem:Gamma-Fourier} that
\[
\Gamma_\alpha(\kappa)^\sigma=\tau(\kappa)^\sigma\sum_{\underline x\in \mathbb Z/q^m\times\mathbb Z/q^n}\kappa^{-1}(\underline x)^\sigma\hat\alpha(\underline\zeta^{\underline x})^\sigma.
\]
We have
\begin{align*}
\tau(\kappa)^\sigma&=\frac{1}{q^{m+n}}\sum_{(x_1,x_2)\in \mathbb Z/q^m\times \mathbb Z/q^n}\kappa(x_1,x_2)^\sigma(\zeta_m^{-x_1}\zeta_n^{-x_2})^\sigma\\
&=\frac{1}{q^{m+n}}\sum_{(x_1,x_2)\in \mathbb Z/q^m\times \mathbb Z/q^n}\kappa(x_1,x_2)^\sigma\zeta_m^{-\chi_{\cyc}(\sigma)x_1}\zeta_n^{-\chi_{\cyc}(\sigma)x_2}\\
&=\frac{\kappa^\sigma(\chi_{\cyc}(\sigma),\chi_{\cyc}(\sigma))^{-1}}{q^{m+n}}\sum_{(x_1,x_2)\in \mathbb Z/q^m\times \mathbb Z/q^n}\kappa(x_1,x_2)^\sigma\zeta_m^{-x_1}\zeta_n^{-x_2}\\
&=\kappa^\sigma(\chi_\mu(\sigma))^{-1}\tau(\kappa^\sigma).
\end{align*}
Since $\alpha$ is an elliptic function measure, we have
\[
\hat\alpha(\underline\zeta^{\underline x})^\sigma=R(\delta(\underline\zeta^{\underline x})^\sigma)=R(\delta(\underline\zeta^{\chi_E(\sigma)\underline x}))=\hat\alpha(\underline\zeta^{\chi_E(\sigma)\underline x})
\]
for some $R\in k(E)$.
Therefore, combining these equations gives
\begin{align*}
\Gamma_\alpha(\kappa)^\sigma&= \kappa^\sigma(\chi_\mu(\sigma))^{-1}\tau(\kappa^\sigma) \sum_{\underline x\in \mathbb Z/q^m\times\mathbb Z/q^n}\kappa^{-1}(\underline x)^\sigma\hat\alpha(\underline\zeta^{\chi_E(\sigma)\underline x})\\
&=\frac{\kappa^\sigma(\chi_E(\sigma))}{\kappa^\sigma(\chi_\mu(\sigma))}\tau(\kappa^\sigma) \sum_{\underline x\in \mathbb Z/q^m\times\mathbb Z/q^n}\kappa^{-1}(\underline x)^\sigma\hat\alpha(\underline\zeta^{\underline x})\\
&=\frac{\kappa^\sigma(\chi_E(\sigma))}{\kappa^\sigma(\chi_\mu(\sigma))}\Gamma_\alpha(\kappa^\sigma)
\end{align*}
where the last equality follows from Lemma~\ref{lem:Gamma-Fourier} applied to $\kappa^\sigma$.
\end{proof}

\section{Algebraic Independence Results}

The main result of this section is Theorem~\ref{thm:Hida-Lam}, where we prove an algebraic independence result of functions on $E_{q^\infty}$ taking values in a \emph{finite field} whose characteristic is distinct from $q$.
The first step is Theorem~\ref{thm: Lamplugh 4.5}, which is an analogue of \cite[Proposition~3.1]{sinnott} (and also \cite[Theorem~4.5]{Lam15}).
This step involves proving an algebraic independence result of functions on $E_{q^\infty}$ taking values in a \emph{general field}, $\mathcal F$.
Let $E$ be an elliptic curve as fixed in Section~\ref{section: preliminaries}.
We suppose that $E$ can be considered as a curve over the field $\mathcal F$ (for example, the residue field of $H$ modulo a prime ideal).
Suppose that $q>3$ is a rational prime that splits in $\mathcal O_K$ and $\charac(\mathcal F)\neq q$.
This result essentially says that endomorphisms in $\End(E_{\mathfrak q^\infty})\times \End(E_{{\mathfrak q^*}^\infty})$ which are independent over $\End_{\mathcal F}(E)$, are algebraically independent.

The following lemma is required for the proof of Theorem~\ref{thm: Lamplugh 4.5}.

\begin{Lemma}
\label{prop: Lam15 4.4}
Let $\Phi_1, \ldots, \Phi_s$ be non-trivial morphisms from $E^n$ to $E$ of the form
\[
\Phi_j: (P_i)_{i=1}^n \mapsto \sum_{i=1}^n \alpha_{ij}(P_i)
\]
where $\alpha_{ij}\in \End_{\mathcal F}(E)$ for all $1\leq i \leq n$ and $1\leq j \leq s$.
Suppose that the only relation of the kind $\alpha\Phi_k = \beta \Phi_{\ell}$ for $\alpha, \beta\in \End_{\mathcal F}(E)$ and $k\neq \ell$, is when $\alpha = \beta =0$.
If $r_1, \ldots, r_s\in \mathcal F(E)$ with $\sum_{j=1}^s r_j \circ \Phi_j = 0$, then each $r_j$ is a constant function.
\end{Lemma}

\begin{proof}
See {\cite[Proposition~4.4]{Lam15}}.
\end{proof}

\begin{Th}
\label{thm: Lamplugh 4.5}
Let $\mathcal F$ be any field as above, and $E$ an elliptic curve defined over $\mathcal F$ such that $\End_{\mathcal F}(E)\simeq \mathcal O_K$.
Suppose that $\underline{\eta}_1, \ldots, \underline{\eta}_s \in \End(E_{\mathfrak q^\infty})\times \End(E_{{\mathfrak q^*}^\infty})$ are such that $\alpha\underline{\eta}_k = \beta\underline{\eta}_{\ell}$ for $k\neq \ell$ and some $\alpha, \beta\in \End_{\mathcal F}(E)$ only when $\alpha=\beta=0$.
Consider the function 
\[
R = \sum_{j=1}^s r_j \circ \underline{\eta}_j: E_{\mathfrak q^\infty}\times E_{{\mathfrak q^*}^\infty}\rightarrow \overline{\mathcal F} \]
where $r_j \in \mathcal F(E)$ and $\overline{\mathcal F}$ denotes an algebraic closure of $\mathcal F$.
If $R(Q)=0$ for all $Q\in E_{\mathfrak q^\infty}\times E_{{\mathfrak q^*}^\infty}$, then all $r_j$'s are constant functions.
\end{Th}

\begin{proof}
We recall that $\End(E_{\mathfrak q^\infty})\simeq \mathcal O_{\mathfrak q}$, $\End(E_{{\mathfrak q^*}^\infty})\simeq \mathcal O_{{\mathfrak q^*}}$ and $\End_{\mathcal F}(E)\simeq \mathcal O_K$.
Consider a free $\mathcal O_K$ submodule $A$ of $\mathcal O_{\mathfrak q}\times\mathcal O_{{\mathfrak q^*}}$ of rank $n$ that contains $\underline{\eta}_j$ for $1\leq j\leq s$.
Let $\{\underline{\varepsilon}_{i}\}_{i=1}^n$ be an $\mathcal O_K$-basis of $A$.
Then, there exist unique $\alpha_{ij}\in \mathcal O_K$ such that
\[
\underline{\eta}_j = \sum_{i=1}^n \alpha_{ij}\underline{\varepsilon}_i.
\]
Define the map 
\[
\iota: E_{\mathfrak q^\infty}\times E_{{\mathfrak q^*}^\infty}= E_{q^\infty}\rightarrow E^{n}\textrm{ given by } Q \mapsto \left(\underline{\varepsilon}_i Q \right)_{i=1}^n.\]
For each $1\leq j \leq s$, denote the morphism
\[
\Phi_j : E^{n} \rightarrow E; \qquad \left( P_i\right)_{i=1}^n \mapsto \sum_{i=1}^n \alpha_{ij}P_i.
\]

We have assumed that
\[
\sum_{j=1}^s r_j \circ \Phi_j(\mathcal{Q}) =0 \qquad \textrm{for all }\mathcal{Q}\in \iota(E_{\mathfrak q^\infty}\times E_{{\mathfrak q^*}^\infty})\subseteq E^n.
\]
Hence, the above equality must hold for all $\mathcal{Q}$ in the Zariski closure of $\iota(E_{\mathfrak q^\infty}\times E_{{\mathfrak q^*}^\infty})$.
It follows from basic facts about Zariski closed subgroups of $E^{n}$ (see \cite[Lemmas~1 and 3]{Sch87}) that either the Zariski closure of $\iota(E_{q^\infty})$ is $E^{n}$ or there exist $\alpha_i \in \mathcal O_K$ (not all zero) such that
\[
\sum_{i=1}^n \alpha_i \underline{\varepsilon}_i (Q) = 0 \textrm{ for all }Q\in E_{q^\infty}.
\]
If the latter holds, it means that $\sum_{i=1}^n \alpha_i \underline{\varepsilon}_i =0$.
However, this contradicts the fact that $\underline{\varepsilon}_1, \ldots ,\underline{\varepsilon}_n$ is a basis for $A$.
Thus, the Zariski closure of $\iota(E_{q^\infty})$ is $E^{n}$.
Lemma~\ref{prop: Lam15 4.4} implies that each $r_i$ is a constant function.
\end{proof}

To prove the main result in this section, we need a strengthened version of Theorem~\ref{thm: Lamplugh 4.5}.
This is achieved by combining the following Diophantine approximation result (Lemma~\ref{lem:approx}) with a special case of a lemma due to Hida (Lemma~\ref{lem:Hida}), which we record below.

\begin{Lemma}
\label{lem:approx}
Given $\underline{\beta}_1, \ldots, \underline{\beta}_d\in\mathcal O_{\mathfrak q}\times \mathcal O_{{\mathfrak q^*}}$ for any integer $d\geq 1$, and a positive constant $c\leq 1$, there exists an integer $N$ such that for all $n\geq N$, there exist algebraic integers $b_1, \ldots, b_d\in \mathcal O_K$ and a unit $u \in \mathcal O_{\mathfrak q}^{\times}\times \mathcal O_{{\mathfrak q^*}}^\times$ satisfying
\begin{align*}
 v_{\mathfrak p} (u \underline{\beta}_i - b_i)&\geq n \textrm{ for }\mathfrak p\in \{\mathfrak q, {\mathfrak q^*}\} \textrm{ and}\\
 N_{K/\mathbb Q}(b_i) &< c \cdot q^{2n}.
\end{align*}
\end{Lemma}
\begin{proof}
See {\cite[Lemma~2.3.9]{Lam-thesis}}.
\end{proof}

\begin{Lemma}
\label{lem:Hida}
Let $r$ be a positive integer.
Let $X=\bigcup_{i=1}^k X_i$ be a proper subset of $\mathbb G_{m/\overline\mathbb Q_q}^2$ such that
\begin{itemize}
 \item[(i)] $X$ is Zariski closed.
 \item[(ii)]For each $i$, there exists a closed subscheme $Y_i$ that is stable under $t\mapsto t^{p^{rn}}$ for all $n\in\mathbb Z$, such that $X_i=\underline{\zeta}Y_i$ for certain $\underline{\zeta}\in\mu_{q^\infty}^2$;
\end{itemize}
There exists $P$, which is a $p^r$-power, and an infinite sequence of integers $0<n_1<n_2<\cdots$ such that for all $j\ge1$,
\[
\Xi_j\cap X=\emptyset,
\]
where $\Xi_j$ is defined by
\[
\left\{\left(\frac{P^x}{q^{n_j}},\frac{P^y}{q^{n_j}}\right)\mod\mathbb{Z}_q^2:x,y\in\mathbb Z\right\}\subset (\mathbb Q_q/\mathbb{Z}_q)^2
\]
after identifying $\mu_{q^\infty}^2$ with $(\mathbb Q_q/\mathbb{Z}_q)^2$ under an appropriate choice of basis.
\end{Lemma}

\begin{proof}
See {\cite[Lemma~3.4]{hida1}}.
\end{proof}
\begin{rem}\label{rk:Hida}On studying he proof of the above lemma, we see that $P\equiv 1 \mod q$.
If we write $P=1+q^vu$, where $q\nmid u$, then $\abs{\Xi_j}=q^{2(n_j-v)}$.\end{rem}

\begin{Th}\label{thm:Hida-Lam}
Let $\mathbb F$ be a finite field.
Suppose that $\underline{\eta}_1, \ldots, \underline{\eta}_s \in \End(E_{\mathfrak q^\infty})\times \End(E_{{\mathfrak q^*}^\infty})$ are such that the only relation of the kind $\alpha\underline{\eta}_k = \beta\underline{\eta}_{\ell}$ for $k\neq \ell$ and $\alpha, \beta\in \End_{\mathbb F}(E)$ is when $\alpha=\beta=0$.
Consider the function
\[
R = \sum_{i=1}^s r_i \circ \underline{\eta}_i: E_{\mathfrak q^\infty}\times E_{{\mathfrak q^*}^\infty}\rightarrow \overline{\mathbb F}
\]
where $r_i \in \mathbb F(E)$.
We identify $E_{q^\infty}$ with $\mu_{q^\infty}^2\subset \mathbb G_{m/\overline\mathbb Q_q}^2$.
Then either $\{Q\in E_{q^\infty}:R(Q)\ne 0\}$ is Zariski dense in $\mathbb G_{m/\overline\mathbb Q_q}^2$ or $R$ is identically zero.
In the latter case, all $r_i$'s are constant functions.
\end{Th}
\begin{proof}
Suppose that $\{Q\in E_{q^\infty}:R(Q)\ne 0\}$ is not Zariski dense and that $R$ is not identically zero.
We take $P$ to be a large enough $p$-power so that $R$ is defined over $\mathbb F_P$ (the finite field of cardinality $P)$.
Let $X$ be the Zariski closure of $\{Q\in E_{q^\infty}:R(Q)\ne 0\}$ in $\mathbb G_{m/\overline\mathbb Q_q}^2$.
Then, $X$ is a proper subset of $\mathbb G_{m/\overline\mathbb Q_q}^2$ and $X^P\subseteq X$.

Let $\log_q:\widehat\mathbb G_{m/\overline\mathbb Q_q}^2\rightarrow \widehat\mathbb G_{a/\overline\mathbb Q_q}^2$ be the $q$-adic logarithm map.
We decompose $\log_q(X)$ into a finite union of closed subsets, each of which is stable under the multiplication by $P$.
This allows us to write $X$ as a finite union of closed subschemes $X_i$ of the form $\underline\zeta Y_i$, where $\underline\zeta \in \mu_{q^\infty}^2$ and $Y_i$ is stable under $t\mapsto t^P$.
Therefore, Lemma~\ref{lem:Hida} applies.
In particular, there exists a sequence of integers $0<n_1<n_2<\cdots$ and a collection of subsets $\Xi_j$ of $q^{n_j}$-torsion points in $E_{q^\infty}$ on which $R$ vanishes, with $\abs{\Xi_j}=q^{2(n_j-v)}$ for some fixed integer $v$.

Define 
\[
\delta := \max_{1\leq i \leq s}\deg (r_i).
\]
We apply Lemma~\ref{lem:approx} to $\underline{\eta}_1, \ldots, \underline{\eta}_s$ and $c =\displaystyle \frac{1}{q^{2v} \cdot s \cdot \delta }$.
There exists an integer $N$ such that for all $n_j\geq N$, there are algebraic integers $b_1, \ldots, b_s \in \mathcal O_K$ and $u\in \mathcal O_{\mathfrak q}^\times \times \mathcal O_{{\mathfrak q^*}}^\times$ (depending on $n_j$) satisfying
\begin{align*}
 v_{\mathfrak p} (u \underline{\eta}_i - b_i)&\geq n_j \textrm{ for }\mathfrak p \in \{\mathfrak q, {\mathfrak q^*}\} \textrm{ and}\\
 \abs{ N_{K/\mathbb Q}(b_i)} &< c \cdot q^{2n_j}.
\end{align*}
In particular, the rational function
\[
R_{n_j} := \sum_{i=1}^s r_i \circ b_i \in \mathbb F(E)
\]
agrees with $R\circ u$ on $E_{q^{n_j}}$.
Thus, it vanishes on $\Xi_j$.
Moreover,
\[
\deg(R_{n_j})\le \sum_{i=1}^s\delta\cdot N_{K/\mathbb Q}(b_i)<s\delta\cdot c\cdot q^{2n_j}=q^{2n_j-2v}=\abs{\Xi_j}.
\]
Therefore, $R_{n_j}=0$ and thus $R$ is zero on $E_{q^{n_j}}$.
But $n_j$ can be arbitrarily large.
This implies that $R$ is identically zero, which is a contradiction.
This concludes the first assertion of the theorem.
The last assertion follows immediately from Theorem~\ref{thm: Lamplugh 4.5}.
\end{proof}

\section{A theorem on two-variable Gamma transforms}

The purpose of this section is to prove a two-variable version of \cite[Theorem~5.1]{Lam15} (which in turn generalizes a result of Sinnott \cite[Theorem~3.1]{sinnott}).
Our proof utilizes crucially Theorem~\ref{thm:Hida-Lam} from the previous section.
Throughout, we use the same notation introduced in Sections~\ref{section: preliminaries} and \ref{section: distribution and measures}.

\begin{Th}\label{thm:valuation-gamma-transform}
Let $\alpha$ be an elliptic function measure for $E$ defined over $k$ on $\mathbb{Z}_q^2$ that is supported on $(\mathbb{Z}_q^\times)^2$, and satisfies $\alpha\circ\omega=\alpha$ for all $\omega\in\mu_K^2$.
Let $R$ denote the corresponding rational function (so that $\hat\alpha=R\circ \delta$ as in Definition~\ref{defn:ellipticmeasure}), and let $n=\ord_\pi(R)$ (as in Definition~\ref{defn:ord}).
Then for a Zariski dense set of finite-order characters $\kappa$ of $(1+q\mathbb{Z}_q)^2$, we have
\[
\ord_\pi\left(\Gamma_\alpha(\kappa)\right)=n.
\]
\end{Th}
\begin{rem}\label{rk:Zar}
We view $\Hom(\mathbb{Z}_q^2,\mu_{q^\infty})$ as a subset of $\mathbb G_{m/\overline\mathbb Q_q}^2$ by sending $\kappa$ to $(\kappa(1,0),\kappa(0,1))$.
A set of finite-order characters is called \emph{Zariski dense} if its image in $\mathbb G_{m/\overline\mathbb Q_q}^2$ is a dense subset under the Zariski topology.
\end{rem}

The following lemma is a key technical ingredient of the proof of Theorem~\ref{thm:valuation-gamma-transform}.

\begin{Lemma}
\label{lem:beta}
Let $\alpha$ be an elliptic function measure as in the statement of Theorem~\ref{thm:valuation-gamma-transform}.
Define
\[
\beta=\sum_\eta \left(\alpha\circ \eta\right)|_{(1+q\mathbb{Z}_q)^2},
\]
where $\eta$ runs over a set of representatives for $(\mu_{q-1}/\mu_K)^2$ and $\alpha\circ\eta$ is defined as in Definition~\ref{def:dist}(i).
For each $y=(y_1,y_2){\in \mu_{q-1}^2}$, we write
\[
\beta_{y}=\beta|_{y_1(1+q^M\mathbb{Z}_q)\times y_2(1+q^M\mathbb{Z}_q)},
\]
where $M\ge1$ is the integer such that $\mu_{q^\infty}\cap k= \mu_{q^M}$.
Let $\kappa=(\kappa_1,\kappa_2)$ be a finite-order character of $(1+q\mathbb{Z}_q)^2$.
Suppose that there exist integers $m,n\ge M$ satisfying
\[
\ker(\kappa_1)=1+q^{m+M}\mathbb{Z}_q,\quad \ker(\kappa_2)=1+q^{n+M}\mathbb{Z}_q.
\]
Let $\underline\zeta=(\zeta_1,\zeta_2)\in \mu_{q^\infty}$ such that 
\[
\zeta_1^{q^m}=\kappa_1(1+q^m),\quad \zeta_2^{q^n}=\kappa_2(1+q^n).
\]
Then $\Gamma_\beta(\kappa)\in\pi\mathfrak{O}$ if and only if $\hat\beta_{y}(\underline \zeta^{y^{-1}})\in \pi\mathfrak{O}$ for all $y\in \mu_{q-1}^2$.
\end{Lemma}

\begin{proof}
Suppose that $\Gamma_\beta(\kappa)\in\pi\mathfrak{O}$.
Let $\sigma\in \Gal(J/k)$ and $\underline\xi\in \mu_{q^\infty}^2$.
Recall from the proof of Lemma~\ref{lem:Gamma-sigma} that 
\[
\hat\alpha(\underline\xi)^\sigma=\hat\alpha(\underline\xi^{\chi_E(\sigma)}).
\]
Since Fourier transform is additive, we have equivalently
\[
\hat\beta(\underline\xi)^\sigma=\hat\beta(\underline\xi^{\chi_E(\sigma)}).
\]
Furthermore, Lemma~\ref{lem:Gamma-sigma} asserts that
\[
\Gamma_\beta(\kappa)^ \sigma=\frac{\kappa^\sigma(\chi_E(\sigma))}{\kappa^\sigma(\chi_\mu(\sigma))}\Gamma_\beta(\kappa^\sigma).
\]
Thus, $\ord_\pi\left(\Gamma_\beta(\kappa^\sigma)\right)$ is independent of $\sigma\in\Gal(J/k)$ because $\Gamma_\beta(\kappa)\in\pi\mathfrak{O}$ by assumption and $\kappa$ takes values in the group of roots of unity.
In particular, $\Gamma_\beta(\kappa^\sigma)\in \pi \mathfrak{O}$ for all $\sigma\in\Gal(J/k)$ under our hypothesis that $\Gamma_\beta(\kappa)\in\pi\mathfrak{O}$.

Let $N=\max(m,n)$.
Write $k_{N-1}$ to denote the $(N-1)$-th layer of the $\mathbb{Z}_q$-extension $J/k$, and set $H=\Gal(k_{N-1}/k)$.
Let $y\in (1+q\mathbb{Z}_q)^2$.
We have
\begin{align*}
 \sum_{\sigma\in H}\kappa^\sigma(y)^{-1}\Gamma_\beta(\kappa^\sigma)&= \sum_{\sigma\in H}\kappa^\sigma(y)^{-1}\int_{(1+q\mathbb{Z}_q)^2}\kappa^\sigma(x) d\beta(x) \\
 &=\sum_{\sigma\in H}\int_{(1+q\mathbb{Z}_q)^2}\kappa^\sigma(x/y)d\beta(x)\\
 &=\int_{(1+q\mathbb{Z}_q)^2}\Tr_{k_{N-1}/k}\circ\kappa(x/y)d\beta(x) \\
 &=q^{N-1}\int_{y_1(1+q^m\mathbb{Z}_q)\times y_2(1+q^n\mathbb{Z}_q)}\kappa(x/y)d\beta(x) .
\end{align*}
Note that $q^{N-1}$ is a unit in $\mathfrak{O}$ (since $q\ne p$).
Therefore, $\Gamma_\beta(\kappa)\in\pi\mathfrak{O}$ implies that
\[
\int_{y_1(1+q^m\mathbb{Z}_q)\times y_2(1+q^n\mathbb{Z}_q)}\kappa(x/y)d\beta(x) \in\pi\mathfrak{O}.
\]
Let $x=(x_1,x_2)=y(1+q^mz_1,1+q^nz_2)=(y_1(1+q^mz_1),y_2(1+q^nz_2))$, where $z_1,z_2\in\mathbb{Z}_q$.
Then 
\[
\kappa(x/y)=\kappa(1+q^mz_1,1+q^nz_2)=\kappa((1+q^m)^{z_1},(1+q^n)^{z_2})=\zeta_1^{x_1/y_1-1}\zeta_2^{x_2/y_2-1}.
\]
Thus, we deduce that
\[
\int_{y_1(1+q^m\mathbb{Z}_q)\times y_2(1+q^n\mathbb{Z}_q)}\zeta_1^{x_1/y_1}\zeta_2^{x_2/y_2}d\beta(x) \in\pi\mathfrak{O}.
\]
If we replace $y$ by $yt=(y_1t_1,y_2t_2)$ and $(\zeta_1,\zeta_2)$ by $(\zeta_1^{t_1},\zeta_2^{t_2})$ for any $t=(t_1,t_2)\in (1+q^M\mathbb{Z}_q)^2$, the same containment holds.
Hence, summing over $t\in(1+q^M\mathbb{Z}_q)^2/(1+q^m\mathbb{Z}_q)\times (1+q^n\mathbb{Z}_q) $, we deduce that
\[
\hat\beta_y(\underline\zeta^{y^{-1}})=\int_{y(1+q\mathbb{Z}_q)^2}\zeta_1^{x_1/y_1}\zeta_2^{x_2/y_2}d\beta(x) \in\pi\mathfrak{O}.
\]

The converse follows from Lemma~\ref{lem:Gamma-Fourier} and the fact that the Gauss sum $\tau(\kappa)$ is a $\pi$-adic unit (which is a consequence of the fact that its conductor is coprime to $p$).
\end{proof}

\begin{proof}[Proof of Theorem~\ref{thm:valuation-gamma-transform}]
Without loss of generality, we assume that $n=\ord_\pi(R)=0$.
Let $\beta$ be as defined in the statement of Lemma~\ref{lem:beta}, and $w_K$ denote the number of elements in $\mu_K$ (which is coprime to $p>3$).
We have
\[
\frac{1}{w_K^2}\Gamma_\alpha(\kappa)=\Gamma_\beta(\kappa).
\]
Let us write 
\[
\alpha_{\eta y}=\alpha|_{\eta_1y_1(1+q^M\mathbb{Z}_q)\times \eta_2y_2(1+q^M\mathbb{Z}_q)}
\]
for $\eta=(\eta_1,\eta_2)\in \mu_{q-1}^2$ and $y=(y_1,y_2)\in (1+q\mathbb{Z}_q)^2$.
Note that $\alpha_{\eta y}$ is an elliptic function measure since it is a restriction of $\alpha$.
Furthermore, we write $R_{\eta y}$ for the rational function on $E$ attached to $\alpha_{\eta y}$ (meaning that $\hat\alpha_{\eta y}= R_{\eta y}\circ\delta$ as functions on $\mu_{q^\infty}^2$).
As can be seen in the proof of Lemma~\ref{lem:EFM}, $R_{\eta y}$ takes values in $\mathfrak{O}$.
Let $\tilde R_{\eta y}$ denote the function $R_{\eta y}$ modulo $\pi$.

Suppose that the set of characters $\kappa$ with $\ord_\pi(\Gamma_\alpha(\kappa))= 0$ is not Zariski dense.
Note that for all $\kappa$, we have $\ord_\pi(\Gamma_\alpha(\kappa))=\ord_\pi(\Gamma_\beta(\kappa))$ by Lemma~\ref{lem:integral-elliptic-function} and the fact that $p\nmid w_K$.
Equivalently, the set of characters $\kappa$ such that $\Gamma_\beta(\kappa)\not\in \pi\mathfrak{O}$ is not Zariski dense.
By Lemma~\ref{lem:beta}, the set of elements $Q\in E_{q^\infty}$ such that
\[
\sum_{\eta\in(\mu_{q-1}/\mu_K)^2} \tilde R_{\eta y}([\eta^{-1}]\circ Q)\ne 0
\]
is not Zariski dense.

Applying Theorem~\ref{thm:Hida-Lam}, it follows that $\tilde R_{\eta y}$ is a constant function.
Let $c_{\eta y}$ denote a constant of $\mathfrak{O}$ lifting $\tilde R_{\eta y}$ and let $\delta_0$ denote the Dirac measure of $\mathbb{Z}_q^2$ concentrated at $(0,0)$.
By definition, the Fourier transform $\hat\delta_0$ sends all $\underline{\zeta}\in \mu_{q^\infty}$ to $1$.
Therefore, the Fourier transform of $\alpha_{\eta y}-c_{\eta y}\delta_0$ takes values in $\pi\mathfrak{O}$.
In particular,
\[
\ord_{\pi}(\alpha_{\eta y}-c_{\eta y}\delta_0)>0.\]
However, if we restrict the measure $\alpha_{\eta y}-c_{\eta y}$ to $(\mathbb{Z}_q^\times)^2$, it agrees with $\alpha_{\eta y}$.
Thus,
\[
\ord_{\pi}(\alpha_{\eta y})=\ord_{\pi}(\alpha_{\eta y}-c_{\eta y}\delta_0)>0.\]
This contradicts our hypothesis that $\ord_\pi(R)=0$.
\end{proof}

\section{Proof of Theorem~\ref{thmA}}
\label{S:L}

In this section we apply Theorem~\ref{thm:valuation-gamma-transform} to study $\pi$-adic valuations of special values of $L$-functions and prove Theorem~\ref{thmA} stated in the introduction.

\subsection{Notation on ray class fields and CM elliptic curves}

We keep the notation introduced in Section~\ref{section: preliminaries}.
Recall that $K$ is a fixed imaginary quadratic field, and $H$ is its Hilbert class field.

\begin{Defi}
Let $\mathfrak a$ be an integral ideal of $K$.
\begin{itemize}
 \item We write $\mathscr{R}(\mathfrak a)$ for the ray class field of $K$ with conductor $\mathfrak a$.
 \item Given another ideal $\mathfrak b$ of $K$ which is coprime to $\mathfrak a$, we write $(\mathfrak b,\mathscr{R}(\mathfrak a))\in\Gal(\mathscr{R}(\mathfrak a)/K)$ for the Artin symbol of $\mathfrak b$.
 \item Given a character $\rho$ on $\Gal(\mathscr{R}(\mathfrak a)/K)$, we shall write $\rho(\mathfrak b)$ and $\rho\left((\mathfrak b,\mathscr{R}(\mathfrak a))\right)$ interchangeably.
\end{itemize}
\end{Defi}

Recall from \S\ref{section: preliminaries} that $E$ is an elliptic curve with complex multiplication by $\mathcal O_K$ with good reduction at the primes above $p$ and $q$.
Let $\omega_E$ denote the N\'eron differential for $E_{/{\mathscr{R}(\mathfrak f)}}$ and $\mathcal{L}=\Omega_\infty \mathcal O_K$ be its period lattice.
Note that $\Omega_{\infty}$ is uniquely determined up to a root of unity in $K$.

Given an ideal $\mathfrak b$ of $K$ coprime to $\mathfrak f$, there exists {$\Lambda(\mathfrak b)\in \mathscr{R}(\mathfrak f)^\times$ such that
\begin{equation}
 \mathcal{L}_{\mathfrak b} = \Lambda(\mathfrak b)\mathfrak b^{-1}\mathcal{L}
\label{eq:defnLambda}
\end{equation}
is the lattice associated with $E^{(\mathfrak b, \mathscr{R}(\mathfrak f))}$, as given by \cite[(16) on p.~42]{dS87} (see also \cite[D{\'e}finition, p.~198]{GS81}).
For simplicity, we shall write $E^{(\mathfrak b)}$ for the CM elliptic curve $E^{(\mathfrak b,\mathscr{R}(\mathfrak f))}$} and denote by 
\[
\lambda(\mathfrak b):E\rightarrow E^{(\mathfrak b)}
\]
the unique isogeny given by \cite[(15) on p.~42]{dS87}.

Consider the complex analytic isomorphism of complex Lie groups
\begin{equation}
\label{eqn: xi iso}
\xi_\mathfrak b : \mathbb C/\mathcal{L}_\mathfrak b \xrightarrow{\sim}E^{(\mathfrak b)}(\mathbb C) \ \textrm{given by} \ \xi_\mathfrak b(z) = \left( \wp(z,\mathcal{L}_\mathfrak b), \wp^\prime(z,\mathcal{L}_\mathfrak b)\right),
\end{equation}
where $\wp$ is the Weierstrass $\wp$-function and $\wp^\prime$ is the corresponding derivative.
We have the Weierstrass equation
\begin{equation}\label{eq:weierstrass}
 y^2=4x^3-g_2(\mathcal{L}_\mathfrak b)x-g_3(\mathcal{L}_\mathfrak b)
\end{equation}
describing $E^{(\mathfrak b)}$.

When $\mathfrak b=\mathcal O_K$, we shall write $\xi_1$ in place of $\xi_{\mathcal O_K}$.
We recall the following relation:
\begin{equation}\label{eq:xi-transform}
 \xi_\mathfrak b\left(\Lambda(\mathfrak b)z\right)=\lambda(\mathfrak b)(\xi_1(z))
\end{equation}
as discussed in \cite[commutative diagram (21) on p.~43]{dS87} and \cite[Proposition~4.10]{GS81}.

\subsection{Review on \texorpdfstring{$L$}{}-functions}
\begin{Defi}
\label{imprimitive L function}
Let $\mathfrak h$ be any integral ideal of $K$.
Let $\epsilon$ be any Hecke character of $K$ with conductor dividing some power of $\mathfrak h$.
The \emph{imprimitive $L$-function} of $\epsilon$ modulo $\mathfrak h$ is defined as follows
\[
L_{\mathfrak h}(\epsilon,s) = \sum_{\gcd(\mathfrak a, \mathfrak h)=1} \frac{\epsilon(\mathfrak a)}{(N\mathfrak a)^s}.
\]
\end{Defi}
Let $\epsilon$ be a Hecke character over $K$ {of infinity type $(a,b)$}.
Denote by $L(\epsilon,s)$ the \emph{primitive Hecke $L$-function} of $\epsilon$.
Recall that the imprimitive (or partial) $L$-function differs from the primitive (or classical) $L$-function by a finite number of Euler factors.
We can further define the \emph{primitive algebraic Hecke $L$-function},
\[
L^{\alg}(\overline\epsilon) := \frac{L\left({\overline\epsilon,a+b}\right)}{{(2\pi)^b\Omega_{\infty}^{b-a}} }.
\]
If $\Psi$ and $\kappa$ are as in the statement of Theorem~\ref{thmA}, then
\[
L^{\alg}\left(\overline{\Psi\kappa}\right)=L^{\alg}\left(\overline{\varphi^{k-j}\kappa}\chi_0 N^j\right)= \frac{L\left(\overline{\varphi^{k-j}\kappa}\chi_0 N^j,k+j\right)}{(2\pi)^j\Omega_{\infty}^{k-j} }=\frac{L\left(\overline{\varphi^{k-j}\kappa}\chi_0,k\right)}{(2\pi)^j\Omega_{\infty}^{k-j} },
\]
where $\varphi$ and $\chi_0$ are given as in \S\ref{section: preliminaries}.

Henceforth, we assume that $\kappa$ is of conductor $\mathfrak q^{m+1} {\mathfrak q^*}^{n+1}$ and set $ F_{m,n}=\mathscr{R}(\mathfrak h)$ with $\mathfrak h = \mathfrak g \mathfrak q^{m+1} {\mathfrak q^*}^{n+1}$.
Let $\mathfrak g$ be an auxiliary principal ideal that is divisible $\mathfrak f$ and is relatively prime to $pq$.
Then $\upsilon=\kappa\overline\chi_0$ is a character of $\Gal\left( \mathscr{R}(\mathfrak h)/K\right)$.
The imprimitive $L$-function of $\overline{\upsilon \varphi^{k-j}}$ modulo $\mathfrak h$ can be written as
\[
L_{\mathfrak h}\left(\overline{\upsilon \varphi^{k-j}},s \right) = \sum_{\tau \in \Gal(\mathscr{R}(\mathfrak h)/K)} \overline{\upsilon}(\tau)\sum_{(\mathfrak b,\mathscr{R}(\mathfrak h))=\tau} \frac{\overline{\varphi^{k-j}}(\mathfrak b)}{(N\mathfrak b)^{s}},
\]
where the second sum runs over integral ideals $\mathfrak b$ of $\mathcal O_K$ such that $\gcd(\mathfrak b,\mathfrak h)=1$.
We define the following partial imprimitive L-functions:
\begin{Defi}
Let $\mathfrak h$ and $\varphi$ be as above.
For $\tau\in\Gal(\mathscr{R}(\mathfrak h)/K)$, we define
\[
L_{\mathfrak h}\left(\overline{\varphi^{k-j}}, s , (\mathfrak b,\mathscr{R}(\mathfrak h))\right)=\sum_{\substack{\mathfrak b\unlhd\mathcal O_K\\(\mathfrak b,\mathscr{R}(\mathfrak h))=\tau\\\gcd(\mathfrak b,\mathfrak h)=1}} \frac{\overline{\varphi^{k-j}}(\mathfrak b)}{(N\mathfrak b)^{s}} .
\]
\end{Defi}
In particular, we have
\[
L_{\mathfrak h}\left(\overline{\upsilon\varphi^{k-j}},s \right) = \sum_{\tau \in \Gal(\mathscr{R}(\mathfrak h)/K)} \overline{\upsilon}(\tau)L_\mathfrak h(\overline{\varphi^{k-j}}, s , \tau).
\]

\begin{rem}
The (primitive and imprimitive) $L$-functions we have discussed so far only converge on some right half-plane.
However, they admit analytic continuations to the entire complex plane.
In order to prove Theorem~\ref{thmA}, we shall relate $L^{\alg}\left(\overline{\upsilon\varphi^{k-j}}\right)$ to Gamma transforms of certain elliptic function measure that we construct in the following subsection.
\end{rem}

Let $F = \mathscr{R}(\mathfrak g q)$ and write $\Delta =\Gal(F/K)$.
Since $\mathfrak f\mid \mathfrak g$, we have
\[
F = \mathscr{R}(\mathfrak g q) = K\left(j(E), h(E_{\mathfrak g q})\right) = H\left(x(E_{\mathfrak g q})\right).
\]
Here $h$ denotes a Weber function and we may choose this to be the $x$-coordinates on a Weierstrass model for the elliptic curve.
Set $F_\infty=\bigcup_{n\ge1} \mathscr{R}(\mathfrak g q^n)$; this is a $\mathbb Z_q^2$-extension of $F$.
Recall that $K_\infty$ is the $\mathbb{Z}_q^2$-extension of $K$, we fix an isomorphism 
\[
\Gal(F_\infty/K)\simeq \Gal(F/K) \times \Gal(F_\infty/F) \simeq \Gal(F/K) \times \Gal(K_\infty/K) \simeq \Delta \times \mathbb{Z}_q^2.
\]
By definition, $\upsilon=\kappa\overline\chi_0$ is a character of $\Gal(\mathscr{R}(\mathfrak f)\cdot K_\infty/K)$, which is a quotient of $\Gal(F_\infty/K)$.
Our hypothesis that $q\nmid [\mathscr{R}(\mathfrak f):K]$ allows us to regard $\kappa$ (resp. $\overline\chi_0$) as a character of $\mathbb{Z}_q^2$ (resp. $\Delta$).
Then $\kappa$ (resp. $\upsilon$) may be regarded as a character of $\Gal(F_{m,n}/F)$ (resp. $\Delta\times \Gal(F_{m,n}/F)$).

\begin{Defi}
Given an ideal $\mathfrak c$ of $\mathcal O_K$ that is coprime to $\mathfrak h$, let $\tau_\mathfrak c$ denote $(\mathfrak c,F_{m,n})=(\mathfrak c,\mathscr{R}(\mathfrak h))$.
\end{Defi}

We conclude this subsection with the following lemma on the Galois action on partial imprimitive $L$-values.

\begin{Lemma}
\label{lem:action-on-L}
Let $\mathfrak b$ be an ideal of $\mathcal O_K$ coprime to $\mathfrak h$ such that $(\mathfrak b,\mathscr{R}(\mathfrak f))=1$.
For any $\rho\in \mathfrak h^{-1}\mathcal{L}/\mathcal{L}$ and any integral ideal $\mathfrak c$ of $\mathcal O_K$ that is coprime to $\mathfrak h$, we have
\[
\tau_\mathfrak b\cdot \frac{L_{\mathfrak h}\left(\overline{\varphi^{k-j}}, k ,\tau_\mathfrak c\right)}{(2\pi)^j\rho^{k-j}}=\frac{L_{\mathfrak h}\left(\overline{\varphi^{k-j}}, k ,\tau_{\mathfrak b\mathfrak c} \right)}{(2\pi)^j\rho^{k-j}}.
\]
\end{Lemma}

\begin{proof}
Equation \eqref{eq:L-Eisen} in the appendix tells us that
\begin{equation}\label{eq:L-Eisen0}
\frac{L_{\mathfrak h}\left(\overline{\varphi^{k-j}}, k, \tau_\mathfrak c\right)}{(2\pi)^j\rho^{k-j}} 
= \frac{(N\mathfrak h\sqrt{d_K})^{-j}\Lambda(\mathfrak c)^{k-j} }{(k-1)!\varphi(\mathfrak c)^{k-j}} E_{j,k}\left(\rho, \mathcal{L} \right)^{\tau_\mathfrak c}.
\end{equation}
Since $\tau_\mathfrak b$ acts trivially on $\mathscr{R}(\mathfrak f)$, we deduce that
\begin{align*}
\tau_\mathfrak b\cdot \frac{L_{\mathfrak h}\left(\overline{\varphi^{k-j}}, k, \tau_\mathfrak c\right) }{(2\pi)^j\rho^{k-j}}&=\frac{ {(N\mathfrak h\sqrt{d_K})^{-j}}\Lambda(\mathfrak c)^{k-j}}{(k-1)!\varphi(\mathfrak c)^{k-j}} E_{j,k}\left( \rho, \mathcal{L} \right)^{\tau_{\mathfrak b\mathfrak c}}.
\end{align*}
On replacing $\mathfrak c$ by $\mathfrak b\mathfrak c$ in \eqref{eq:L-Eisen0}, we have
\[ \frac{L_{\mathfrak h}\left(\overline{\varphi^{k-j}}, k, {\tau_{\mathfrak b\mathfrak c}}\right) }{(2\pi)^j\rho^{k-j}}
=\frac{(N\mathfrak h\sqrt{d_K})^{-j}\Lambda(\mathfrak b\mathfrak c)^{k-j}}{(k-1)!\varphi(\mathfrak b\mathfrak c)^{k-j}} E_{j,k}\left( \rho, \mathcal{L} \right).\]

The hypothesis that $(\mathfrak b,\mathscr{R}(\mathfrak f))=1$ implies that $\varphi(\mathfrak b)=\Lambda(\mathfrak b)$ by \cite[(18) on p.~42]{dS87}.
Thus, equation (17) in \emph{op. cit.} tells us that
\[
\frac{ \Lambda(\mathfrak b\mathfrak c)}{\varphi(\mathfrak b\mathfrak c)}=\frac{ \Lambda(\mathfrak c)^{\tau_{\mathfrak b}}\Lambda(\mathfrak b)}{\varphi(\mathfrak c)\varphi(\mathfrak b)}=\frac{ \Lambda(\mathfrak c)}{\varphi(\mathfrak c)}.
\]
Hence the result follows.
\end{proof}

\subsection{A rational function with a canonical divisor}
\label{S:elliptic-function}
The goal of this section is to generalize the construction of a rational function on a CM elliptic curve from \cite[Section~6.3]{Lam15}.
In order to consider Hecke characters of more general infinity-type, we introduce a new derivative operator, which did not make an appearance in \emph{loc. cit.}
This allows us to carry out step (3) outlined in the introduction.
The notation introduced in the previous section will continue to be utilized.

Let $\mathfrak b$ be an integral ideal of $K$ that is coprime to $\mathfrak f$.
We fix an auxiliary ideal $\mathfrak a$ of $\mathcal O_K$ that is coprime to $6\mathfrak h$ and that $(\mathfrak a,\mathscr{R}(\mathfrak f))=1$.
Define the rational function $\zeta_{\mathfrak b,\mathfrak a}$ on $E^{(\mathfrak b)}$ by 
\begin{equation}
\label{eqn: zeta rational function}
\zeta_{\mathfrak b,\mathfrak a}(P) = \prod_Q
\left( x(P)-x(Q)\right)^{-1},
\end{equation}
where $Q$ runs over a set of representatives of $E_{\mathfrak a}^{(\mathfrak b)}\setminus \{0\} \pmod{\pm 1}$.
There exists a constant $c(\mathfrak b,\mathfrak a)\in H^\times$ such that the function \[\gamma_{\mathfrak b,\mathfrak a}(P) := c(\mathfrak b,\mathfrak a)\zeta_{\mathfrak b,\mathfrak a}(P)\]
has the property that for all $\beta\in \End\left(E^{(\mathfrak b)}\right)$ with $\gcd(\beta,\mathfrak a)=1$,
\[
\gamma_{\mathfrak b,\mathfrak a}(\beta(P)) = \prod_{R\in \ker(\beta)} \gamma_{\mathfrak b,\mathfrak a}(P \oplus R)
\](see \cite[Appendix]{coates91}).

We can write
\[
\mathcal{L}_{\mathfrak b} = \mathbb Z \omega_{1,\mathfrak b} + \mathbb Z \omega_{2,\mathfrak b}
\]
such that $\frac{\omega_{1,\mathfrak b}}{\omega_{2,\mathfrak b}}$ lies in the upper half plane.
We define the constant (see \cite[(4) on p.~48]{dS87})
\[
A(\mathcal{L}_{\mathfrak b}) := \frac{1}{2\pi i}\left(\omega_{1,\mathfrak b}\overline{\omega_{2,\mathfrak b}} - \overline{\omega_{1,\mathfrak b}}\omega_{2,\mathfrak b}\right).
\]
As in \cite[p.~57, (4)]{dS87}, let 
\[
\partial=-\frac{\partial}{\partial z},\quad \mathcal D_\mathfrak b=-A(\mathcal{L}_\mathfrak b)^{-1}\left(\overline{z}\frac{\partial}{\partial z}+\overline{\omega}_{1,\mathfrak b}\frac{\partial}{\partial \omega_{1,\mathfrak b}}+\overline{\omega}_{2,\mathfrak b}\frac{\partial}{\partial \omega_{2,\mathfrak b}}\right).
\]
For integers $0\le -j<k$, define the derivative operator $\mathscr{D}_{j,k}$ on $\mathbb C(E^{(\mathfrak b)})$ by
\[
\mathscr{D}_{j,k}(f) = \mathcal D_\mathfrak b^{-j}\partial^{k+j}\log f(z),
\]
where $z$ is a complex variable after identifying $E^{(\mathfrak b)}$ with $\mathbb C/\mathcal{L}_\mathfrak b$ via $\xi_\mathfrak b$ as given by \eqref{eq:xi-transform}.

\begin{Lemma}
\label{lemma: Lam14 3.1.4}
Let $Q$ be a primitive $\mathfrak h$-division point on $E$ and $\rho \in \mathfrak h^{-1}\mathcal{L}\setminus \mathcal{L}$.
Then there exist $\sigma\in\Gal(F_{m,n}/H)$ and $\zeta\in \mu_K$ (which we identify with $\Aut(E)$) such that
\[
Q = \zeta\left( \xi_1(\rho)^{\sigma}\right).
\]
Fix $\mathfrak c_0$ to be an ideal of $\mathcal O_K$ coprime to $\mathfrak h$ such that $\tau_{\mathfrak c_0}=\sigma$.
Suppose that $(\mathfrak c_0,\mathscr{R}(\mathfrak f))=1$.
Let $\mathfrak b$ and $\mathfrak c$ be ideals of $\mathcal O_K$ coprime to $\mathfrak h$ with $(\mathfrak c,\mathscr{R}(\mathfrak f))=1$.
Then
\begin{multline}
\mathscr{D}_{j,k}(\gamma_{\mathfrak b,\mathfrak a})\circ\lambda(\mathfrak b)(Q^{\tau_\mathfrak c}) = -(k-1)! \left( (N\mathfrak a)- \Lambda(\mathfrak a)^{k-j}\tau_\mathfrak a\right) \left(\frac{\varphi(\mathfrak b)}{ \Lambda(\mathfrak b)}\right)^{k-j}\times\\
\left(\frac{N\mathfrak h\sqrt{d_K}}{2\pi}\right)^j \frac{L_{\mathfrak h}\left(\overline{ \varphi^{k-j}}, k, \tau_{\mathfrak b\mathfrak c\mathfrak c_0}\right)}{(\zeta\rho)^{k-j}}.
\end{multline}
\end{Lemma}

\begin{proof}
By Class Field Theory, we have
\[
(\mathcal{O}_K/\mathfrak h)^\times/\mu_K \simeq \Gal(F_{m,n}/H)\hookrightarrow \Aut(E[\mathfrak h])\]
(see \cite[Chapter 2, proof of Theorem~2.3]{silverman}).
It follows that $\Aut(E[\mathfrak h])$ is generated by the image of $\Gal(F_{m,n}/H)$ and $\mu_K$.
The first assertion now follows just as in the proof of \cite[Lemma~3.1.4]{Lam-thesis} or \cite[Lemma~6.4]{Lam15}.

 In the appendix, we prove in \eqref{eq:app} that with $P=\xi_\mathfrak b(\Lambda(\mathfrak b)\rho)$
\[
\mathscr{D}_{j,k}(\gamma_{\mathfrak b,\mathfrak a})(P) = -(k-1)! \left( (N\mathfrak a)- \Lambda(\mathfrak a)^{k-j}\tau_\mathfrak a\right) \left(\frac{\varphi(\mathfrak b)}{ \Lambda(\mathfrak b)}\right)^{k-j}
\left(\frac{N\mathfrak h\sqrt{d_K}}{2\pi}\right)^j \frac{L_{\mathfrak h}\left(\overline{ \varphi^{k-j}}, k, \tau_{\mathfrak b}\right)}{\rho^{k-j}}.
\]
On replacing $P$ (resp. $\rho$) by $\zeta P$ (resp. $\zeta\rho$), we deduce that
\[
\mathscr{D}_{j,k}(\gamma_{\mathfrak b,\mathfrak a})(\zeta P) = -(k-1)! \left( (N\mathfrak a)- \Lambda(\mathfrak a)^{k-j}\tau_\mathfrak a\right) \left(\frac{\varphi(\mathfrak c)}{ \Lambda(\mathfrak c)}\right)^{k-j}
\left(\frac{N\mathfrak h\sqrt{d_K}}{2\pi}\right)^j \frac{L_{\mathfrak h}\left(\overline{ \varphi^{k-j}}, k, \tau_{\mathfrak b}\right)}{(\zeta\rho)^{k-j}}.\]
If we let $\tau_{\mathfrak c\mathfrak c_0}$ act on both sides of this equation, Lemma~\ref{lem:action-on-L} tells us that
\[
\mathscr{D}_{j,k}(\gamma_{\mathfrak b,\mathfrak a})(\zeta P^{\tau_{\mathfrak c\mathfrak c_0}} )= -(k-1)! \left( (N\mathfrak a)- \Lambda(\mathfrak a)^{k-j}\tau_\mathfrak a\right) \left(\frac{\varphi(\mathfrak c)}{ \Lambda(\mathfrak c)}\right)^{k-j}
\left(\frac{N\mathfrak h\sqrt{d_K}}{2\pi}\right)^j \frac{L_{\mathfrak h}\left(\overline{ \varphi^{k-j}}, k, \tau_{\mathfrak b\mathfrak c\mathfrak c_0}\right)}{(\zeta\rho)^{k-j}}.\]
The result now follows from \eqref{eq:xi-transform}.
\end{proof}

Define 
\[
\rho_{m,n}=\frac{\Omega_{\infty}}{g\nu^{m+1}{\nu^*}^{n+1}}\in \mathbb C^\times,
\]
where $g$, $\nu$, $\nu^*$ are fixed generators of $\mathfrak g$, $\mathfrak q$ and $\mathfrak q^*$ respectively (such generators exist since these ideals are assumed to be principal).
Then $\xi_1(\rho_{m,n})$ is a primitive $\mathfrak h$-division point of $E$ (since $\mathfrak h=\mathfrak g\mathfrak q^{m+1}{\mathfrak q^*}^{n+1}$).

Let $V$ (respectively $Q_{m,n}$) be a fixed primitive $\mathfrak g$-division (respectively $\mathfrak q^{m+1} {\mathfrak q^*}^{n+1}$-division) point on $E$.
By Lemma~\ref{lemma: Lam14 3.1.4}, there exist $\zeta\in \mu_K$ and $\sigma_0=\tau_{\mathfrak c_0}$, where $\mathfrak c_0$ is an ideal of $K$, coprime to $\mathfrak h$, depending on $V$ and $Q_{m,n}$, such that
\[
V\oplus Q_{m,n}=\zeta(\xi_1(\rho_{m,n})^{\sigma_0}).
\]
Since $(\mathfrak g,q)=1$, there is an  isomorphism of groups
\[
\Aut\left(E[\mathfrak g\mathfrak q^{m+1}(\mathfrak q^*)^{n+1}]\right)\simeq
\Aut\left(E[\mathfrak g]\right)\times
\Aut\left(E[\mathfrak q^{m+1}(\mathfrak q^*)^{n+1}]\right),
\]
which in turn induces the decomposition
\[
\Gal(F_{m,n}/H)\simeq\Gal(F_{m,n}/\mathscr{R}(\mathfrak g))\times\Gal(\mathscr{R}(\mathfrak g)/H).
\]
Therefore, we may choose $V$ so that $(\mathfrak c_0,\mathscr{R}(\mathfrak f))=1$ for all $m$ and $n$.

By Lemma~\ref{lemma: Lam14 3.1.4}, given any ideals $\mathfrak b$ and $\mathfrak c$ of $\mathcal O_K$ coprime to $\mathfrak h$ such that $(\mathfrak c,\mathscr{R}(\mathfrak f))=1$, we have
\begin{align}&\ \mathscr{D}_{j,k}(\gamma_{\mathfrak b,\mathfrak a})\circ \lambda(\mathfrak b)\left((V\oplus Q_{m,n})^{\tau_\mathfrak c}\right)\notag\\
 =&\ -(k-1)!\left(N(\mathfrak a) - \Lambda(\mathfrak a)^{k-j} \tau_\mathfrak a\right)\left(\frac{\varphi(\mathfrak b)^{k-j}}{\Lambda(\mathfrak b)^{k-j}}\left(\frac{N\mathfrak h \sqrt{d_K}}{2\pi}\right)^j \cdot\frac{L_{\mathfrak h}\left(\overline{\varphi^{k-j}}, k ,\tau_{\mathfrak b\mathfrak c\mathfrak c_0}\right)}{(\zeta\rho_{m,n})^{k-j}}\right)\notag\\
 =&\ \frac{-(k-1)!\varphi(\mathfrak b)^{k-j}}{\Lambda(\mathfrak b)^{k-j}}\left(\frac{N\mathfrak h \sqrt{d_K}}{2\pi}\right)^j\cdot\frac{N(\mathfrak a)L_{\mathfrak h}\left(\overline{\varphi^{k-j}}, k ,\tau_{\mathfrak b\mathfrak c\mathfrak c_0} \right)-\varphi(\mathfrak a)^{k-j}L_{\mathfrak h}\left(\overline{\varphi^{k-j}}, k ,\tau_{\mathfrak a\mathfrak b\mathfrak c\mathfrak c_0}\right)}{(\zeta\rho_{m,n})^{k-j}}.
 \label{eq:temp-formula}
\end{align}

We fix $\{\mathfrak b_i:i\in I\}$ to be a set of representatives of integral ideals in $K$ such that $\Gal(\mathscr{R}(\mathfrak f)/K)=\{(\mathfrak b_i,\mathscr{R}(\mathfrak f)):i\in I\}$.
Recall that $\Gal(F_{m,n}/K)\simeq \Delta\times \Gal(F_{m,n}/F)$, where $\Delta=\Gal(F/K)$.
Then,
\[
[\mathscr{R}(q\mathfrak g):\mathscr{R}(\mathfrak f)]\sum_{\sigma\in\Gal(F_{m,n}/\mathscr{R}(\mathfrak g))}\kappa^{-1}(\sigma)\sum_{i\in I}\chi_0(\mathfrak b_i)=\sum_{\eta\in\Gal(F_{m,n}/K)}\upsilon^{-1}(\eta).
\]
Let us regard  $\kappa$ as a character of $\Gal(F_{m,n}/\mathscr{R}(\mathfrak g))\simeq \Gal(F_{m,n}/F)\times \Gal(F/\mathscr{R}(\mathfrak g))$ sending the elements of $\Gal(F/\mathscr{R}(\mathfrak g))$ to $1$.
We deduce from \eqref{eq:temp-formula} that
\begin{equation}
\begin{split}
\label{rational function and L function expression}
&\ \sum_{\sigma\in\Gal(F_{m,n}/\mathscr{R}(\mathfrak g))}\kappa^{-1}(\sigma)\sum_{\delta\in\Gal(\mathscr{R}(\mathfrak g)/\mathscr{R}(\mathfrak f)),i\in I}\frac{\chi_0(\mathfrak b_i)\Lambda(\mathfrak b_i)^{k-j}}{\varphi(\mathfrak b_i)^{k-j}}\mathscr{D}_{j,k}(\gamma_{\mathfrak b_i,\mathfrak a})\circ\lambda(\mathfrak b_i)(V^\delta\oplus Q_{m,n}^\sigma) \\
=&\ -(k-1)!\left( \frac{N\mathfrak h \sqrt{d_K}}{2\pi}\right)^j \sum_{\eta\in\Gal(F_{m,n}/K)}\upsilon^{-1}(\eta)\frac{N(\mathfrak a)L_{\mathfrak h}\left(\overline{\varphi^{k-j}}, k ,\eta \tau_{\mathfrak c_0} \right)-\varphi(\mathfrak a)^{k-j} L_{\mathfrak h}\left(\overline{\varphi^{k-j}}, k ,\eta\tau_{\mathfrak a\mathfrak c_0}\right)}{(\zeta\rho_{m,n})^{k-j}}\\
=&\ -(k-1)!\left( \frac{N\mathfrak h \sqrt{d_K}}{2\pi}\right)^j\frac{N(\mathfrak a)\upsilon(\sigma_0)L_\mathfrak h\left(\overline{\varphi^{k-j}\upsilon},k\right)-\varphi(\mathfrak a)^{k-j}\upsilon(\sigma_0\tau_\mathfrak a)L_\mathfrak h\left(\overline{\varphi^{k-j}\upsilon},k\right)}{(\zeta\rho_{m,n})^{k-j}}\\
=&\ -(k-1)!\left( \frac{N\mathfrak h \sqrt{d_K}}{2\pi}\right)^j\left(N(\mathfrak a)-\varphi(\mathfrak a)^{k-j}\upsilon(\tau_\mathfrak a)\right)\upsilon(\sigma_0)\frac{L_\mathfrak h\left(\overline{\varphi^{k-j}\upsilon},k\right)}{(\zeta\rho_{m,n})^{k-j}}.
\end{split}
\end{equation}

The above calculations lead us to define the following rational function on $E$.

\begin{Defi}
\label{defi: rational function main}
Let $\mathfrak a$ be an ideal of $\mathcal O_K$ chosen as above.
Let $V$ be a primitive $\mathfrak g$-division point of $E$, we define a rational function on $E$
 sending $P\in E$ to 
 \[
\vartheta_{\fa,V}^\Psi(P)=\sum_{\delta\in\Gal(\mathscr{R}(\mathfrak g)/\mathscr{R}(\mathfrak f)),i\in I}\frac{\chi_0(\mathfrak b_i)\Lambda(\mathfrak b_i)^{k{-j}}}{\varphi(\mathfrak b_i)^{k-j}}\mathscr{D}_{j,k}(\gamma_{\mathfrak b_i,\mathfrak a})\circ\lambda(\mathfrak b_i)( V^\delta\oplus P).
\]
\end{Defi}

\subsection{Gamma transforms and \texorpdfstring{$L$}{}-values}
We can associate with $\vartheta_{\fa,V}^\Psi$ an elliptic function measure, $\alpha$ on $\mathbb{Z}_q^2$ via Lemma~\ref{lem:EFM}.
The measure $\alpha$ depends on $\Psi$ and our choice of $\mathfrak a$ and $V$.
We further define $\alpha^* = \alpha|_{(\mathbb{Z}_q^\times)^2 }$.

We now relate the Gamma transform of $\alpha^*$ to special values of imprimitive algebraic $L$-functions.
Recall that $p$ is a rational prime satisfying $(p) = \mathfrak p {\mathfrak p^*}$ in $\mathcal O_K$ with $\mathfrak p \neq \mathfrak p^*$ and $\gcd(p, 6q)=1$.
As before, set $\pi$ to be the uniformizer of the local field $k$, which is a finite unramified extension of $\mathbb Q_p$ containing $\mathbb Q_p(E_{q\mathfrak g})$.

\begin{Lemma}\label{thm: Lam15 6.6}
Let $\kappa$ be as before.
Then
\[
\ord_{\pi}\left( \Gamma_{\alpha^{*}} ( \kappa)\right) = \ord_{\pi}\left((k-1)!\left( N(\mathfrak a) - {\varphi}(\mathfrak a)^{k-j}\upsilon(\tau_\mathfrak a)\right)L_{\mathfrak h}^{\alg}(\overline{\upsilon\varphi^{k-j}}) \right),
\]
where $\upsilon=\kappa\overline\chi_0$.
\end{Lemma}

\begin{proof}
Let $\underline{\zeta}=(\zeta_1, \zeta_2)\in \mu_{q^\infty}^2$ and set $Q_{m,n} = \delta(\underline{\zeta})$ in our construction above.
Using Lemma~\ref{lem:Gamma-Fourier} in conjunction with \eqref{rational function and L function expression}, yields
\begin{align*}
\Gamma_{\alpha^*}(\kappa) &= \tau(\kappa)\sum_{\sigma\in \Gal(F_{m,n}/\mathscr{R}(\mathfrak g))} \chi^{-1}(\sigma)\vartheta_{\fa,V}^\Psi(Q_{m,n}^\sigma)\\
&={-(k-1)!\tau(\kappa)\left( \frac{N\mathfrak h \sqrt{d_K}}{2\pi}\right)^j}\left(N(\mathfrak a)-\varphi(\mathfrak a)^{k-j}\upsilon(\tau_\mathfrak a)\right)\upsilon(\sigma_0)\frac{L_\mathfrak h\left(\overline{\varphi^{k-j}\upsilon},k\right)}{(\zeta\rho_{m,n})^{k-j}}.
\end{align*}
Standard facts about Gauss sums tell us that $\ord_{\pi}(\tau(\kappa))= 0$ since the conductor of $\kappa$ is coprime to $p$.
Finally, as $\upsilon$ is a finite character, $\upsilon(\sigma_0)$ is a root of unity.
By our choice of $\mathfrak h$, we also know that $N\mathfrak h$ is coprime to $p$.
This completes the proof of the lemma.
\end{proof}
We now study the factor $N(\mathfrak a) - {\varphi}(\mathfrak a)^{k-j}\upsilon(\tau_{\mathfrak a})$.
Recall that $\tau_\mathfrak a$ denotes $(\mathfrak a,F_{m,n})$, and thus depends on $m$ and $n$, a priori.
However, we may regard it as an element of $\Gal(F_\infty/K)$ since the Artin symbols $\tau_\mathfrak a$ are compatible under restriction as $\mathfrak h$ varies over ideals dividing $\mathfrak g q^\infty$.

\begin{Lemma}
\label{lemma: factor has trivial pi-adic valuation}
For a Zariski dense set of $\kappa$, we have 
\[
\ord_\pi\left(N(\mathfrak a)-\varphi^{k-j}(\mathfrak a)\upsilon(\tau_\mathfrak a)\right)=0.
\]
\end{Lemma}

\begin{proof}
Suppose the contrary.
Let $(\zeta_1,\zeta_2)={(\kappa(\tau_{\mathfrak a,\mathfrak q}),\kappa(\tau_{\mathfrak a,{\mathfrak q^*}}))}\in \mu_{q^\infty}^2$, where $\tau_{\mathfrak a,\mathfrak{l}}$ denotes the restriction of $\tau_\mathfrak a$ to $\Gal(\mathscr{R}(\mathfrak g\mathfrak{l}^\infty)/\mathscr{R}(\mathfrak g))$.
Then, 
\[
\ord_\pi\left(N(\mathfrak a)-{\varphi}^{k-j}(\mathfrak a){\upsilon}(\tau_\mathfrak a)\right)=0
\]
if and only if 
\[
N(\mathfrak a){\varphi}(\mathfrak a)^{j-k}\not\equiv \zeta_1\zeta_2\mod \pi\mathfrak{O}
\]
since $(\mathfrak a,\mathscr{R}(\mathfrak f))=1$, which implies that $\chi_0(\tau_\mathfrak a)=1$.
Note that the left-hand side is independent of $\kappa$.
In particular, this condition is invariant under the map $(\zeta_1,\zeta_2)\mapsto(\zeta_1,\zeta_2)^{p^r}$, where $p^r$ is the cardinality of the residue field of $\mathfrak{O}$.

Our assumption that the set of $\kappa$ satisfying the stated property above is not Zariski dense allows us to apply Lemma~\ref{lem:Hida}.
Let $P$ be the power of $p^r$ given by the said lemma.
In particular, under the isomorphism $\mu_{q^\infty}^2\simeq (\mathbb Q_q/\mathbb{Z}_q)^2$, there exists an arbitrary large $n$ such that 
\begin{equation}\label{eq:mod-pi}
 N(\mathfrak a){\varphi}(\mathfrak a)^{j-k}\equiv \zeta_1\zeta_2\mod \pi\mathfrak{O}
\end{equation}
for all $(\zeta_1,\zeta_2)$ which can be identified with $\displaystyle\left(\frac{P^x}{q^n},\frac{P^{y}}{q^n}\right)$, where $x,y\in \mathbb Z$.
In particular, Remark~\ref{rk:Hida} tells us that there are $q^{2(n-v)}$ such elements, where $v=\ord_q(P-1)$.

Note that $q$-power roots of unity modulo $\pi\mathfrak{O}$ are distinct since $p\ne q$.
Suppose that the left-hand side of \eqref{eq:mod-pi} modulo $\pi$ is a $q^m$-th root of unity, where $m<n$.
Then, for each $q^n$-th root of unity $\zeta_1$, there are exactly $q^{m}$ choices of $q^n$-th roots of unity $\zeta_2$ such that \eqref{eq:mod-pi} holds.
This gives us at most $q^{n+m}$ choices of $(\zeta_1,\zeta_2)\in\mu_{q^n}^2$.
But this is a contradiction as soon as $n+m<2(n-v)$.
\end{proof}

\begin{rem}
\label{the Zariski dense set is open}
For a given ideal $\mathfrak a$, denote the Zariski dense set of characters described in Lemma~\ref{lemma: factor has trivial pi-adic valuation} by $Z_{\mathfrak a}$.
This set is defined by the equation 
\[
f(\mathfrak a) := N(\mathfrak a)-{\varphi}^{k-j}(\mathfrak a){\upsilon}(\tau_\mathfrak a) \not\equiv 0 \mod \pi\mathfrak{O}.
\]
But note that $Z_{\mathfrak a} = \bigcup_{m\in \pi\mathfrak{O}} Z_{m,\mathfrak a}$ where each $Z_{m,\mathfrak a}$ is defined by equation
\[
f_{m}(\mathfrak a) := f(\mathfrak a)- m \neq 0,
\]
as $m$ varies over elements of $\pi\mathfrak{O}$.
Since each $Z_{m,\mathfrak a}$ is Zariski open, we have that $Z_{\mathfrak a}$ is Zariski open.
\end{rem}

Once we combine Lemmas~\ref{thm: Lam15 6.6} and \ref{lemma: factor has trivial pi-adic valuation} with Theorem~\ref{thm:valuation-gamma-transform}, Theorem~\ref{thmA} follows.

\section{Proof of Theorem~\ref{thmB}}
We continue employing the notation introduced in \S\ref{S:L}.
Throughout this section, we assume that $j=0$.
In addition, we assume that the character $\chi_0$ of $\Gal(\mathscr{R}(\mathfrak f)/K)$ from \S\ref{section: preliminaries} satisfies
\begin{equation}
\label{eq:condition-rho}
\ord_\pi\left(\sum_{i\in I}\frac{\chi_0(\mathfrak b_i)}{\varphi(\mathfrak b_i)^k}\right)=0.
\end{equation}

\begin{rem}
\label{remark: sufficient conditions for technical hypothesis}
Note that the $\pi$-adic valuation in \eqref{eq:condition-rho} is always non-negative since $\mathfrak b_i$ are coprime to $\mathfrak p$.
Suppose that $p\nmid {[\mathscr{R}(\mathfrak f):K]}$, then there exists at least {a character $\rho$ of $\Gal(\mathscr{R}(\mathfrak f)/K)$ such that $\rho\chi_0$ satisfies \eqref{eq:condition-rho}.} 
Indeed, 
{\[
\sum_{\rho\in\widehat{\Gal(\mathscr{R}(\mathfrak f)/K)}}\sum_{i\in I}\frac{\rho\chi_0(\mathfrak b_i)}{\varphi(\mathfrak b_i)^k}=[\mathscr{R}(\mathfrak f):K].
\]
Therefore, if $\ord_\pi([\mathscr{R}(\mathfrak f):K])=0$}, at least one of the summands should have zero $\pi$-adic valuation.
\end{rem}

The following lemma generalizes \cite[Lemma~6.7]{Lam15} and is crucial in our proof of Theorem~\ref{thmB}.

\begin{Lemma}
\label{lemma: Lam15 Lemma 6.7}
{Suppose that our auxiliary ideal $\mathfrak a$ is chosen so that $\gcd\left(\mathfrak a, 6q\mathfrak p \mathfrak g\prod_{i\in I}\mathfrak b_i\right)=1$, $(\mathfrak a,\mathscr{R}(\mathfrak f))=1$}, and $\mathfrak a\equiv 1\mod \mathfrak f$.
Then,
\[
\ord_{\pi}\left( \vartheta_{\fa,V}^\Psi\right) ={\ord_\pi\left((k-1)!\right)}.
\]
\end{Lemma}
\begin{proof}
Let us first recall the following facts proved in \cite[proof of Lemma~6.7]{Lam15}.
\begin{itemize}
 \item[(a)] The rational function $\mathscr{D}_{0,k}(\gamma_{\mathfrak b,\mathfrak a})$ on $E^{(\mathfrak b)}$ has poles of order $k$ at all the elements of $P\in E_\mathfrak a^{(\mathfrak b)}\setminus\{0\}$, with leading coefficient with respect to $z-z_P$ equal to $(k-1)!$.
 \item[(b)]Furthermore, $\mathscr{D}_{0,k}(\gamma_{\mathfrak b,\mathfrak a})$ has a pole of order $k$ at $P=0$, with leading coefficient with respect to $z$ equal to $N(\mathfrak a)-1$.
 \item[(c)]The poles described above are the only poles of $\mathscr{D}_{0,k}(\gamma_{\mathfrak b,\mathfrak a})$.
 \item[(d)]Let $x_\mathfrak b$ and $y_\mathfrak b$ be the functions sending a point $P\in E^{(\mathfrak b)}$ to its $x$- and $y$-coordinates given by the Weierstrass equation \eqref{eq:weierstrass}.
 The only zeros of the function $x_\mathfrak b(P)-x_\mathfrak b(R)$ are $P=R$ and $P=\ominus R$.
 If $x_\mathfrak b(R)\ne0$, these are simple zeros and the leading coefficient with respect to $z-z_P$ is given by $y_\mathfrak b(P)$.
\end{itemize}

Let $i\in I$.
Since $\mathfrak a$ is coprime to $\mathfrak b_i$, the isogeny $\lambda(\mathfrak b_i)$ induces an isomorphism $E_\mathfrak a\simeq E_\mathfrak a^{(\mathfrak b_i)}$.
Therefore, by (c), the poles of $\mathscr{D}_{0,k}(\gamma_{\mathfrak b_i,\mathfrak a})\circ \lambda(\mathfrak b_i)$ are precisely the elements in $E_\mathfrak a$.
Recall from Definition~\ref{defi: rational function main} that
\[
\vartheta_{\fa,V}^\Psi(P) = \sum_{i\in I}\frac{\chi_0(\mathfrak b_i)\Lambda(\mathfrak b_i)^k}{\varphi(\mathfrak b_i)^k}{\sum_{\delta\in\Gal(\mathscr{R}(\mathfrak g)/\mathscr{R}(\mathfrak f))}}\mathscr{D}_{0,k}(\gamma_{\mathfrak b_i,\mathfrak a})\circ\lambda(\mathfrak b_i)( V^\delta\oplus P).
\]
In particular, the poles of $\vartheta_{\fa,V}^\Psi(P)$ are given by $U\ominus V^\delta$, where $U\in E_\mathfrak a$ and $\delta\in \Gal(\mathscr{R}(\mathfrak g)/{\mathscr{R}(\mathfrak f)})$.

Let $P$ be a pole of $\mathscr{D}_{0,k}(\gamma_{\mathfrak b_i,\mathfrak a})\circ \lambda(\mathfrak b_i)$.
By \eqref{eq:xi-transform}, the leading coefficient of $\mathscr{D}_{0,k}(\gamma_{\mathfrak b_i,\mathfrak a})\circ \lambda(\mathfrak b_i)$ with respect to $z-z_P$ is that of $\mathscr{D}_{0,k}(\gamma_{1,\mathfrak a})$ multiplied by $\Lambda(\mathfrak b_i)^{-k}$, where $\gamma_{1,\mathfrak a}$ denotes the rational function on $E$ (so corresponding to the choice of $i$ gives $E^{(\mathfrak b_i)}=E$).
Consequently, by (a) the leading coefficient of $\vartheta_{\fa,V}^\Psi$ with respect to $z-z_P$, when $P$ is the pole $U\ominus V^\sigma$ where $U\in E_\mathfrak a\setminus\{0\}$, is given by
\[
(k-1)!\sum_{i\in I}\frac{\chi_0(\mathfrak b_i)}{\varphi(\mathfrak b_i)^k},
\]
which has $\pi$-adic valuation equal to $\ord_\pi\left((k-1)!\right)$ by assumption \eqref{eq:condition-rho}.

Let $i\in I$, $\delta \in \Gal(\mathscr{R}(\mathfrak g)/\mathscr{R}(\mathfrak f))$ and $Q\in E_\mathfrak a \setminus\{0\}$.
By (d), the rational functions (on $E$) given by $x_{\mathfrak b_i}\circ \lambda(\mathfrak b_i)(P\oplus V^\delta)-x_{\mathfrak b_i}\circ \lambda(\mathfrak b_i)(Q)$ and $x(P\oplus V^\delta)-x(Q)$ (where $x$ denotes the $x$-coordinate function on $E$) have the same zeros.
Furthermore, by \eqref{eq:xi-transform}, the leading terms of these two rational functions differ by the constant $\Lambda(\mathfrak b_i)$.
Consequently, these two functions differ by a unit in $\mathfrak{O}$.
Therefore, as in \cite[proof of Lemma~6.7]{Lam15}, we can write
\[
\vartheta_{\fa,V}^\Psi(P)=g(P)\prod_{\substack{\delta\in \Gal(\mathscr{R}(\mathfrak g)/\mathscr{R}(\mathfrak f)) \\ Q\in\left( E_\mathfrak a \setminus\{0\}\right)/ \pm1}}\left((x(P\oplus V^\delta)-x(Q)\right)^{-k},
\]
where $g$ is a rational function on $E$ belonging to
\[
\mathfrak{O}\Big[x\left(\lambda(\mathfrak b_i)(P\oplus V^\delta)\right), y\left(\lambda(\mathfrak b_i)(P\oplus V^\delta)\right) : i\in I, \delta\in \Gal(\mathscr{R}(\mathfrak g)/\mathscr{R}(\mathfrak f))\Big].
\]
In particular $\ord_\pi(g)\ge0$.

As has been established in \cite[proof of Lemma~6.7]{Lam15}, the functions $x(P\oplus V^\delta)-x(Q)$ take values in $\mathfrak{O}^\times$ for almost all $P$.
Furthermore, by comparing leading terms at $P=U\ominus V^\delta$, we deduce that $g$ takes values in $\mathfrak{O}^\times$ at these points.
Thus, $\ord_\pi(g)=0$, which concludes the proof.
\end{proof}

We can now prove Theorem~\ref{thmB}.
Let $\upsilon=\kappa\overline\chi_0$ as before.
By an argument similar to Lemma~\ref{lemma: factor has trivial pi-adic valuation} it suffices to prove the theorem for imprimitive values $L_{\mathfrak h}^{\alg}\left(\overline{\Psi\kappa}\right)$ because for almost all finite-order characters $\kappa$ of $\Gal(K_\infty/K)$, we have
\[
\ord_{\pi}\left( L^{\alg} \left({\overline{\Psi\kappa}}\right)\right) = \ord_{\pi}\left( L_{\mathfrak h}^{\alg}\left({\overline{\Psi\kappa}}\right)\right).
\]
Indeed, for any prime ideal $\mathfrak{r}$ of $K$ and for almost all characters $\kappa$,
\[
\ord_{\pi}\left(1 - \frac{{\overline{\Psi\kappa(\mathfrak{r})}}}{N(\mathfrak{r})^k} \right) =0
\]
as the $q$-power roots of unity modulo $\pi\mathfrak{O}$ are distinct since $p\ne q$.

Lemma~\ref{lemma: Lam15 Lemma 6.7} asserts that {$\ord_{\pi}\vartheta_{\fa,V}^\Psi=\ord_\pi\left((k-1)!\right)$}.
In particular, the associated elliptic function measure $\alpha^*$ satisfies $\ord_\pi\alpha^*={\ord_\pi\left((k-1)!\right)}$.
Therefore, on combining Lemma~\ref{thm: Lam15 6.6} with Theorem~\ref{thm:valuation-gamma-transform}, we deduce that for a Zariski dense set of $\kappa$, we have
\[
\ord_{\pi}\left( \left( N(\mathfrak a) - {\varphi}(\mathfrak a)^k{\upsilon(\tau_\mathfrak a)}\right) L_\mathfrak h^{\alg}\left({\overline{\Psi\kappa}}\right)\right)=0.
\] 
The same argument as in Remark \ref{the Zariski dense set is open} shows that this Zariski dense set is also open.
Since the intersection of two open dense sets is open dense, there exists a dense set of characters $\kappa$ with
\[\ord_{\pi}\left( L^{\alg}\left(\overline{\Psi\kappa}\right)\right)=0.\]

\appendix
\section{appendix}
In this appendix we carry out a technical calculation required in the proof of Lemma~\ref{lemma: Lam14 3.1.4}.
For this calculation, we rely heavily on the work of de Shalit in \cite{dS87}.
In particular, we express special $L$-values in terms of logarithmic derivatives of rational functions.
We do so by relating both of these quantities to values of Eisenstein series.

\subsection{Relating rational functions to Eisenstein series}
As in the main text, let $K$ be an imaginary quadratic field and $H/K$ be the Hilbert class field of $K$.
Let $E_{/H}$ be a CM elliptic curve with CM by $\mathcal{O}_K$ and $\mathcal{L}$ be the associated lattice.
Let $\mathfrak a$ and $\mathfrak b$ be ideals of $K$ such that $\mathfrak b$ is coprime to $6\mathfrak f$.
With respect to $\mathcal{L}_\mathfrak b$, we can define an \emph{elliptic function}, denoted by $\Theta(z;\mathcal{L}_\mathfrak b,\mathfrak a)$, as in \cite[Chapter~II, Section~2.3, (10) on p.~49]{dS87}.
Let $\xi_\mathfrak b$ be the isomorphism of complex Lie groups defined in \eqref{eqn: xi iso}.
It follows from \cite[(16) on p.~54]{dS87} that for any $z\in\mathbb C$ with $P=\xi_\mathfrak b(z)\in E^{(\mathfrak b)}$,
\begin{equation}
 \Theta(z;\mathcal{L}_\mathfrak b,\mathfrak a) =C_{\mathfrak b,\mathfrak a}\cdot \zeta_{\mathfrak b,\mathfrak a}(P)^{12},
\label{eq:Theta-zeta}
\end{equation}
where $\zeta_{\mathfrak b,\mathfrak a}(P)$ is the rational function introduced in \eqref{eqn: zeta rational function} and $C_{\mathfrak b,\mathfrak a}$ is some constant that is independent of $P$ and $z$ (the power of 12 appears because the product in \eqref{eqn: zeta rational function} is taken over $\mathfrak a$-torsions modulo $\pm1$, whereas the product in \cite[(16) on p.~54]{dS87} is taken over all non-trivial $\mathfrak a$-torsions, without modulo $\pm1$).

For integers $k\geq 1$ and $0\leq -j < k$, let $E_{j,k}(z,\mathcal{L}_\mathfrak b)$ be the $(j,k)$-th \emph{Eisenstein series} associated to the lattice $\mathcal{L}_\mathfrak b$ given as in \cite[(5) on p.~57]{dS87}.
Notice that when $k+j\ge3$, we have explicitly
\[
E_{j,k}(z,\mathcal{L}_\mathfrak b) = (k-1)!A(\mathcal{L}_\mathfrak b)^j{\sum_{w\in \mathcal{L}_\mathfrak b}}'\frac{(\overline{z} + \overline{w})^{k-j}}{\abs{z+w}^{2k}}=(k-1)!A(\mathcal{L}_\mathfrak b)^j{\sum_{w\in \mathcal{L}_\mathfrak b}}'\frac{(\overline{z} + \overline{w})^k (z + w)^j}{\abs{z+w}^{2(k+j)}}.
\]
Here, the sum runs over all $w\in \mathcal{L}_\mathfrak b$ except possibly $w=-z$ if $z\in \mathcal{L}_\mathfrak b$.
Further, for each integral ideal $\mathfrak a$, we can define (see \cite[(5) on p.~57]{dS87})
\[
E_{j,k}(z;\mathcal{L}_\mathfrak b,\mathfrak a) = (N\mathfrak a) E_{j,k}(z,\mathcal{L}_\mathfrak b) - E_{j,k}(z, \mathfrak a^{-1}\mathcal{L}_\mathfrak b).
\]

From \eqref{eq:Theta-zeta}, we deduce that, for $k\geq 1$, 
\begin{equation}
\label{Dk and Ez}
\begin{split}
12\mathscr{D}_{j,k}(\gamma_{\mathfrak b,\mathfrak a})(P) &= \mathcal D_\mathfrak b^{-j}\partial^{k+j} \log \Theta(z;\mathcal{L}_\mathfrak b,\mathfrak a)\\
&= -12 E_{j,k}(z,\mathcal{L}_\mathfrak b, \mathfrak a)\quad \textrm{ by \cite[Chapter~II, Section~3.1, (7) on p.~58]{dS87}.}
\end{split}
\end{equation}

\subsection{Relating Eisenstein series to rational \texorpdfstring{$L$}{}-values}
Recall that $\mathfrak f$ is an ideal of $\mathcal O_K$ that is divisible by the conductor of the Hecke character $\varphi$.
Let $\mathfrak m$ be a principal ideal of $\mathcal O_K$ such that $\mathfrak f\mid \mathfrak m$.
Let $\mathfrak c$ be another ideal which is coprime to $\mathfrak m$.
Then for any $\Omega\in \mathbb C^\times$ \cite[Chapter~II, Proposition~3.5, p.~62]{dS87} asserts that
\begin{equation}
\label{3.5 ds}
(N\mathfrak m^{-j})E_{j,k}\left(\Omega, \mathfrak c^{-1}\mathfrak m\Omega\right) = (k-1)! \left( \frac{\sqrt{d_K}}{2\pi}\right)^{j}\Omega^{j-k}\varphi(\mathfrak c)^{k-j} L_{\mathfrak m}(\overline{\varphi^{k-j}}, k, (\mathfrak c, \mathscr{R}(\mathfrak m))).
\end{equation}
Let $\alpha\in\mathcal O_K$ be a generator of our chosen principal ideal $\mathfrak m$.
We choose $\Omega\in\mathbb C^\times$ in \eqref{3.5 ds} to be the period $\Omega_\infty$ so that
\[
\mathcal{L}=\Omega_\infty \mathcal{O}_K.
\]
Let $\rho$ be the primitive $\mathfrak m$-division point on $\mathbb C/\mathcal{L}$ given by $\rho = \frac{\Omega_\infty}{\alpha}$.
Then,
\begin{align*}
E_{j,k}\left( \Omega_\infty, \mathfrak c^{-1}\mathfrak m\Omega_\infty \right) &= E_{j,k}\left( \rho\alpha, \mathfrak c^{-1}\mathfrak m\Omega_\infty \right)\\ 
&= E_{j,k}\left( \rho\alpha, \mathfrak c^{-1}\mathfrak m\mathcal{L} \right)\\
&= \alpha^{j-k}E_{j,k}\left( \rho, \mathfrak c^{-1}\mathcal{L} \right) \quad \textrm{by \cite[Proposition~3.3(i), p.~58]{dS87}}\\ 
&= \alpha^{j-k}\Lambda(\mathfrak c)^{k-j} E_{j,k}\left( \rho, \mathcal{L} \right)^{(\mathfrak c, \mathscr{R}(\mathfrak m))} \quad \textrm{by \cite[Proposition~3.3(iii), p.~58]{dS87}}, 
\end{align*}
where $\Lambda(\mathfrak c)\in H^\times$ is defined as in \eqref{eq:defnLambda}.

Combined with \eqref{3.5 ds}, the above calculation shows that
\begin{align}
\notag(k-1)! L_{\mathfrak m}\left(\overline{\varphi^{k-j}}, k, (\mathfrak c,\mathscr{R}(\mathfrak m))\right) 
&= \left(\frac{\Lambda(\mathfrak c) \Omega_\infty}{\alpha \varphi(\mathfrak c)} \right)^{k-j}\left(\frac{2\pi}{N\mathfrak m\sqrt{d_K}}\right)^j E_{j,k}\left( \rho, \mathcal{L} \right)^{(\mathfrak c,\mathscr{R}(\mathfrak m))}\\
&= \left(\frac{\Lambda(\mathfrak c) \rho}{\varphi(\mathfrak c)}\right)^{k-j} \left(\frac{2\pi}{N\mathfrak m\sqrt{d_K}}\right)^j E_{j,k}\left( \Lambda(\mathfrak c)\rho, \mathcal{L}_{\mathfrak c} \right) \quad \textrm{by \cite[(8), p.~58]{dS87}}
\label{eq:L-Eisen}.
\end{align}

\begin{rem}
In the special case when $H=K$ and $E$ is defined over $K$, (i.e., $K$ has class number 1) we know from \cite[(18) on p.~42]{dS87} that $\Lambda(\mathfrak c) = \varphi(\mathfrak c)$.
Moreover, it is also clear in this case that $\psi = \varphi$.
Therefore, on taking $j=0$, we obtain
\[
L_{\mathfrak m}\left(\overline{\psi^k}, k, (\mathfrak c,\mathscr{R}(\mathfrak m))\right) = \frac{\rho^k}{(k-1)!} E_k\left( \psi(\mathfrak c)\rho, \mathcal{L}_\mathfrak c \right)
\]
(c.f. \cite[Theorem~6.2]{Lam15}).
\end{rem}

\subsection{Relating rational functions to \texorpdfstring{$L$}{}-values}
\label{appendix final calculation}
Our final step is to combine the calculations in the previous two sections to relate the image of the operator $\mathscr{D}_{j,k}$ applied to our chosen rational function to the $(j,k)$-th Eisenstein series.
Let $P$ be an $\mathfrak m$-torsion on $E$.
We know from \eqref{Dk and Ez} that
\begin{align*}
\mathscr{D}_{j,k}(\gamma_{\mathfrak c,\mathfrak a})(P) &= - E_{j,k}(z;\mathcal{L}_\mathfrak c,\mathfrak a)\\
&= -\left( (N\mathfrak a)E_{j,k}(z,\mathcal{L}_\mathfrak c) - E_{j,k}(z,\mathfrak a^{-1}\mathcal{L}_\mathfrak c)\right)\\
&= -\left( (N\mathfrak a)E_{j,k}(z,\mathcal{L}_\mathfrak c) - \Lambda(\mathfrak a)^{k-j}E_{j,k}(z,\mathcal{L}_\mathfrak c)^{(\mathfrak a,\mathscr{R}(\mathfrak m))}\right) \ \textrm{by \cite[Prop.~3.3(iii), p.~58]{dS87}}\\
&= -\left( (N\mathfrak a)- \Lambda(\mathfrak a)^{k-j}(\mathfrak a,\mathscr{R}(\mathfrak m))\right)E_{j,k}(z,\mathcal{L}_\mathfrak c).
\end{align*}
Now, choosing $P=\xi_\mathfrak c\left(\Lambda(\mathfrak c)\rho\right)$,

we deduce that
\begin{align}
 \mathscr{D}_{j,k}(\gamma_{\mathfrak c,\mathfrak a})(P) = -(k-1)! \left( (N\mathfrak a)- \Lambda(\mathfrak a)^{k-j}(\mathfrak a,\mathscr{R}(\mathfrak m))\right)& \left(\frac{\varphi(\mathfrak c)}{\rho \Lambda(\mathfrak c)}\right)^{k-j}\times\notag\\
 &\left(\frac{N\mathfrak m\sqrt{d_K}}{2\pi}\right)^j L_{\mathfrak m}\left(\overline{ \varphi^{k-j}}, k, (\mathfrak c,\mathscr{R}(\mathfrak m))\right),\label{eq:app}
\end{align}
which is the formula that is utilized in the proof of Lemma~\ref{lemma: Lam14 3.1.4}.

\end{document}