---
abstract: 'With respect to the analytic-algebraic dichotomy, the theory of Siegel modular forms of half-integral weight is lopsided; the analytic theory is strong whereas the algebraic lags behind. In this paper, we capitalise on this to establish the fundamental object needed for the analytic side of the Iwasawa main conjecture – the $p$-adic $L$-function obtained by interpolating the complex $L$-function at special values. This is achieved through the Rankin-Selberg method and the explicit Fourier expansion of non-holomorphic Siegel Eisenstein series. The construction of the $p$-stabilisation in this setting is also of independent interest.'
author:
- Salvatore Mercuri
title: '<span style="font-variant:small-caps;">$p$-adic $L$-functions on metaplectic groups</span>'
---

[ oldtitletitle title[oldtitle]{}]{} [ @oldtitletitle title[@oldtitle]{}]{}

Introduction {#intro}
============

Traditionally, $p$-adic $L$-functions have dual constructions – analytic and algebraic – and it is the substance of the Iwasawa main conjecture that these two are equivalent. This conjecture can be formulated for various settings; for example, over $GL_1$, the conjecture asserts that the analytic construction – Kubota-Leopoldt’s $p$-adic interpolation of the Dirichlet $L$-function – is equivalent to Iwasawa’s algebraic $p$-adic $L$-function. The Iwasawa main conjecture for classical modular forms of integral weight is formulated over $GL_2$ and this has been a recent, active research area with its connections to the Birch and Swinnerton-Dyer conjecture, see [@Skinner1] and [@Skinner]. Provided one has both the analytic and algebraic machinery, the Iwasawa main conjecture can be formulated for higher dimensional modular forms and groups, for example [@Wan]

The algebraic theory of half-integral weight modular forms, both classical and metaplectic, has long been inchoate due to the difficulties present in developing the ‘Galois side’. Recent work by M. Weissmann in [@Weissmann] has made progress in this regard by developing $L$-groups for metaplectic covers, the length and methods of which further underline the difficulties present here. The analytic theory is substantial however, and in this paper we give the analytic construction of the $p$-adic $L$-function for Siegel modular forms of half-integral weight and any degree $n$. In [@Mercuri] we gave a similar construction when $n = 1$; in that case the $p$-adic $L$-function was already known to exist by the Shimura correspondence, this is not so for general $n>1$.

The proof found here is adapted from the method of A. Panchishkin found in Chapters 2 and 3 of [@Panchishkin], which proves the existence of the analytic $p$-adic $L$-function for Siegel modular forms of integral weight and even degree $n$. This method makes critical use of the Rankin-Selberg method and reduces the question of $p$-adic boundedness of the $L$-function down to that of the Fourier coefficients of the Eisenstein series that are involved in the Rankin-Selberg integral expression. For full generality, it is assumed that $p$ does not divide the level of the modular form $f$ and a crucial step is to produce another form $f_0$ such that $p$ does divide the level. Significant modifications to the method of [@Panchishkin] were required to make this work in the metaplectic case – this is Section \[pstabsection\]. Outside of this, the success of Panchishkin’s method is facilitated by the work of G. Shimura in developing the Rankin-Selberg integral expression in this setting, [@Shimuraexp], and the arithmeticity of Eisenstein series, [@Shimurabook Chapters 16–17]. Interestingly, the $p$-adic boundedness of the Eisenstein series coefficients is almost immediate in this case, making the final step of the proof simpler than that found in [@Panchishkin].

After preliminary Sections \[modformsection\] and \[Lfunsection\], we establish the $p$-stabilisation in Section \[pstabsection\]. Fairly elementary manipulations on the level of the Rankin-Selberg integral follow in Section \[tracesection\]. Sections \[thetasection\] and \[eisensteinsection\] are devoted to transformation formulae of theta series and Fourier expansions of Eisenstein series – these are relatively well-known. Finally the statement and proof of the main theorem, and the subsequent existence of the $p$-adic $L$-function, are given in Section \[interpolation\].

Siegel modular forms {#modformsection}
====================

This section runs through the very basics of the modular forms that we study and their Fourier expansions are detailed.

For any matrix $A\in M_n(\mathbb{C})$, note the use of the following notation: $A>0$ ($A\geq 0$) to mean that $A$ is positive definite (resp. positive semi-definite), $|A| := \det(A)$, $\|A\| :=|\det(A)|$, and $\widetilde{A} := (A^T)^{-1}$. For any collection $A_1, \dots, A_r$ of matrices, let ${\operatorname{diag}}[A_1, \dots, A_r]$ be the matrix whose $j$th diagonal block is $A_j$ and is zero off the diagonal.

Let ${\operatorname{\mathbb{A}_{\mathbb{Q}}}}$ and $\mathbb{I}_{\mathbb{Q}}$ denote the adele ring and idele group, respectively, of $\mathbb{Q}$. The Archimedean place is denoted by $\infty$ and the non-Archimedean places by $\mathbf{f}$. Recall the adelic norm $$|x|_{\mathbb{A}} = \prod_v|x_v|_v$$ where $x\in{\operatorname{\mathbb{A}_{\mathbb{Q}}}}$ and each valuation, $|\cdot|_v$, is normalised. Let $\mathbb{T}$ denote the unit circle and define three $\mathbb{T}$-valued characters on $\mathbb{C}$, $\mathbb{Q}_p$, and ${\operatorname{\mathbb{A}_{\mathbb{Q}}}}$ respectively by $$\begin{aligned}
    e:z&\mapsto e^{2\pi iz}, \\
    e_p:x&\mapsto e(-\{x\}), \\
    e_{\mathbb{A}}:x&\mapsto e(x_{\infty})\prod_p e_p(x_p),\end{aligned}$$ where $\{x\}$ denotes the fractional part of $x\in\mathbb{Q}_p$. If $x\in {\operatorname{\mathbb{A}_{\mathbb{Q}}}}$ then we also write $e_{\infty}(x) = e(x_{\infty})$.

Write any $\alpha\in GL_{2n}(\mathbb{Q})$ as $$\alpha = \begin{pmatrix} a_{\alpha} & b_{\alpha} \\ c_{\alpha} & d_{\alpha}\end{pmatrix}$$ where $x_{\alpha}\in M_n(\mathbb{Q})$ for $x\in\{a, b, c, d\}$. Define an algebraic group $G$, subgroup $P\leq G$, and the Siegel upper half-space $\mathbb{H}_n$ by $$\begin{aligned}
    G:&= Sp_n(\mathbb{Q}) = \{\alpha\in GL_{2n}(\mathbb{Q})\mid \alpha^T\iota\alpha = \iota\},\ \hspace{20pt} \iota := \begin{pmatrix} 0 & -I_n \\ I_n & 0\end{pmatrix}, \\
    P:&= \{\alpha\in G\mid c_{\alpha} = 0\}, \\
    \mathbb{H}_n :&= \{z = x+iy\in M_n(\mathbb{C})\mid x, y\in M_n(\mathbb{R}), z^T = z, y>0\},\end{aligned}$$ and let $G_{\mathbb{A}}$ denote the adelisation of $G$.

A half-integral weight is an element $k\in\mathbb{Q}$ such that $k-\frac{1}{2}\in\mathbb{Z}$; an integral weight is an element $\ell\in\mathbb{Z}$. The factor of automorphy of half-integral weight involves taking a square root; to guarantee consistency of the choice of root one uses the double metaplectic cover $Mp_n$ of $Sp_n$. The localizations $M_p := Mp_n(\mathbb{Q}_p)$ and the adelisation $M_{\mathbb{A}}$ of $Mp_n(\mathbb{Q})$ can be described as groups of unitary transformations, respectively, on $L^2(\mathbb{Q}_p^n)$ and $L^2({\operatorname{\mathbb{A}_{\mathbb{Q}}}}^n)$ with the exact sequences $$\begin{aligned}
    1\to\mathbb{T}\to M_x\to G_x\to 1,\end{aligned}$$ where $x\in\{p, \mathbb{A}\}$. There are natural projections ${\mathop{\rm pr}\nolimits}_{\mathbb{A}}:M_{\mathbb{A}}\to G_{\mathbb{A}}$ and ${\mathop{\rm pr}\nolimits}_p:M_p\to G_p$, either of which will usually be denoted ${\mathop{\rm pr}\nolimits}$ as the context is clear. On the flip side, there are natural lifts $r:G\to M_{\mathbb{A}}$ and $r_P:P_{\mathbb{A}}\to M_{\mathbb{A}}$ through which we view $G$ and $P_{\mathbb{A}}$ as subgroups of $M_{\mathbb{A}}$.

For any two fractional ideals ${\mathfrak}{x}, {\mathfrak}{y}$ of $\mathbb{Q}$ such that ${\mathfrak}{xy}\subseteq\mathbb{Z}$, congruence subgroups are defined by the following respective subgroups of $G_p, G_{\mathbb{A}}$, and $G$: $$\begin{aligned}
    D_p[{\mathfrak}{x}, {\mathfrak}{y}] :&= \{x\in G_p\mid a_x, d_x\in M_n(\mathbb{Z}_p), b_x\in M_n({\mathfrak}{x}_p), c_x\in M_n({\mathfrak}{y}_p)\}, \\
    D[{\mathfrak}{x}, {\mathfrak}{y}] :&= Sp_n(\mathbb{R})\prod_p D_p[{\mathfrak}{x}, {\mathfrak}{y}], \\
    \Gamma[{\mathfrak}{x}, {\mathfrak}{y}] :&= G\cap D[{\mathfrak}{x}, {\mathfrak}{y}].\end{aligned}$$ Typically these will take the form $\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$ for certain fractional ideals ${\mathfrak}{b}$ and integral ideals ${\mathfrak}{c}$.

One of the key differences in the theory of half-integral weight modular forms is in the congruence subgroups one considers. The factor of automorphy involved can only be defined for a certain subgroup ${\mathfrak}{M}\leq M_{\mathbb{A}}$, and any congruence subgroups $\Gamma$ must therefore be contained in ${\mathfrak}{M}$. This subgroup, ${\mathfrak}{M}$, is defined via the theta series and is given by $$\begin{aligned}
    C_p^{\theta} :&= \{\xi\in D_p[1, 1]\mid (a_{\xi}b_{\xi}^T)_{ii}\in 2\mathbb{Z}_p, (c_{\xi}d_{\xi}^T)_{ii}\in 2\mathbb{Z}_p, 1\leq i\leq n\}, \\
    C^{\theta} :&= Sp_n(\mathbb{R})\prod_p C_p^{\theta}, \\
    {\mathfrak}{M} :&= \{\sigma\in M_{\mathbb{A}}\mid {\mathop{\rm pr}\nolimits}(\sigma)\in P_{\mathbb{A}}C^{\theta}\}.\end{aligned}$$ Typically we shall take ${\mathfrak}{b}$ and ${\mathfrak}{c}$ such that $\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]\leq \Gamma[2, 2]\leq {\mathfrak}{M}$.

The spaces defined supra act on each other as follows. The action of $Sp_n(\mathbb{R})$ on $\mathbb{H}_n$ and the traditional factor of automorphy are given by $$\begin{aligned}
    \gamma\cdot z = \gamma z :&= (a_{\gamma}z+b_{\gamma})(c_{\gamma}z+d_{\gamma})^{-1}, \\
    j(\gamma, z) :&= \det(c_{\gamma}z+d_{\gamma}),\end{aligned}$$ where $\gamma\in Sp_n(\mathbb{R})$ and $z\in\mathbb{H}_n$. If $\alpha\in G_{\mathbb{A}}$ then we extend the above by $\alpha\cdot z = \alpha_{\infty}\cdot z$ and $j(\alpha, z) = j(\alpha_{\infty}, z)$. If $\sigma\in M_{\mathbb{A}}$ with $\alpha = {\mathop{\rm pr}\nolimits}(\sigma)\in G_{\mathbb{A}}$ then set $x_{\sigma} = x_{\alpha}$ for $x\in\{a, b, c, d\}$; the action of $M_{\mathbb{A}}$ on $\mathbb{H}_n$ is given by $\sigma\cdot z = \alpha\cdot z$.

For any $\sigma\in{\mathfrak}{M}$ we can define a holomorphic function $h_{\sigma} = h(\sigma, \cdot):\mathbb{H}_n\to\mathbb{C}$ satisfying the following properties $$\begin{aligned}
    h(\sigma, z)^2 &= \zeta j({\mathop{\rm pr}\nolimits}(\sigma), z)\ \text{for a constant $\zeta = \zeta(\sigma)\in \mathbb{T}$}, \label{h1} \\
    h(tr_P(\gamma), z) &= t^{-1}|\det((d_{\gamma})_{\infty})^{\frac{1}{2}}|\ \text{if $t\in\mathbb{T}$ and $\gamma\in P_{\mathbb{A}}$}, \label{h2}\\
    h(\rho\sigma\tau, z) &= h(\rho, z)h(\sigma, \tau z)h(\tau, z)\
\text{if ${\mathop{\rm pr}\nolimits}(\rho)\in P_{\mathbb{A}}, {\mathop{\rm pr}\nolimits}(\tau)\in C^{\theta}$}. \label{h3}\end{aligned}$$ The proofs for the above three properties can be found in [@Shimuraold pp. 294–295].

If $k$ is a half-integral weight then put $[k] := k-\frac{1}{2}\in\mathbb{Z}$; if $\ell$ is an integral weight then put $[\ell] := \ell$. The factors of automorphy of half-integral weights $k$ and integral weights $\ell$ are given as $$\begin{aligned}
    j_{\sigma}^k(z) :&= h_{\sigma}(z)j({\mathop{\rm pr}\nolimits}(\sigma), z)^{[k]}, \\
    j_{\alpha}^{\ell}(z) :&= j(\alpha, z)^{\ell},\end{aligned}$$ where $\sigma\in {\mathfrak}{M}$, $\alpha\in G_{\mathbb{A}}$, and $z\in\mathbb{H}_n$. Given a function $f:\mathbb{H}_n\to\mathbb{C}$ and an element $\xi\in G_{\mathbb{A}}$ or ${\mathfrak}{M}$, the *slash operator* of an integral or half-integral weight $\kappa\in\frac{1}{2}\mathbb{Z}$ is defined by $$\begin{aligned}
    (f||_{\kappa}\xi)(z) :&= j_{\xi}^k(z)^{-1}f(\xi\cdot z).\end{aligned}$$

\[modform\] Let $\kappa\in\frac{1}{2}\mathbb{Z}$ be an integral or half-integral weight, and let $\Gamma\leq G$ be a congruence subgroup with the assumption that $\Gamma\leq{\mathfrak}{M}$ if $\kappa\notin\mathbb{Z}$. Denote by $C_{\kappa}^{\infty}(\Gamma)$ the complex vector space of $C^{\infty}$ functions $f:\mathbb{H}_n\to\mathbb{C}$ such that $f||_{\kappa}\alpha = f$ for any $\alpha\in\Gamma$. Let $\mathcal{M}_{\kappa}(\Gamma)\subseteq C_{\kappa}^{\infty}(\Gamma)$ denote the subspace of holomorphic functions (with the additional cusp holomorphy condition if $n=1$). Elements of $\mathcal{M}_{\kappa}(\Gamma)$ are called *modular forms of weight $\kappa$, level $\Gamma$*; if $\kappa\notin\mathbb{Z}$ they are also known as *metaplectic modular forms*.

Elements of $C_{\kappa}^{\infty}(\Gamma)$ and $\mathcal{M}_{\kappa}(\Gamma)$ have Fourier expansions summing over positive semi-definite symmetric matrices, the precise forms for which are given later in this section. The subspace $\mathcal{S}_{\kappa}(\Gamma)\subseteq\mathcal{M}_{\kappa}(\Gamma)$ is characterised by all forms $f$ such that the Fourier expansion of $f||_{\kappa}\sigma$ sums over positive definite symmetric matrices, for any $\sigma\in G_{\mathbb{A}}$ if $\kappa\in \mathbb{Z}$, or for any $\sigma\in{\mathfrak}{M}$ if $\kappa\notin\mathbb{Z}$. Write $$\mathcal{X}_{\kappa} = \bigcup_{\Gamma}\mathcal{X}_{\kappa}(\Gamma)$$ where $\mathcal{X}\in\{\mathcal{M}, \mathcal{S}\}$ and the union is taken over all congruence subgroups of $G$ (that are contained in ${\mathfrak}{M}$ if $\kappa\notin\mathbb{Z}$).

Take a fractional ideal ${\mathfrak}{b}$ and integral ideal ${\mathfrak}{c}$ and put $\Gamma = \Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$; when $\kappa\notin\mathbb{Z}$ assume that ${\mathfrak}{b}^{-1}\subseteq 2\mathbb{Z}$ and ${\mathfrak}{bc}\subseteq 2\mathbb{Z}$, then $\Gamma \leq{\mathfrak}{M}$ in this case. Take a Hecke character $\psi:\mathbb{I}_{\mathbb{Q}}/\mathbb{Q}^{\times}\to\mathbb{T}$ of $\mathbb{Q}$ such that $\psi_p(a) = 1$ if $a\in\mathbb{Z}_p^{\times}$ and $a\in 1+{\mathfrak}{c}_p$, and such that $\psi_{\infty}(x)^n = {\operatorname{sgn}}(x_{\infty})^{n[\kappa]}$. Modular forms of character $\psi$ are defined by $$\begin{aligned}
    C_{\kappa}^{\infty}(\Gamma, \psi) :&= \{F:\mathbb{H}_n\to\mathbb{C}\in C_{\kappa}^{\infty}\mid F||_{\kappa}\gamma = \psi_{{\mathfrak}{c}}(|a_{\gamma}|)F\ \text{for all $\gamma\in\Gamma$}\}, \\
    \mathcal{M}_{\kappa}(\Gamma, \psi) :&= \mathcal{M}_{\kappa}\cap C_{\kappa}^{\infty}(\Gamma, \psi), \\
    \mathcal{S}_{\kappa}(\Gamma, \psi) :&= \mathcal{S}_{\kappa}\cap\mathcal{M}_{\kappa}(\Gamma, \psi).\end{aligned}$$

Understand ${\mathop{\rm pr}\nolimits}= {\operatorname{id}}$ if $\kappa\in\mathbb{Z}$. If $f\in\mathcal{M}_{\kappa}(\Gamma, \psi)$ then its adelisation $f_{\mathbb{A}}:{\mathop{\rm pr}\nolimits}^{-1}(G_{\mathbb{A}}) \to\mathbb{C}$ is $$f_{\mathbb{A}}(x) := \psi_{{\mathfrak}{c}}(|d_w|)(f||_{\kappa}w)(\mathbf{i})$$ where $x = \alpha w$ for $\alpha\in G$ and $w\in {\mathop{\rm pr}\nolimits}^{-1}(D[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}])$, and $\mathbf{i} = iI_n$. To give the precise Fourier expansions of these forms, define the following spaces of symmetric matrices: $$\begin{aligned}
    S:&=\{\xi\in M_n(\mathbb{Q})\mid \xi^T=\xi\},
&&\hspace{10pt}S_+:=\{\xi\in S\mid \xi\geq 0\}, \\
    S^{\triangledown} :&= \{\xi\in S\mid \xi_{ii}\in\mathbb{Z}, \xi_{ij}\in\tfrac{1}{2}\mathbb{Z}, i<j\}, && \hspace{10pt}S_+^{\triangledown} := S^{\triangledown}\cap S_+, \\
    S({\mathfrak}{r}):&=S\cap M_n({\mathfrak}{r}) ,
&&S_{\mathbf{f}}({\mathfrak}{r}):=\prod_{p\in\mathbf{f}}S({\mathfrak}{r})_p,\end{aligned}$$ for any fractional ideal ${\mathfrak}{r}$ of $\mathbb{Q}$. Let $S_{\mathbb{A}} = S_{\infty}S_{\mathbf{f}}$ denote the adelisation of $S$.

Take a congruence subgroup $\Gamma = \Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$ (contained in ${\mathfrak}{M}$ if $\kappa\notin\mathbb{Z}$), a modular form $f\in\mathcal{M}_{\kappa}(\Gamma, \psi)$, and matrices $q\in GL_n({\operatorname{\mathbb{A}_{\mathbb{Q}}}})$, $s\in S_{\mathbb{A}}$. The Fourier expansion of $f_{\mathbb{A}}$ is given as $$f_{\mathbb{A}}\left(r_P\begin{pmatrix} q & s\tilde{q} \\ 0 & \tilde{q}\end{pmatrix}\right) = |q_{\infty}|^{[k]}\|q_{\infty}\|^{\kappa-[\kappa]}\sum_{\tau\in S_+}c_f(\tau, q)e_{\infty}({\operatorname{tr}}(\mathbf{i}q^T\tau q))e_{\mathbb{A}}({\operatorname{tr}}(\tau s)),$$ for some $c_f(\tau, q) = c(\tau, q; f)\in\mathbb{C}$ satisfying the following properties $$\begin{aligned}
    &c_f(\tau, q)\neq 0\ \text{only if $e_{\mathbb{A}}({\operatorname{tr}}(q^T\tau qs))=1$ for all $s\in S_{\mathbf{f}}({\mathfrak}{b}^{-1})$}, \label{c1} \\
    &c_f(b^T\tau b, q) = |b|^{[\kappa]}\|b\|^{\kappa-[\kappa]}c_f(\tau, bq)\ \text{for any $b\in GL_n(\mathbb{Q})$}, \label{c3} \\
    &\psi_{\mathbf{f}}(|a|)c_f(\tau, qa) = c_f(\tau, q)\ \text{for any ${\operatorname{diag}}[a, \tilde{a}]\in D[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$}.  \label{c4}\hspace{80pt}\end{aligned}$$

The proof of the above expansion and properties can be found in Proposition 1.1 of [@Shimurahalf]. The coefficients $c_f(\tau, 1)$ are the Fourier coefficients of $f$; by property \[c1\], the modular form $f\in\mathcal{M}_{\kappa}(\Gamma, \psi)$ has Fourier expansion $$f(z) = \sum_{\tau\in{\mathfrak}{b}S_+^{\triangledown}}c_f(\tau, 1)e({\operatorname{tr}}(\tau z)).$$ If $F\in C_{\kappa}^{\infty}(\Gamma, \psi)$, then it has Fourier expansion of the form $$F(z) = \sum_{\tau\in {\mathfrak}{b}S^{\triangledown}} c_F(\tau, 1; y) e({\operatorname{tr}}(\tau x)).$$

The complex $L$-function that one associates to an eigenform can be expressed as an integral, the precise form of which is given in terms of the Petersson inner product. Fix ${\mathfrak}{b}$; for any two $f, g\in C_{\kappa}^{\infty}(\Gamma, \psi)$ this inner product is defined by $$\langle f, g\rangle_{{\mathfrak}{c}} = \langle f, g\rangle := {\operatorname{Vol}}(\Gamma{\backslash}\mathbb{H}_n)^{-1}\int_{\Gamma{\backslash}\mathbb{H}_n}f(z)\overline{g(z)}\Delta(z)^{\kappa}d^{\times}z$$ in which $$\Delta(z) := \det({\operatorname{\mathfrak{Im}}}(z)), \hspace{15pt} d^{\times}z := \Delta(z)^{-n-1}\bigwedge_{i\leq j}(dx_{ij}\wedge dy_{ij}),$$ for any $z = (x_{ij}+iy_{ij})_{i, j=1}^n\in\mathbb{H}_n$. This integral is convergent whenever one of $f, g$ belongs to $\mathcal{S}_{\kappa}(\Gamma, \psi)$.

Complex $L$-function {#Lfunsection}
====================

The standard complex $L$-function is defined in this section and it is ended by stating the known Rankin-Selberg integral expression.

For any Hecke character ${\operatorname{\varphi}}$ of $\mathbb{Q}$ let ${\operatorname{\varphi}}^*(p) = {\operatorname{\varphi}}^*(p\mathbb{Z})$ denote the associated ideal character. Though the integral expression can be stated for any half-integral weight $k$, we take $k\geq n+1$ to ease up on notation – we shall be making this assumption later on anyway. For a prime $p$, the association of the Satake $p$-parameters – $(\lambda_{p, 1}, \dots, \lambda_{p, n})\in\mathbb{C}^n$ – to a non-zero Hecke eigenform $f\in\mathcal{S}_k(\Gamma, \psi)$ is well-known (see, for example, [@Shimurahalf p. 46]). The standard $L$-function of $f$, twisted by a Hecke character $\chi$ of conductor ${\mathfrak}{f}$, is then defined by $$\begin{aligned}
    L_p(t) :&= \begin{cases} \displaystyle\prod_{i=1}^n (1-p^n\lambda_{p, i}t) &\text{if $p\mid{\mathfrak}{c}$}, \\ 
        \displaystyle\prod_{i=1}^n (1-p^n\lambda_{p, i}t)(1-p^n\lambda_{p, i}^{-1}t) &\text{if $p\nmid{\mathfrak}{c}$}; \end{cases} \\
    L_{\psi}(s, f, \chi) :&= \prod_p L_p\left((\psi^{{\mathfrak}{c}}\chi^*)(p)p^{-s}\right)^{-1},\end{aligned}$$ in which $$\psi^{{\mathfrak}{c}}(x) := \left(\frac{\psi}{\psi_{{\mathfrak}{c}}}\right)(x).$$ The Rankin-Selberg integral expression, (4.1) in [@Shimuraexp p. 342], is given there in generality; we restate it now for our purposes. Fix $\tau\in {\mathfrak}{b}S_+^{\triangledown}$ such that $c_f(\tau, 1)\neq 0$, let ${\mathfrak}{t}$ be an integral ideal satisfying $h^T(2\tau)^{-1}h\in 4{\mathfrak}{t}^{-1}$ for any $h\in\mathbb{Z}^n$, and let $\rho_{\tau}$ be the quadratic character associated to the extension $\mathbb{Q}(i^{[n/2]}\sqrt{|2\tau|})$. Choose $\mu\in\{0, 1\}$ such that $(\psi\chi)_{\infty}(x) = {\operatorname{sgn}}(x_{\infty})^{[k]+\mu}$.

The key ingredients of the integral are three modular forms: the eigenform $f$, a theta series $\theta_{\chi}$, and a normalised non-holomorphic Eisenstein series $\mathcal{E}(z, s)$. The definition of the theta series, taken from [@Shimuraexp (2.1)], is $$\begin{aligned}
    \theta_{\chi}^{\mu}(z; \tau) = \theta_{\chi}(z) := \sum_{x\in M_n(\mathbb{Z})}(\chi_{\infty}\chi^*)^{-1}(|x|)|x|^{\mu}e({\operatorname{tr}}(x^T\tau xz)), \label{theta}\end{aligned}$$ where we understand tht $(\chi_{\infty}\chi^*)(0) = 1$ if ${\mathfrak}{f} = \mathbb{Z}$. This has weight $\frac{n}{2}+\mu$, level $\Gamma[2, 2{\mathfrak}{tf}^2]$ determined by Proposition 2.1 of [@Shimuraexp], character $\rho_{\tau}\chi^{-1}$, and coefficients in $\mathbb{Q}(\chi)$.

The Eisenstein series of weight $\kappa\in\frac{1}{2}\mathbb{Z}$ is now defined more generally. Let $\Gamma' = \Gamma[{\mathfrak}{x}^{-1}, {\mathfrak}{xy}]$ be a congruence subgroup, contained in ${\mathfrak}{M}$ if $\kappa\notin\mathbb{Z}$, and let ${\operatorname{\varphi}}$ be a Hecke character such that ${\operatorname{\varphi}}_{\infty}(x) = {\operatorname{sgn}}(x_{\infty})^{[\kappa]}$ and ${\operatorname{\varphi}}_p(a) = 1$ if $a\in\mathbb{Z}_p^{\times}$ and $a\in 1+{\mathfrak}{y}_p$. This series is defined by $$\begin{aligned}
    E(z, s; \kappa, {\operatorname{\varphi}}, \Gamma') := \sum_{\alpha\in P\cap\Gamma'{\backslash}\Gamma'}{\operatorname{\varphi}}_{{\mathfrak}{y}}(|a_{\gamma}|)(\Delta^{s-\frac{\kappa}{2}}||_{\kappa}\alpha)(z), \label{eisenstein}\end{aligned}$$ for variables $z\in\mathbb{H}_n$ and $s\in\mathbb{C}$. This sum is convergent for ${\operatorname{\mathfrak{Re}}}(s)>\frac{n+1}{2}$ and can be continued meromorphically to all of $s\in\mathbb{C}$ by a functional equation in $s\mapsto \frac{n+1}{2}-s$. This series belongs to $C_{\kappa}^{\infty}(\Gamma, {\operatorname{\varphi}}^{-1})$ and is normalised by a product of Dirichlet $L$-functions as follows. Let ${\mathfrak}{a}$ be any integral ideal and define $$\begin{aligned}
    \begin{split}
        L_{{\mathfrak}{a}}(s, {\operatorname{\varphi}}) :&= \prod_{p\nmid{\mathfrak}{a}}(1-{\operatorname{\varphi}}^*(p)p^{-s})^{-1}; \\
        \Lambda_{{\mathfrak}{a}}^{n, \kappa}(s, {\operatorname{\varphi}}) :&= \begin{cases}
            L_{{\mathfrak}{a}}(2s, {\operatorname{\varphi}})\displaystyle\prod_{i=1}^{[n/2]}L_{{\mathfrak}{a}}(4s-2i, {\operatorname{\varphi}}^2) &\text{if $\kappa\in\mathbb{Z}$}, \\
            \displaystyle\prod_{i=1}^{[(n-1)/2]}L_{{\mathfrak}{a}}(4s-2i+1, {\operatorname{\varphi}}^2) &\text{if $\kappa\notin\mathbb{Z}$}.
            \end{cases}
    \end{split}\end{aligned}$$ The normalised Eisenstein series is given by $$\begin{aligned}
    \mathcal{E}(z, s; \kappa, {\operatorname{\varphi}}, \Gamma') := \overline{\Lambda_{{\mathfrak}{y}}^{n, \kappa}(s, \bar{{\operatorname{\varphi}}})}E(z, \bar{s}; \kappa, {\operatorname{\varphi}}, \Gamma'). \label{normalisedeisen}\end{aligned}$$

Set $\eta := \psi\bar{\chi}\rho_{\tau}$. In this setting, the integral expression of [@Shimuraexp (4.1)] becomes: $$\begin{aligned}
    \begin{split}
        L_{\psi}(s, f, \bar{\chi}) &= \left[\Gamma_n\left(\tfrac{s-n-1+k+\mu}{2}\right)2c_f(\tau, 1)\right]^{-1}N({\mathfrak}{b})^{\frac{n(n+1)}{2}}|4\pi\tau|^{\frac{s-n-1+k+\mu}{2}} \\
    &\times \left(\tfrac{\Lambda_{{\mathfrak}{c}}}{\Lambda_{{\mathfrak}{y}}}\right)\left(\tfrac{2s-n}{4}\right)\prod_{q\in\mathbf{b}}g_q\left((\psi^{{\mathfrak}{c}}\bar{\chi}^*)(q)q^{-s}\right)\langle f, \theta_{\bar{\chi}}\mathcal{E}(\cdot, \tfrac{2s-n}{4})\rangle_{{\mathfrak}{y}} V,
    \end{split}\label{RS}\end{aligned}$$ in which $\Lambda_{{\mathfrak}{a}}(s) := \Lambda_{{\mathfrak}{a}}^{n, k-\frac{n}{2}-\mu}(s, \eta)$; ${\mathfrak}{y}:= {\mathfrak}{c}\cap(2{\mathfrak}{t}{\mathfrak}{f}^2)$; $\mathbf{b}$ is the finite set of primes $q\nmid{\mathfrak}{c}$ such that ${\operatorname{ord}}_q(|\tau|)\neq 0$ and $g_q\in\mathbb{Q}[t]$ satisfy $g_q(0) = 1$; $$\mathcal{E}(z, s) := \mathcal{E}\left(z, s; k-\tfrac{n}{2}-\mu, \bar{\eta}, \Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{by}]\right);$$ and $V := {\operatorname{Vol}}(\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{by}]{\backslash}\mathbb{H}_n)$.

$p$-stabilisation {#pstabsection}
=================

If $p$ divides the level of $f$ then the behaviour of the $p$th Hecke operator becomes simpler; this is a key tool in the construction of the $p$-adic $L$-function. For full generality we begin with a form $f$, assume that $p\nmid{\mathfrak}{c}$ does not divide the level, and we construct a form $f_0$ – called the $p$-stabilisation of $f$ – for which $p$ does divide the level, and the $L$-functions remain mostly unchanged. In [@Mercuri] we constructed $f_0$ explicitly in the case $n=1$ which was possible through explicit formulae on the action of the Hecke operators involved on the Fourier coefficients. For general $n$, it is easier to work with abstract Hecke rings, the Satake isomorphism, and certain Hecke polynomials, however at the end of this section, we show how all this abstract Hecke yoga reduces to the explicit form found in [@Mercuri], when $n=1$.

Let $({\mathfrak}{b}^{-1}, {\mathfrak}{bc})\subseteq 2\mathbb{Z}\times 2\mathbb{Z}$ be ideals as before; put $\Gamma = \Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$ and $D = D[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}]$. Then define $$\begin{aligned}
    {\text{{{\fontfamily{pag}\selectfont}O}}}_p :&= GL_n(\mathbb{Z}_p),   &&{\text{{{\fontfamily{pag}\selectfont}O}}}:= \prod_p GL_n(\mathbb{Z}_p), \\
    {\text{{{\fontfamily{pag}\selectfont}X}}}_p :&= M_n(\mathbb{Z}_p)\cap GL_n(\mathbb{Q}_p), &&{\text{{{\fontfamily{pag}\selectfont}X}}}:= GL_n(\mathbb{Q})_{\mathbf{f}}\cap\prod_p {\text{{{\fontfamily{pag}\selectfont}X}}}_p, \\
    Z_0 :&= \{{\operatorname{diag}}[\tilde{q}, q]\mid q\in {\text{{{\fontfamily{pag}\selectfont}X}}}\}, &&Z:= D[2, 2]Z_0D[2, 2].\end{aligned}$$

For appropriate groups, $\Delta$ and $\Xi$, the abstract Hecke ring $\mathcal{R}(\Delta, \Xi)$ denotes the ring of formal finite sums $\sum_{\xi}c_{\xi}\Delta\xi\Delta$ where $c_{\xi}\in\mathbb{C}$ and $\xi\in \Xi$. Consider the Hecke ring $\mathcal{R}(V, W)$ defined in [@Shimurahalf p. 39] and let $\mathcal{R}$ denote the factor ring of $\mathcal{R}(V, W)$ defined in [@Shimurahalf p. 41] or analogously to \[factorring\] below – this is the adelic Hecke ring which acts on forms in $\mathcal{M}_k(\Gamma, \psi)$, and it is factored in order to give the Satake isomorphism. We need the use of a slightly different Hecke ring and we define this more explicitly. Let $D_0 := D\cap P_{\mathbb{A}}$ and $\Gamma_0 = \Gamma\cap P$; define $$\begin{aligned}
    W_0 :&= \{(\alpha, t)\mid t\in \mathbb{T}, {\mathop{\rm pr}\nolimits}(\alpha)\in G_{\mathbf{f}}\cap D_0Z_0D_0\} \\
    V_0 :&= \{(\alpha, 1)\mid {\mathop{\rm pr}\nolimits}(\alpha)\in G_{\mathbf{f}}\cap D_0\},\end{aligned}$$ and then let $\mathcal{R}_0$ denote the factor ring $$\begin{aligned}
    \mathcal{R}_0 := \mathcal{R}(V_0, W_0)/\langle V_0(\alpha, 1)V_0-tV_0(\alpha, t)V_0\mid (\alpha, t)\in W_0\rangle. \label{factorring}\end{aligned}$$ All of the above Hecke rings, $\mathcal{R}(V, W), \mathcal{R}(V_0, W_0), \mathcal{R}$, and $\mathcal{R}_0$, act on the space of modular forms $\mathcal{M}_k(\Gamma, \psi)$. The action of $D{\operatorname{diag}}[\tilde{q}, q]D\in\mathcal{R}(V, W)$ is given by decomposing into single cosets $$G\cap(D{\operatorname{diag}}[\tilde{q}, q]D) = \bigsqcup_{\alpha}\Gamma\alpha,$$ where $\alpha\in G\cap D{\operatorname{diag}}[\tilde{q}, q]D$ and summing over the actions of $\alpha$ on $f$ by the slash operator involving an extended factor of automorphy $J^k(\alpha, z) = J^{\frac{1}{2}}(\alpha, z)j(\alpha, z)^{[k]}$ – see Sections 2, 3, and 4 of [@Shimurahalf] for the details here. The actions of the other rings are analogous.

Assume that $p\nmid{\mathfrak}{c}$ and let $\mathcal{R}_p$ and $\mathcal{R}_{0p}$ denote the local Hecke rings of $\mathcal{R}$ and $\mathcal{R}_0$ respectively; the local rings are generated by $D_p{\operatorname{diag}}[\tilde{q}, q]D_p$ and $(D_0)_p{\operatorname{diag}}[\tilde{q}, q](D_0)_p$ respectively, for $q\in{\text{{{\fontfamily{pag}\selectfont}X}}}_p$. Let $W_n$ be the Weyl group of transformations generated by $x_i\mapsto x_i^{-1}; x_j\mapsto x_j$ for $j\neq i$, and let $\mathbb{C}[x_1^{\pm}, \dots, x_n^{\pm}]^{W_n}$ denote the ring of Weyl-invariant complex polynomials. The Satake map $\omega_p:\mathcal{R}_{0p}\to\mathbb{C}[x_1^{\pm}, \dots, x_n^{\pm}]^{W_n}$ is defined through the composition of two maps as follows $$\begin{aligned}
    \omega_p:\mathcal{R}_{0p}\xrightarrow{\Phi_p} \mathcal{R}({\text{{{\fontfamily{pag}\selectfont}O}}}_p, GL_n(\mathbb{Q}_p))\xrightarrow{\omega_{0p}}\mathbb{C}[x_1^{\pm}, \dots, x_n^{\pm}]^{W_n}.\end{aligned}$$

#### The map $\Phi_p$.

If $\sigma = {\operatorname{diag}}[\tilde{q}, q]$ with $q\in{\text{{{\fontfamily{pag}\selectfont}X}}}_p$ then by Lemma 2.1 of [@Shimuraint] we have the decomposition $$\begin{aligned}
    (D_{0})_p\sigma (D_{0})_{p} =\bigsqcup_{x\in X}\bigsqcup_{s\in Y_x}\bigsqcup_{d\in R_x}D_p\alpha_{d, s}, \hspace{20pt}\alpha_{d, s} = \begin{pmatrix} \tilde{d} & sd \\ 0 & d\end{pmatrix}, \label{satakecoset}\end{aligned}$$ where $X\subseteq GL_n(\mathbb{Q}_p)$, $R_x\subset x{\text{{{\fontfamily{pag}\selectfont}O}}}_p$ represents ${\text{{{\fontfamily{pag}\selectfont}O}}}_p{\backslash}{\text{{{\fontfamily{pag}\selectfont}O}}}_p x{\text{{{\fontfamily{pag}\selectfont}O}}}_p$, and $Y_x\subseteq S_p$. To define $\Phi_p$ we extend, by $\mathbb{C}$-linearity, the map $$\Phi_p((D_0)_p\sigma(D_0)_p) := \sum_{d, s}J(r_P(\alpha_{d, s}))^{-1}{\text{{{\fontfamily{pag}\selectfont}O}}}_pd,$$ where $J(\alpha) := J^{\frac{1}{2}}(\alpha, i)$ for $\alpha\in {\mathop{\rm pr}\nolimits}^{-1}(D[2, 2]Z_0D[2, 2])$ (see [@Shimurahalf (2.7) and Lemma 2.4] for the precise definition and characterisation of $J(\alpha)$).

#### The map $\omega_{0p}$.

Note that any coset ${\text{{{\fontfamily{pag}\selectfont}O}}}_pd$ with $d\in GL_n(\mathbb{Q}_p)$ contains an upper triangular matrix of the form $$\begin{aligned}
    \begin{pmatrix} 
    p^{a_{d_1}} & \star & \cdots & \star \\ 
    0 & p^{a_{d_2}} & \cdots & \star \\ \vdots & \vdots & \ddots & \vdots \\ 
    0 & 0 & \cdots & p^{a_{d_n}}
    \end{pmatrix},  \label{dform}\end{aligned}$$ with $a_{d_i}\in\mathbb{Z}$, and then define $$\omega_{0p}({\text{{{\fontfamily{pag}\selectfont}O}}}_pd) := \prod_{i=1}^n (p^{-i}x_i)^{a_{d_i}}.$$ Through the decomposition ${\text{{{\fontfamily{pag}\selectfont}O}}}_p x{\text{{{\fontfamily{pag}\selectfont}O}}}_p = \bigsqcup_d {\text{{{\fontfamily{pag}\selectfont}O}}}_pd$ and $\mathbb{C}$-linearity, we extend this to obtain $\omega_{0p}$.

By Lemma 1.1.3 of [@Andrianov] there exists an embedding ${\varepsilon}_p:\mathcal{R}_p\to\mathcal{R}_{0p}$; the restriction of $\omega_p$ to ${\varepsilon}_p(\mathcal{R}_p)$ defines the usual Satake isomorphism – see Section 4 of [@Shimurahalf]. The full map $\omega_p$ is not necessarily injective, however. Assume also that $p\nmid{\mathfrak}{b}^{-1}$. There exists an element $U_p\in\mathcal{R}_{0p}$ – called the *Frobenius* element – defined by $$U_p = \Gamma_0\begin{pmatrix} p^{-1}I_n & 0 \\ 0 & pI_n\end{pmatrix} \Gamma_0 =  \bigsqcup_{u\in S({\mathfrak}{b}^{-1}/p^2{\mathfrak}{b}^{-1})}\Gamma_0\begin{pmatrix} p^{-1}I_n & p^{-1}u \\ 0 & pI_n\end{pmatrix}.$$ If $n=1$, it is well-known that $U_p$ corresponds to the $p$th Hecke operator when $p\mid{\mathfrak}{c}$; for general $n>1$, this is no longer true. Denote the centraliser of $U_p$ in $\mathcal{R}_{0p}$ by $\mathcal{C} := \{ A\in \mathcal{R}_{0p}\mid U_pA = AU_p\}$; the map $\Phi_p$ restricted to $\mathcal{C}$ is injective – use Proposition 2.1.1, Lemma 2.1.1 of [@Andrianov p. 86] along with the argument of Lemma 4.3 of [@Shimurahalf] to see why. Note that $\omega_p(U_p) = p^{\frac{n(n+1)}{2}}x_1\cdots x_n$.

The definition of the $p$-stabilisation is now achieved through factorisations of a certain Hecke polynomial. This polynomial is an element $\widetilde{R}_n\in\mathbb{C}[x_1^{\pm}, \dots, x_n^{\pm}][z]$ and is formed of variations of the usual symmetric square polynomial as follows: $$\begin{aligned}
    \widetilde{R}_n(x_1, \dots, x_n; z) = \widetilde{R}_n(z) :&= \prod_{\delta_i\in\{\pm 1\}}(1-p^{\frac{n(n+1)}{2}}x_1^{\delta_1}\cdots x_n^{\delta_n}z).\end{aligned}$$ This has degree $2^n$ and has an immediate decomposition of the form $$\widetilde{R}_n(z) = \sum_{m=0}^{2^n}(-1)^m\widetilde{T}_mz^m,$$ where the coefficients $\widetilde{T}_m = \widetilde{T}_m(x_1, \dots, x_n)\in\mathbb{C}[x_1^{\pm}, \dots, x_n^{\pm}]$. These coefficients are clearly invariant under the group of Weyl transformations so, by the Satake isomorphism, there exists a polynomial $$\begin{aligned}
    R_n(z) = \sum_{m=0}^{2^n}(-1)^mT_mz^m \label{RTfact}\end{aligned}$$ whose coefficients $T_m\in\mathcal{R}_p$ satisfy $\widetilde{T}_m = \omega_p(T_m)$. If $n=1$ then notice, from [@Shimurahalf (5.4a)], that $\omega_p(T_1) = \omega_p(T(p))$, so $T_1 = T(p)$ as operators on $f$ in this case.

The polynomial $R(z)$ has a factorisation involving the Frobenius element $U_p$ – this part is similar to the methods found in [@Andrianov pp. 90–91] and [@Panchishkin pp. 42–50].

\[usum\] With $T_m$ and $U_p$ defined as above $$\sum_{m=0}^{2^n}(-1)^mT_mU_p^{2^n-m} = 0.$$

Denote the sum on the left-hand side by $Y$, this belongs to $\mathcal{R}_{0p}$. It is easy to check that $\widetilde{R}_n(z) = (p^{\frac{n(n+1)}{2}}z)^{2^n}\widetilde{R}_n((p^{n(n+1)}z)^{-1})$ so, immediately from \[RTfact\], we have $$\begin{aligned}
    T_m = p^{(n+1)(m-2^{n-1})}T_{2^n-m}. \label{Trel}\end{aligned}$$ By \[Trel\] above and the same argument of Proposition 2.1.2 in [@Andrianov pp. 88–89] (in this case the degree of $T_j$ is the degree of $\widetilde{T}_j$ in the variable $x_1\cdots x_n$, i.e. it is $j$) we therefore have $T_m U_p^{2^n-m}\in\mathcal{C}$ and so we just need to show that $\omega_p(Y) = 0$. For this, note $$\begin{aligned}
    \omega_p(Y) &= \omega_p(U_p)^{2^n}\sum_{m=0}^{2^n}(-1)^m\widetilde{T}_m\cdot (\omega_p(U_p)^{-1})^m \\
    &= \omega_p(U_p)^{2^n}\widetilde{R}_n(\omega_p(U_p)^{-1})\end{aligned}$$ which is zero, since $(1-\omega_p(U_p)z) = (1-p^{\frac{n(n+1)}{2}}x_1\cdots x_nz)$ is a factor of $\widetilde{R}_n(z)$.

For any $0\leq m\leq 2^n$, define $$V_{m, p} = V_m := \sum_{\ell=0}^m (-1)^{\ell}T_{\ell}U_p^{m-\ell}\in\mathcal{R}_{0p}.$$

\[Rfact\] The Hecke polynomial $R(z)$ can be factorised as $$\begin{aligned}
    R(z) = \left(\sum_{m=0}^{2^n-1}V_mz^m\right)(1-U_pz) \label{Rfacteq}\end{aligned}$$

By definition $V_0 = 1$ and by Lemma \[usum\] $V_{2^n-1}U_p = -T_{2^n}$. For integers $1\leq m\leq 2^n-2$ we have $$V_m-V_{m-1}U_p = \sum_{\ell=0}^m (-1)^{\ell}T_{\ell}U_p^{m-\ell}-\sum_{\ell=0}^{m-1}(-1)^{\ell}T_{\ell}U_p^{m-1-\ell}U_p = (-1)^mT_m.$$ Expanding the right hand side of \[Rfacteq\] therefore gives the factorisation \[RTfact\], which concludes the proof.

\[pstab\] Let $f\in\mathcal{M}_k(\Gamma, \psi)$ be a non-zero Hecke eigenform with Satake $p$-parameters $(\lambda_{p, 1}, \dots, \lambda_{p, n})$ for $p\nmid{\mathfrak}{c}$. Then the $p$-stabilisation of $f$ is defined by $$\begin{aligned}
    f_0 := \sum_{m=0}^{2^n-1}\left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)^{-m}f|V_{m, p}. \label{pstabform}\end{aligned}$$

It is not clear from this method that $f_0\neq 0$, and therefore we need to make the crucial assumption that there exists $\tau$ such that $c(\tau, 1; f_0)\neq 0$ later on. Note that Böchere and Schmidt give, in [@Boch Section 9], an alternative construction for the $p$-stabilisation of a Siegel modular form of integral weight which method does guarantee that $f_0\neq 0$, if $f\neq 0$.

\[pstabprop\] If $f\in\mathcal{M}_k(\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}], \psi)$ is an eigenform and $p\nmid{\mathfrak}{c}$, then $f_0\in\mathcal{M}_k(\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}_0], \psi)$ where ${\mathfrak}{c}_0 = {\mathfrak}{c}p^{2(2^n-1)}$. Moreover, $$f_0|U_p = \left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)f_0.$$

Define $W_p := \Gamma_0\left(\begin{smallmatrix} pI_n & 0 \\ 0 & p^{-1}I_n\end{smallmatrix}\right)\Gamma_0\in\mathcal{R}_{0p}$; clearly $f|W_p = p^{nk}f(p^2z)$ has level $\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}p^2]$ and therefore, as operators, $W_p\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}p^2] = W_p$. Let $\alpha^{\iota} := \det(\alpha)\alpha^{-1}$ denote the standard involution on invertible matrices and extend this to $\mathcal{R}_{0p}$ by putting $(\Gamma_0\alpha\Gamma_0)^{\iota} = \Gamma_0\alpha^{\iota}\Gamma_0$. The elements $W_p$ and $U_p$ satisfy $U_p = W_p^{\iota}$ and, by the argument of [@Panchishkin p. 49], we have $U_p\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}p^2] = U_p$ as operators as well. So $f|U_p\in\mathcal{M}_k(\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}p^2], \psi)$ and the first property follows by definition of $f_0$ and $V_m$.

Set $\lambda_0 := p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}$. The second property is given by the calculation $$\begin{aligned}
    f_0|[\lambda_0-U_p] &= \lambda_0 f_0|[1-\lambda_0^{-1}U_p] \\
        &=\lambda_0 f\bigg|\left[\sum_{m=0}^{2^n-1}(\lambda_0^{-1})^mV_{m, p}\right][1-\lambda_0^{-1}U_p] \\
    &=\lambda_0f|R_n(\lambda_0^{-1}),\end{aligned}$$ where Proposition \[Rfact\] was invoked in the last line and Definition \[pstab\] in the penultimate. This is zero since we have $f|R_n(\lambda_0^{-1}) = \widetilde{R}_n(\lambda_{p, 1}, \dots, \lambda_{p, n}; \lambda_0^{-1})f$ and $(1-\lambda_0z)$ is a factor in $\widetilde{R}_n(\lambda_{p, 1}, \dots, \lambda_{p, n}; z)$.

For $q\neq p$, the $q$th Hecke operator commutes with $V_{m, p}$. Therefore $f_0$ and $f$ share the same eigenvalues away from $p$, and we then have the following corollary.

\[pstablfun\] If $\chi$ is a Hecke character of $p$-power conductor, then $L_{\psi}(s, f, \chi) = L_{\psi}(s, f_0, \chi)$.

In [@Mercuri] we showed, if $n=1$, that the $p$-stabilisation of $f$ takes the form $$\begin{aligned}
    f_0(z) := f(z) - \left(\frac{-1}{p}\right)^{[k]}p^{-\frac{1}{2}}\lambda_{p, 1}^{-1}\left(f\otimes\left(\frac{\cdot}{p}\right)\right)(z) - p^{k-1}\lambda_{p, 1}^{-1}f(p^2z), \label{pstab1}\end{aligned}$$ where, for any Dirichlet character ${\operatorname{\varphi}}$ of conductor $F$, $$(f\otimes{\operatorname{\varphi}})(z) := \sum_{n=1}^{\infty}{\operatorname{\varphi}}(n)c_f(n, 1)e(nz)$$ denotes the twist of $f$ by ${\operatorname{\varphi}}$. If ${\mathfrak}{bc}\subseteq{\mathfrak}{b}^{-1}$, then $f\otimes{\operatorname{\varphi}}\in\mathcal{S}_k(\Gamma[{\mathfrak}{b}^{-1}, F^2{\mathfrak}{bc}], \psi{\operatorname{\varphi}}^2)$, so we can see immediately that $f_0\in\mathcal{S}_k(\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}p^2], \psi)$ and this matches Proposition \[pstabprop\].

By definition and [@Shimurahalf (5.4a)] we have $V_{1, p} = U_p-T_1 = U_p-T(p)$ in this case, so $f_0$ in \[pstab\] becomes $$f_0 = f+(p\lambda_{p, 1})^{-1}f|V_{1, p} = f+p^{-1}\lambda_{p, 1}^{-1}f|U_p-p^{-1}\lambda_{p, 1}^{-1}\Lambda(p)f,$$ where $\Lambda(p)$ denotes the eigenvalue of $f$ under $T_p$. By Lemma 3.1 (c), this is precisely the form of \[pstab1\] above.

Tracing the Rankin-Selberg Integral {#tracesection}
===================================

Given the relationship, established in Corollary \[pstablfun\], between $L(s, f, \chi)$ and $(s, f_0, \chi)$ the focus can be shifted to the latter. The level, ${\mathfrak}{y}$, of the Rankin-Selberg integral \[RS\] will depend on $\chi$, which dependence we naturally seek to avoid; this section removes this dependence and this method hinges now on the fact that $p$ divides the level of $f_0$.

Fix $\tau\in {\mathfrak}{b}S_+^{\triangledown}$ such that $c(\tau, 1; f_0)\neq 0$. Recalling ${\mathfrak}{t}$ as an integral ideal such that $h^T(2\tau)^{-1}h\in 4{\mathfrak}{t}^{-1}$ and define the matrix $$\begin{aligned}
\label{tauhat}
    \hat{\tau} := N({\mathfrak}{t})(2\tau)^{-1}\in M_n(\mathbb{Z}).\end{aligned}$$ Take a Dirichlet character $\chi$ of modulus $p^{\ell}$ and conductor $p^{\ell_{\chi}}$ with $0\leq \ell_{\chi}\leq \ell\in\mathbb{Z}$, a $\mu\in\{0, 1\}$ such that $(\psi_{\infty}\chi)(-1) = (-1)^{[k]+\mu}$, and put $\eta := \psi\bar{\chi}\rho_{\tau}$.

This section involves many levels and liftings of modular forms through these levels so first we define and clarify these schematically. Fix ${\mathfrak}{b}$, think of $f_0$ as a form of level $\Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{c}_0]$ and put $${\mathfrak}{y}_{\chi} := [{\mathfrak}{tc}p^{\ell_{\chi}}]^2.$$ If we assume that $2^n-1\leq \ell$, then the levels involved are $\Gamma_{\alpha} := \Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{y}_{\alpha}]$ where the integral ideals ${\mathfrak}{y}_{\alpha}$ are indexed by $\alpha\in\{r, \ell\in\mathbb{Z}\mid \ell_{\chi}\leq\ell\leq r\}\cup\{0\}$. They are defined below, arranged in order of divisibility: $$\begin{split}
        &\vdots \\
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &{\mathfrak}{y}_{r} := {\mathfrak}{y}_{0}p^{2r} \\
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &\vdots \\
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &{\mathfrak}{y}_{\ell} := {\mathfrak}{y}_{0}p^{2\ell} \\
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &\vdots \\
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &{\mathfrak}{y}_{1} := {\mathfrak}{y}_0p^{2} \\ 
        &\rotatebox[origin=c]{270}{$\subseteq$} \\
        &{\mathfrak}{y}_0 := {\mathfrak}{t}^2{\mathfrak}{cc}_0,
    \end{split}\label{tower1}$$ On the other hand, if $0\leq \ell<2^n-1$ then we have ${\mathfrak}{y}_0\subseteq {\mathfrak}{y}_{\ell}$ and we could just take ${\mathfrak}{y}_0$ – independent of $\chi$ – as the level of the Rankin-Selberg integral. Generally we lift all the way up to ${\mathfrak}{y}_r$ first, and then we shall trace the Rankin-Selberg integral back down to ${\mathfrak}{y}_0$ which process is given in the rest of this section. This is so that we can treat all characters uniformly. In specific cases, i.e. when $\ell\geq\ell_{\chi}\geq 2^n-1$, one need not lift up to $r$ in the first place and such a case is given as an example at the end of this section. In general, we obtain $$\begin{aligned}
    \begin{split}
    L_{\psi}(s, f_0, \bar{\chi}) &= \left[\Gamma_n\left(\tfrac{s-n-1+k+\mu}{2}\right)2c(\tau, 1; f_0)\right]^{-1}N({\mathfrak}{b})^{n(n+1)}|4\pi\tau|^{\frac{s-n-1+k+\mu}{2}} \\
    &\hspace{10pt}\times \left(\frac{\Lambda_{{\mathfrak}{c}_0}}{\Lambda_{{\mathfrak}{y}_0}}\right)\left(\tfrac{2s-n}{4}\right)\prod_{q\in\mathbf{b}}g_q\left((\psi^{{\mathfrak}{c}_0}\bar{\chi})(q)q^{-s}\right)\left\langle f_0, \theta_{\bar{\chi}}\mathcal{E}(\cdot, \tfrac{2s-n}{4})\right\rangle_{{\mathfrak}{y}_{r}}V_{r}
    \end{split} \label{f0RS}\end{aligned}$$ in which $V_{r} := {\operatorname{Vol}}(\Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{y}_{r}]{\backslash}\mathbb{H}_n)$.

Write $Y_{\alpha} := N({\mathfrak}{b})\sqrt{N({\mathfrak}{y}_{\alpha})}\in\mathbb{Z}$; then $Y_0 = N({\mathfrak}{tbc})p^{2^n-1}$, and notice that $Y_{\chi} = Y_0p^{\ell_{\chi}}$, $Y_{\ell} = Y_0p^{\ell}$, and $Y_r = Y_0p^r$ whenever $r, \ell, \ell_{\chi}\geq 2^n-1$.

The definition of the trace map on modular forms is well-known; with ${\mathfrak}{b}$ fixed, the map ${\mathop{\rm Tr}\nolimits}_{{\mathfrak}{c}}^{{\mathfrak}{c}'}$ for any ${\mathfrak}{c}'\subseteq{\mathfrak}{c}$ takes modular forms of level $\Gamma' = \Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{c}']$ and character $\psi$ down to forms of level $\Gamma = \Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{c}]$ and character $\psi$ and is defined by writing $\Gamma = \bigsqcup_{\gamma}\Gamma'\gamma$ and acting by the slash operator. If $g\in\mathcal{M}_{\frac{n}{2}+\mu}(\Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{y}_r], \chi\rho_{\tau})$, then put $F_g(z, s) := g(z)\mathcal{E}(z, \frac{2s-n}{4}; k-\frac{n}{2}-\mu, \bar{\eta}, \Gamma_r)$ and we have $$\begin{aligned}
    \label{tracedef}
    {\mathop{\rm Tr}\nolimits}_{{\mathfrak}{y}_0}^{{\mathfrak}{y}_r}(F_g) = \sum_{u\in S(\mathbb{Z}/p^{2r}\mathbb{Z})} F_g\big\|_k\left(\begin{smallmatrix} I_n & 0 \\ N({\mathfrak}{y}_0)u & I_n\end{smallmatrix}\right) \sum_{u\in S({\mathfrak}{b}^{-2}/p^{2r}{\mathfrak}{b}^{-2})}F_g\big\|_k\left(\begin{smallmatrix} I_n & 0 \\ Y_0^2u & I_n\end{smallmatrix}\right).\end{aligned}$$ Let $M\in\mathbb{Z}$ and define the matrix $$\iota_M := \begin{pmatrix} 0 & -M^{-1}I_n \\ MI_n & 0\end{pmatrix},$$ which belongs to $P_{\mathbb{A}}\iota$ and is therefore in ${\mathfrak}{M}$. We associate to $\iota_M$ the operator $W(M)$ acting on modular forms of weight $\kappa\in\frac{1}{2}\mathbb{Z}$ by $h|W(M) := h||_k\iota_M$.

\[tracereduce\] Let $\chi$ be of modulus $p^{\ell}$ and take $g$ and $F_g$ as above. If $r\geq\max\{\ell, 2^n-1\}$ is an integer, then $$\left\langle f_0, F_g(\cdot, s)\right\rangle_{{\mathfrak}{y}_r} = (-1)^{n[k]}p^{\frac{rn}{2}}\left\langle f_0, H_g(\cdot, s)|U_p^rW(Y_0)\right\rangle_{{\mathfrak}{y}_0},$$ where $H_g := F_g|W(Y_r)$.

Straight by definition we have $$\left\langle f_0, F_g(\cdot, s)\right\rangle_{{\mathfrak}{y}_r} = \left\langle f_0, {\mathop{\rm Tr}\nolimits}_{{\mathfrak}{y}_0}^{{\mathfrak}{y}_r}(F_g)\right\rangle_{{\mathfrak}{y}_0}.$$ To finish we note that $W(Y_0)^2 = (-1)^{n[k]}$ and we claim $${\mathop{\rm Tr}\nolimits}_{{\mathfrak}{n}_0}^{{\mathfrak}{y}_{r}}(F_{g})|W(Y_0) = p^{\frac{rn}{2}}H_g|U_p^{r},$$ the proof of which is twofold. That the matrices corresponding to the operators match up is given by the rote calculation $$\begin{pmatrix} I_n & 0 \\ Y_0^2u & I_n\end{pmatrix}\iota_{Y_0} = \iota_{Y_r}\begin{pmatrix} p^{-r}I_n & -p^{-r}u \\ 0 & p^{r}I_n\end{pmatrix},$$ for $u\in S({\mathfrak}{b}^{-2}/p^{2r}{\mathfrak}{b}^{-2})$ and in which we used that $Y_r = Y_0p^r$. To prove the claim we need to check the various factors of automorphies of half-integral weight match up. As an element of $\mathcal{R}_{0p}$, the operator $U_p$ acts with factor of automorphy $J^k(\alpha, z) = J^{\frac{1}{2}}(\alpha, z)j(\alpha, z)^{[k]}$, which we discussed towards the beginning of the previous section and which is defined in Section 2 of [@Shimurahalf]. Given the well-known cocycle relation of $j(\alpha, z)$ it suffices to check that identity $$\begin{aligned}
 \label{tracefactors}
    h\left(\begin{pmatrix} I_n & 0 \\ Y_0^2u & I_n\end{pmatrix}, \iota_{Y_0}z\right)h(\iota_{Y_0}, z) = h(\iota_{Y_r}, \alpha_u z)J^{\frac{1}{2}}\left(\alpha_u, z\right),\end{aligned}$$ where $\alpha_u = \left(\begin{smallmatrix} p^{-r}I_n & -p^{-r}u \\ 0 & p^rI_n\end{smallmatrix}\right)$. By considering $\iota_M \in \iota P_{\mathbb{A}}$ and using a combination of the weak automorphy property \[h3\] and Equation (2.5) of [@Shimuratheta p. 1025] we see that $h(\iota_M, z) = |iMz|^{\frac{1}{2}}$. By Lemma 2.4 of [@Shimurahalf] we have $J^{\frac{1}{2}}(\alpha_u, z) = 1$. Finally, by use of Lemma 2.2 of [@Shimuratheta], we have $$h\left(\begin{pmatrix} I_n & 0 \\ Y_0^2u & I_n\end{pmatrix}, \iota_{Y_0}z\right) = |Y_0^2uz+I_n|^{\frac{1}{2}}.$$ So the left-hand side of \[tracefactors\] above is $|Y_0i(z-u)|^{\frac{1}{2}}$ and the right-hand side is $$|Y_0p^{-r}i(z-u)|^{\frac{1}{2}},$$ whence the claim.

As an example, assume that $\chi$ is primitive, $r = \ell = \ell_{\chi}\geq 2^n-1$, and $g = \theta_{\bar{\chi}}$. Letting $H_{\chi}:=H_{\theta_{\bar{\chi}}}$ and $V_{\chi} := {\operatorname{Vol}}(\Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{y}_{\chi}]{\backslash}\mathbb{H}_n)$, the integral expression of \[f0RS\] combined with the above proposition gives $$\begin{aligned}
    \begin{split}
        L_{\psi}(s, f_0, \bar{\chi}) &= \left[\Gamma_n\left(\tfrac{s-n-1+k+\mu}{2}\right)2c(\tau, 1; f_0)\right]^{-1}N({\mathfrak}{b})^{n(n+1)}|4\pi\tau|^{\frac{s-n-1+k+\mu}{2}}(-1)^{n[k]}p^{\frac{rn}{2}} \\
    &\hspace{10pt}\times\left(\frac{\Lambda_{{\mathfrak}{c}_0}}{\Lambda_{{\mathfrak}{y}_0}}\right)\left(\tfrac{2s-n}{4}\right)\prod_{q\in\mathbf{b}}g_q\left((\psi^{{\mathfrak}{c}_0}\bar{\chi})(q)q^{-s}\right) \left\langle f_0, H_{\chi}|U_p^rW(Y_0)\right\rangle_{{\mathfrak}{y}_0}V_{\chi}. \label{tracedf0RS}
    \end{split}\end{aligned}$$

A transformation formula of theta series {#thetasection}
========================================

Transformation formulae for theta series of the form $\theta_{\chi}|W(Y_{\chi})$ when $\chi$ is a primitive Dirichlet character are generally well-known entities. The precise formula of this section is encompassed by the generality of both Theorem A3.3 and Proposition A3.17 of [@Shimurabook]; what follows is a concrete derivation and calculation of the integrals found in the aforementioned results. Theorem A3.3 of [@Shimurabook] gives the existence of a $\mathbb{C}$-linear automorphism $\lambda\mapsto{{}^{\sigma}\!{{\lambda}}}$ of ${\mathfrak}{M}$ on the space of “Schwartz functions on $M_n(\mathbb{Q}_{\mathbf{f}})$”, and it gives formulae of this action by $P_{\mathbb{A}}$ and the inversion $\iota = \left(\begin{smallmatrix} 0 & -I_n \\ I_n & 0\end{smallmatrix}\right)$. This is relevant since a more general class of theta series is defined by Schwartz functions $\lambda$ by $$\begin{aligned}
    \theta(z, \lambda) := \sum_{x\in M_n(\mathbb{Q})} \lambda(x_{\mathbf{f}})|x|^{\mu}e(x^T\tau xz) \label{thetaschwartz}\end{aligned}$$ for a fixed $\tau\in S_+$ and $\mu\in\{0, 1\}$. If $\chi$ is a Hecke character of conductor ${\mathfrak}{f}$, then $\lambda := \prod_p\lambda_p$ and $$\begin{aligned}
    \lambda_p(y) :=\begin{cases}    
        1 &\text{if $y\in M_n(\mathbb{Z}_p)$ and $p\nmid{\mathfrak}{f}$}, \\
        \chi_p(|y|) &\text{if $y\in GL_n(\mathbb{Z}_p)$ and $p\mid{\mathfrak}{f}$}, \\
        0 &\text{otherwise},
    \end{cases} \label{schwartzchi}\end{aligned}$$ gives the series $\theta(z, \lambda) = \theta_{\chi}^{\mu}(z; \tau)$ of \[theta\].

Assume that $\chi$ is a Hecke character of conductor $p^{\ell_{\chi}}$ and let $\iota_{\chi} = \iota_{Y_{\chi}}$. Since $W(Y_{\chi})\in C^{\theta}$, Proposition A3.17 of [@Shimurabook] says that $\theta(z, \lambda)|W(Y_{\chi}) = \theta\left(z, {{}^{\iota_{\chi}^{-1}}\!{{\lambda}}}\right)$ and so we calculate ${{}^{\iota_{\chi}^{-1}}\!{{\lambda}}}$. Note $\iota_{\chi}^{-1} = \iota\sigma$ where $$\sigma := \begin{pmatrix} -Y_{\chi}I_n & 0 \\ 0 & -Y_{\chi}^{-1}\end{pmatrix}\in P,$$ and so ${{}^{\iota_{\chi}^{-1}}\!{{\lambda}}} = {{}^{\iota}\!{{({{}^{\sigma}\!{{\lambda}}})}}}$. Let $d = \frac{n^2}{2}$ if $n$ is even, $d = 0$ if $n$ is odd and let $d_py$ be the Haar measure on $M_n(\mathbb{Q}_p)$ such that the measure of $bM_n(\mathbb{Z}_p)$ is $|b|_p^{n^2/2}$ for any $b\in\mathbb{Q}$. Theorem A3.3 (5), and equation (A3.3) of [@Shimurabook], and the definition of $\lambda$ in \[schwartzchi\] above gives $$\begin{aligned}
 {{}^{\iota}\!{{({{}^{\sigma}\!{{\lambda}}})}}}_p(x) &= i^d\chi_{\infty}(-1)^n|Y_{\chi}|_p^{\frac{n^2}{2}}|\det(2\tau)|_p^{\frac{n}{2}}\int_{Y_{\chi}^{-1}GL_n(\mathbb{Z}_p)}\chi_p(|Y_{\chi}y|)e_p(-{\operatorname{tr}}(x^T2\tau y))d_{p}y \\
    &=i^d\chi_{\infty}(-1)^n|\det(2\tau)|_p^{\frac{n}{2}}\int_{GL_n(\mathbb{Z}_p)}\chi_p(|y|)e_p\left(-\tfrac{{\operatorname{tr}}(x^T2\tau y)}{Y_{\chi}}\right)d_py,\end{aligned}$$ making the change of variables $y\mapsto Y_{\chi}y$ in the last line. By the definitions of $Y_{\chi} = N({\mathfrak}{tbc})p^{\ell_{\chi}}$ and $\hat{\tau}$ of \[tauhat\] we have ${{}^{\iota}\!{{({{}^{\sigma}\!{{\lambda}}})}}}_p(x)$ is equal to $$\begin{aligned}
     i^d\chi_{\infty}(-1)^n|\det(2\tau)|_p^{\frac{n}{2}}\sum_{a\in GL_n(\mathbb{Z}/p^{\ell_{\chi}}\mathbb{Z})}\chi_p(|a|)e\left(\tfrac{{\operatorname{tr}}(x^T\hat{\tau}^{-1}a)}{N({\mathfrak}{bc})p^{\ell_{\chi}}}\right)\int_{p^{\ell_{\chi}}GL_n(\mathbb{Z}_p)} e_p\left(-\tfrac{{\operatorname{tr}}(x^T\hat{\tau}^{-1}y)}{N({\mathfrak}{bc})p^{\ell_{\chi}}}\right)d_py.\end{aligned}$$ The integral in the above equation is non-zero if and only if the integrand is a constant function in $y$ – i.e. if and only if $x\in |N({\mathfrak}{bc})|_p^{-1}\hat{\tau}M_n(\mathbb{Z}_p)$ – at which point it is $p^{-\ell_{\chi}\frac{n^2}{2}}$. Likewise by the same process, if $q\neq p$, we have ${{}^{\iota}\!{{({{}^{\sigma}\!{{\lambda}}})}}}_q(x)\neq 0$ if and only if $x\in |N({\mathfrak}{bc})|_q^{-1}\hat{\tau}M_n(\mathbb{Z}_q)$ at which point it is $|\det(2\tau)|_q^{\frac{n}{2}}$. Therefore ${{}^{\iota}\!{{({{}^{\sigma}\!{{\lambda}}})}}}(x)\neq 0$ if and only if $x\in N({\mathfrak}{bc})\hat{\tau}M_n(\mathbb{Z})$ at which point we have $$\begin{aligned}
    {{}^{\iota}\!{{({{}^{\sigma}\!{{\lambda}}})}}}(x) &= i^d\chi_{\infty}(-1)^n|2\tau|^{-\frac{n}{2}}p^{-\ell_{\chi}\frac{n^2}{2}}G_n(N({\mathfrak}{bc})^{-1}\hat{\tau}^{-1}x, \bar{\chi}), \label{actionlambda}\end{aligned}$$ where, for any Hecke character ${\operatorname{\varphi}}$ of conductor ${\mathfrak}{f}$ and $X\in M_n(\mathbb{Z})$, $$G_n(X, {\operatorname{\varphi}}) := \sum_{a\in GL_n(\mathbb{Z}/N({\mathfrak}{f})\mathbb{Z})}{\operatorname{\varphi}}_{{\mathfrak}{f}}^{-1}(|a|)e\left(\tfrac{{\operatorname{tr}}(X^Ta)}{N({\mathfrak}{f})}\right)$$ denotes the degree $n$ Gauss sum and $G_n({\operatorname{\varphi}}) = G_n(I_n, {\operatorname{\varphi}})$. If ${\operatorname{\varphi}}$ is a primitive Dirichlet character then $G_n(X, {\operatorname{\varphi}}) = {\operatorname{\varphi}}(|X|)^{-1}G_n({\operatorname{\varphi}})$ if $(|X|, N({\mathfrak}{f}))=1$ and $G_n(X, {\operatorname{\varphi}}) = 0$ if $(|X|, N({\mathfrak}{f}))\neq 1$. So, under the assumption that $\chi$ is a primitive Dirichlet character and $x\in N({\mathfrak}{bc})\hat{\tau}M_n(\mathbb{Z})$, \[actionlambda\] becomes $$\begin{aligned}
    {{}^{\iota}\!{{({{}^{\sigma}\!{{\lambda}}})}}}(x) &= i^d\chi(-1)^n|2\tau|^{-\frac{n}{2}}p^{-\ell_{\chi}\frac{n^2}{2}}\chi(|N({\mathfrak}{bc})^{-1}\hat{\tau}^{-1}x|)G_n(\bar{\chi}). \label{actionlambda2}\end{aligned}$$ Hence, by the calculation \[actionlambda2\], the transformation formula $\theta(z, \lambda)|W(Y_{\chi}) = \theta\left(z, {{}^{\iota_{\chi}^{-1}}\!{{\lambda}}}\right)$ on theta series with Schwartz functions translates, when $\chi$ is a primitive Dirichlet character, to $$\theta_{\chi}|W(Y_{\chi}) = \frac{i^d\chi(-1)^n}{|2\tau|^{\frac{n}{2}}}p^{-\ell_{\chi}\frac{n^2}{2}}G_n(\bar{\chi})\sum_{x\in N({\mathfrak}{bc})\hat{\tau}M_n(\mathbb{Z})} \chi(|N({\mathfrak}{bc})^{-1}\hat{\tau}^{-1}x|)|x|^{\mu}e(x^T\tau xz)$$ and this becomes, by writing $x = N({\mathfrak}{bc})\hat{\tau} x'$, the desired formula $$\begin{aligned}
    \theta_{\chi}^{\mu}(z; \tau)|W(Y_{\chi}) = \chi(-1)^n\frac{i^d[N({\mathfrak}{tbc})]^{n\mu}}{|2\tau|^{\frac{n}{2}+\mu}}p^{-\ell_{\chi}\frac{n^2}{2}}G_n(\bar{\chi})\theta_{\bar{\chi}}^{(\mu)}\left(N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^2\frac{z}{2}; \hat{\tau}\right), \label{thetatrans}\end{aligned}$$ where we have written $N(\sqrt{a}) = |N({\mathfrak}{a})^{\frac{1}{2}}|$ for any integral ideal ${\mathfrak}{a}$.

Fourier expansions of Eisenstein series {#eisensteinsection}
=======================================

The holomorphic projection map $\mathbf{Pr}:C_{\kappa}^{\infty}(\Gamma)\to\mathcal{M}_{\kappa}(\Gamma, \psi)$ and its explicit action on Fourier coefficients is well-known when $2n<\kappa\in\mathbb{Z}$ – see Theorem 4.2 of [@Panchishkin p. 71]. This has a simple extension to the half-integral weight case with the formulae remaining unchanged, and we did this in [@Mercuri2 Theorem 3.1].

Given Proposition \[tracereduce\] and the transformation formula \[thetatrans\], it will be germane to give explicit expressions for the Fourier coefficients of $\mathbf{Pr}(H_{\theta_{\chi}^{\star}}|U_p^r)$, where $$\begin{aligned}
    \theta_{\chi}^{\star}(z) := \theta_{\chi}^{\mu}\left(N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^2\frac{z}{2}; \hat{\tau}\right). \label{thetastar}\end{aligned}$$ To ease up on notation, let $\mathcal{E}^{\star}(z, s) := \mathcal{E}(z, s; k-\frac{n}{2}-\mu, \bar{\eta}, \Gamma_r)|W(Y_r)$ and let $$\delta := n\pmod{2}\in\{0, 1\}.$$

The projection map is only applicable for certain values $s$ at which the Eisenstein series satisfies growth conditions; restriction to the set of special values, $\Omega_{n, k}$, at which the standard $L$-function satisfies algebraicity results guarantees this and this set is given by $$\begin{aligned}
    \Omega_{n, k}^+ :&= \left\{m\in\tfrac{1}{2}\mathbb{Z}\bigg|\tfrac{k-m-\mu}{2}\in\mathbb{Z}, n<m\leq k-\mu\right\}, \\
    \Omega_{n, k}^- :&= \left\{m\in\tfrac{1}{2}\mathbb{Z}\bigg|\tfrac{m+k-\mu-1}{2}\in\mathbb{Z}, 2n+1-k+\mu\leq m\leq n\right\}, \\
    \Omega_{n, k} :&= \Omega_{n, k}^-\cup\Omega_{n, k}^+.\end{aligned}$$

\[eisenexp\] For any $\varsigma\in S_+$, define $$V_{\varsigma} := \left\{(\varsigma_1, \varsigma_2)\in M_n(\mathbb{Z})\times S_+\mid \frac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^2}{2}\varsigma_1^T\hat{\tau}\varsigma_1 + \varsigma_2 = \varsigma\right\}.$$ Assume that $k>2n$, $\chi$ is a Dirichlet character, and $m\in\Omega_{n, k}$. For any $\beta\in\mathbb{Z}$, there exists a polynomial $P(\varsigma, \varsigma'; \beta)\in\mathbb{Q}[\varsigma_{ij}, \varsigma_{ij}'\mid 1\leq i\leq j\leq n]$, defined on $\varsigma, \varsigma'\in S_+$; a finite subset $\mathbf{c}$ of primes; polynomials $f_{\sigma, q}\in\mathbb{Z}[t]$, defined for each $\sigma\in S_+$ and $q\in\mathbf{c}$, whose coefficients are independent of $\chi$; and a factor $$\begin{aligned}
    C_{\pm}^{\star}(\sigma, m) :&= Zi^{-n(k-\frac{n}{2}-\mu)}N({\mathfrak}{b}^2{\mathfrak}{y}_r)^{n\left(\frac{3n+1}{2}+\mu-k-m\right)} 2^{n(k-\mu+\frac{3}{2})}\pi^{n\left(\frac{m+k-n-\mu}{2}\right)} \\
    &\hspace{10pt}\times\Gamma_n\left(\tfrac{m+k-n-\mu}{2}\right)^{-1}|\sigma|^{m_{\pm}}\prod_{q\in\mathbf{c}}f_{\sigma, q}(\bar{\eta}(q)q^{\frac{n-\delta}{2}-m}),\end{aligned}$$ where $m_+ = m - n - \frac{1}{2}$, $m_- = 0$, $Z = 1$ if $n$ is odd and $Z = e^{\frac{\pi in}{4}}$ if $n$ is even, such that if $m\in\Omega_{n, k}{\backslash}\{n+\frac{1}{2}\}$ (and $m\neq n+\frac{3}{2}$ if $n>1$ and $(\psi^*\bar{\chi})^2 = 1$), then $\mathbf{Pr}(\theta_{\chi}^{\star}\mathcal{E}^{\star}(\cdot, \frac{2m-n}{4})|U_p^r)$ has non-zero Fourier coefficients only when $\sigma>0$ at which $$c\left(\sigma, 1; \mathbf{Pr}(\theta_{\chi}^{\star}\mathcal{E}^{\star}(\cdot, \tfrac{2m-n}{4})|U_p^r)\right) = \sum_{(\sigma_1, \sigma_2)\in V_{p^r\sigma}}\chi(|\sigma_1|)|\sigma_1|^{\mu}C_+^{\star}(\sigma_2, m)P\left(\sigma_1, p^r\sigma; \tfrac{k-m-\mu}{2}\right)$$ if $m\in\Omega_{n, k}^+$ whereas $$c\left(\sigma, 1; \mathbf{Pr}(\theta_{\chi}^{\star}\mathcal{E}^{\star}(\cdot, \tfrac{2m-n}{4})|U_p^r)\right) = \sum_{(\sigma_1, \sigma_2)\in V_{p^r\sigma}}\chi(|\sigma_1|)|\sigma_1|^{\mu}C_-^{\star}(\sigma_2, m)P\left(\sigma_1, p^r\sigma; \tfrac{k+m-\mu-1-2n}{2}\right)$$ if $m\in\Omega_{n, k}^-$.

Furthermore, the polynomial $P(\varsigma, \varsigma'; \beta)$ satisfies $P(\varsigma, \varsigma'; \beta) \equiv |\varsigma|^{\beta}\pmod{\varsigma_{ij}'}$.

When $k\in\mathbb{Z}$ and $n$ is even the above kind of result is well-known, see for example Theorem 4.6 of [@Panchishkin p. 77]. Since the definition of the projection map remains unchanged, we can obtain the above in a similar manner, by using results on the Fourier development of integral and half-integral weight Eisenstein series as follows.

Let $\kappa\in\frac{1}{2}\mathbb{Z}$ and let $\mathcal{E}_{\kappa}(z, s) = \mathcal{E}(z, s; \kappa, \bar{\eta}, \Gamma[{\mathfrak}{x}^{-1}, {\mathfrak}{xy}])$, assuming that $N({\mathfrak}{x})$ and $N({\mathfrak}{y})$ are both squares, that $({\mathfrak}{x}^{-1}, {\mathfrak}{xy})\subseteq 2\mathbb{Z}\times 2\mathbb{Z}$ if $\kappa\notin\mathbb{Z}$, and set $Y:=\sqrt{N({\mathfrak}{xy})}$. If $2s+\frac{n}{2}\in\Omega_{n, k}$, $s\neq\frac{n+1}{4}$, and $s\neq \frac{n+3}{4}$ if $n>1$ and $\eta^2 = 1$, then by Proposition 17.6 of [@Shimurabook], the analytic continuation of the Eisenstein series, and the fact that $\iota\Gamma[{\mathfrak}{x}^{-1}, {\mathfrak}{xy}]\iota = \Gamma[{\mathfrak}{xy}, {\mathfrak}{x}^{-1}]$, we have $$\mathcal{E}_{\kappa}(z, s)|\iota =
    \displaystyle\sum_{0<\sigma\in ({\mathfrak}{xy})^{-1}S_+^{\triangledown}} c^{\pm}(\sigma, y, s)e(\sigma x) \hspace{20pt} \text{if $2s+\tfrac{n}{2}\in\Omega_{n, k}^{\pm}$,}$$ where, if $\sigma>0$, we have by Propositions 16.9 and 16.10 of [@Shimurabook] that $$\begin{aligned}
    c^{\pm}(\sigma, y, s) :&= ZN({\mathfrak}{xy})^{-\frac{n(n+1)}{2}}|y|^{s-\frac{\kappa}{2}}\xi(y, \sigma; s+\tfrac{\kappa}{2}, s-\tfrac{\kappa}{2})\prod_{q\in\mathbf{c}}f_{Y^2\sigma, q}(\bar{\eta}(q)q^{-2s-\frac{\delta}{2}}), \\
    \xi(g, h; s, s') :&= \int_{S_{\infty}}e(-hx)|x-ig|^{-s}|x-ig|^{-s'}dx,\end{aligned}$$ defined for $0< g\in S_{\infty}, h\in S_{\infty}$, and $s, s'\in\mathbb{C}$. The above integral converges for large enough ${\operatorname{\mathfrak{Re}}}(s+s')$, but is continued analytically via the hyperconfluent geometric function $\omega(g, h; s, s')$ of [@Shimurahyp]. Through this analytic continuation one can represent $\xi(y, \sigma, s+\frac{\kappa}{2}, s-\frac{\kappa}{2})$, for certain values of $s$, in terms of the differential operator $$\begin{aligned}
    R(g; \ell, s') :&= (-1)^{\ell n}e^{{\operatorname{tr}}(g)}|z|^{\ell+s'}\det\left[\frac{\partial_n}{\partial_ng}\right]^{\ell} (e^{-{\operatorname{tr}}(g)}|z|^{-s'}), \\
    \frac{\partial_n}{\partial_ng} :&= \left(\frac{1+\delta_{ij}}{2}\frac{\partial}{\partial g_{ij}}\right)_{i, j=1}^n, \end{aligned}$$ where $\ell\in\mathbb{Z}$. This is obtained by using, in order, the relation (17.11) and analytic continuation of the hyperconfluent geometric function of [@Shimurahyp Theorem 3.1]; the properties (4.7.K) and (4.10) of [@Shimurahyp]and the definitions (3.23)–(3.24) of [@Panchishkin p. 63]; and, finally, Proposition 3.2 of [@Shimurahyp] to get $$\begin{aligned}
    \begin{split}
    c^+(\sigma, y, s) &= C(\sigma, s)|4\pi \sigma y|^{s-\frac{\kappa}{2}}R\left(4\pi\sigma y; \tfrac{\kappa}{2}-s, \tfrac{n+1-\kappa}{2}-s\right)e^{-2\pi{\operatorname{tr}}(\sigma y)}, \\
    c^-(\sigma, y, s) &= C(\sigma, s)|4\pi\sigma y|^{\frac{n+1-\kappa}{2}-s}R\left(4\pi \sigma y; s+\tfrac{\kappa-n-1}{2}, s-\tfrac{\kappa}{2}\right)e^{-2\pi{\operatorname{tr}}(\sigma y)}, \\
    C(\sigma, s) :&= Zi^{-n\kappa}N({\mathfrak}{xy})^{-\frac{n(n+1)}{2}}2^{n(\kappa+\frac{n+3}{2})}\pi^{n(s+\frac{\kappa}{2})}\Gamma_n(s+\tfrac{\kappa}{2})^{-1} \\
    &\hspace{10pt}\times |\sigma|^{s-\frac{\kappa - n-1}{2}}\prod_{q\in\mathbf{c}}f_{Y^2\sigma, q}(\bar{\eta}(q)q^{-2s-\frac{\delta}{2}}).
    \end{split} \label{cpm}\end{aligned}$$ Now, since $\mathcal{E}_{\kappa}(z, s)|W(Y) = Y^{-n\kappa}\mathcal{E}_{\kappa}(z, s)|\iota\big|_{z = Y^2z}$, we have that $$\mathcal{E}_{\kappa}(z, s)|W(Y) = \sum_{0<\sigma\in{\mathfrak}{x}^{-1}S_+^{\triangledown}}c_{{\mathfrak}{y}}^{\pm}(\sigma, y, s)e(\sigma x),$$ where $c_{{\mathfrak}{y}}^{\pm}(\sigma, y, s) := Y^{-n\kappa}c^{\pm}(Y^{-2}\sigma, Y^2y, s)$ are given explicitly by $$\begin{aligned}
    c_{{\mathfrak}{y}}^+(\sigma, y, s) &= Y^{-n\kappa}C(Y^{-2}\sigma, s)|4\pi\sigma y|^{s-\frac{\kappa}{2}}R\left(4\pi\sigma y; \tfrac{\kappa}{2}-s, \tfrac{n+1-\kappa}{2}-s\right)e^{-2\pi{\operatorname{tr}}(\sigma y)}, \\
    c_{{\mathfrak}{y}}^-(\sigma, y, s) &= Y^{-n\kappa}C(Y^{-2}\sigma, s)|4\pi\sigma y|^{\frac{n+1-\kappa}{2}-s}R\left(4\pi\sigma y;  s-\tfrac{n+1-\kappa}{2}, s-\tfrac{\kappa}{2}\right)e^{-2\pi{\operatorname{tr}}(\sigma y)}.\end{aligned}$$ Now let $g\in\mathcal{M}_{\kappa'}$ be a holomorphic modular form such that $c_g(\sigma, 1)=0$ if $|\sigma|=0$. Then for any $2s+\frac{n}{2}\in\Omega_{n, k}$ the Fourier coefficients of $g[\mathcal{E}_{\kappa}(z, s)|W(Y)]$ are non-zero only for $\sigma>0$. Put $C_{{\mathfrak}{y}}^+(\sigma, s) := Y^{-n\kappa}|\sigma|^{s-\frac{\kappa}{2}}C(Y^{-2}\sigma, s)$ and $C_{{\mathfrak}{y}}^-(\sigma, s) := Y^{-n\kappa}|\sigma|^{\frac{n+1-\kappa}{2}-s}C(Y^{-2}\sigma, s)$. By analogy to Theorem 4.6 of [@Panchishkin p. 77], the application of $\mathbf{Pr}$ along with \[cpm\], these coefficients are given by $$\begin{aligned}
    c(\sigma, 1; \mathbf{Pr}([g\cdot\mathcal{E}_{\kappa}(\cdot, s)|W(Y)]|U_p^r)) = \sum_{\sigma_1+\sigma_2=p^r\sigma}c_g(\sigma_1, 1)C_{{\mathfrak}{y}}^+(\sigma, s)P\left(\sigma_1, p^r\sigma; \tfrac{\kappa}{2}-s\right) \label{projexpplus}\end{aligned}$$ when $2s+\frac{n}{2}\in\Omega_{n, k}^+$, $s\neq \frac{n+1}{4}$, $s\neq\frac{n+3}{4}$ if $\eta^2 = 1$, and $\sigma>0$, whereas $$\begin{aligned}
    c(\sigma, 1; \mathbf{Pr}([g\cdot\mathcal{E}_{\kappa}(\cdot, s)|W(Y)]|U_p^r)) = \sum_{\sigma_1+\sigma_2=p^r\sigma}c_g(\sigma_1, 1)C_{{\mathfrak}{y}}^-(\sigma, s)P\left(\sigma_1, p^r\sigma; s+\tfrac{\kappa-n-1}{2}\right) \label{projexpminus}\end{aligned}$$ if $2s+\frac{n}{2}\in\Omega_{n, k}^-$ and $\sigma>0$.

Specialising \[projexpplus\] and \[projexpminus\] above to the case ${\mathfrak}{x} = {\mathfrak}{b}^2$, ${\mathfrak}{y} = {\mathfrak}{y}_r$, $\kappa = k-\frac{n}{2}-\mu$, $\kappa' = \frac{n}{2}+\mu$, $g = \theta_{\chi}^{\star}$, and $s = \frac{2m-n}{4}$ for $m\in\Omega_{n, k}$ gives Proposition \[eisenexp\] upon putting $C_{\pm}^{\star}(\sigma, m) := C_{{\mathfrak}{y}_r}^{\pm}(\sigma, \frac{2m-n}{4})$.

$p$-adic interpolation {#interpolation}
======================

$p$-adic measures and the main theorem
--------------------------------------

Though complex $L$-functions are defined on variables $s\in\mathbb{C}$, they can equally be viewed as Mellin transforms of the continuous characters $\mathbb{R}_{>0}\to\mathbb{C}^{\times}; y\mapsto y^s$. In this latter vantage point, $p$-adic $L$-functions can naturally be constructed as Mellin transforms of continuous characters on $\mathbb{Z}_p^{\times}$ with respect to a $p$-adic measure.

Fix a prime $p\nmid{\mathfrak}{c}$, let $\mathbb{C}_p := \widehat{\overline{\mathbb{Q}}}_p$ denote the completion of the algebraic closure of $\mathbb{Q}_p$, and fix an embedding $\iota_p:\overline{\mathbb{Q}}\hookrightarrow\mathbb{C}_p$. The $p$-adic norm naturally extends to $\mathbb{C}_p$ and its ring of integers is given by $$\mathcal{O}_p := \{x\in\mathbb{C}_p\mid |x|_p\leq 1\}.$$ The domain of the $p$-adic $L$-function will be $$X_p := \{x\in {\operatorname{Hom}}(\mathbb{Z}_p^{\times}, \mathbb{C}_p^{\times})\mid x\ \text{is continuous}\}.$$ The discussion in [@Panchishkin pp. 23–25] concerning the decomposition of $X_p$ tells us that any $\mathbb{C}_p$-analytic function $F$ on $X_p$ is uniquely determined by its values $F(\chi_0 \chi)$ for a fixed $\chi_0\in X_p$ and $\chi$ ranging over non-trivial elements of $X_p^{{\operatorname{tors}}}$. This torsion subgroup can be identified as the group of primitive Dirichlet characters having $p$-power conductor. So to define a $p$-adic measure, it is enough to give its values on $\chi x_p^m$ where $\chi$ is a non-trivial primitive Dirichlet character of $p$-power conductor such that $\chi^2\neq 1$, $m\in\Omega_{n, k}{\backslash}\{n+\frac{1}{2}\}$, and $$\begin{aligned}
    x_p^m:\mathbb{Z}_p^{\times}&\to\mathbb{C}_p^{\times} \\
    y&\mapsto y^m.\end{aligned}$$

\[distribution\] Let $LC(\mathbb{Z}_p^{\times}, \mathbb{C}_p)$ denote the $\mathbb{C}_p$-module of all locally constant functions $\mathbb{Z}_p^{\times}\to\mathbb{C}_p$, and let $A$ be a $\mathbb{C}_p$-module. An $A$-valued distribution on $\mathbb{Z}_p^{\times}$ is an $A$-linear homomorphism $$\nu:LC(\mathbb{Z}_p^{\times}, \mathbb{C}_p)\to A,$$ which we denote by $$\nu(\phi) = \int_{\mathbb{Z}_p^{\times}}\phi d\nu,$$ for any $\phi\in LC(\mathbb{Z}_p^{\times}, \mathbb{C}_p)$.

When $A =\mathbb{C}$ these are called *complex distributions*, whereas when $A=\mathbb{C}_p$ they are *$p$-adic distributions*.

Since $\mathbb{Z}_p^{\times} = \varprojlim(\mathbb{Z}/p^i\mathbb{Z})^{\times}$ is a profinite group, taken with respect to the natural projections $\pi_{ij}:(\mathbb{Z}/p^i\mathbb{Z})^{\times}\to(\mathbb{Z}/p^j\mathbb{Z})^{\times}$ for each $i\geq j$, to any distribution there associates a system of functions $\nu_i:(\mathbb{Z}/p^i\mathbb{Z})^{\times}\to A$ satisfying $$\nu_j(y) = \sum_{x\in \pi_{ij}^{-1}(y)}\nu_i(x), \hspace{20pt} y\in (\mathbb{Z}/p^j\mathbb{Z})^{\times}.$$ This association works by noting that each $\phi\in LC(\mathbb{Z}_p^{\times}, \mathbb{C}_p)$ factors through some $(\mathbb{Z}/p^i\mathbb{Z})^{\times}$ and by $$\int_{\mathbb{Z}_p^{\times}}\phi d\nu = \sum_{x\in \mathbb{Z}/p^i\mathbb{Z}}\phi(x)\nu_i(x).$$ The compatibility criterion of [@Panchishkin p. 17] tells us when we can run the above process backwards.

\[compat\] Let $\{\nu_i:(\mathbb{Z}/p^i\mathbb{Z})^{\times}\to A\}_{i=1}^{\infty}$ be an arbitrary system of functions. If, for any fixed $j\in\mathbb{Z}$ and any function $\phi_j\in LC((\mathbb{Z}/p^j\mathbb{Z})^{\times}, A)$, the sum $$\sum_{y\in (\mathbb{Z}/p^i\mathbb{Z})^{\times}}\phi_j(\pi_{ij}(y))\nu_i(y)$$ is independent of $i$ for large enough $i\geq j$, then there exists a distribution $\nu$ on $\mathbb{Z}_p^{\times}$ associated to $\{\nu_i\}_i$.

\[measure\] Let $C(\mathbb{Z}_p^{\times}, \mathbb{C}_p)$ denote the topological $\mathbb{C}_p$-module of all continuous functions $\mathbb{Z}_p^{\times}\to\mathbb{C}_p$. A *$p$-adic measure* is a $\mathbb{C}_p$-module homomorphism $$\nu:C(\mathbb{Z}_p^{\times}, \mathbb{C}_p)\to\mathbb{C}_p.$$

So distributions are generally quite easy to define; $p$-adic measures arise from $p$-adic distributions that are $p$-adically bounded. Hence defining a distribution interpolating $L$-values is relatively trivial and showing that these expressions are bounded is the crux of the matter. To do this, we will invoke the abstract Kummer congruences, which criterion is well-known in generality; we give a specialisation of it.

\[kummer\] Suppose, for an index set $I$, that $\{f_i\}_{i\in I}\subseteq C(\mathbb{Z}_p^{\times}, \mathcal{O}_p)$ is such that ${\operatorname{span}}_{\mathbb{C}_p}\{f_i\mid i\in I\}$ is dense in $C(\mathbb{Z}_p^{\times}, \mathbb{C}_p)$. For a given system $\{a_i\}_{i\in I}\subseteq\mathcal{O}_p$, there exists an $\mathcal{O}_p$-module homomorphism $\nu:C(\mathbb{Z}_p^{\times}, \mathcal{O}_p)\to\mathcal{O}_p$ such that $$\int_{\mathbb{Z}_p^{\times}}f_id\nu = a_i$$ if and only if, for any finite subset $S\subseteq I$ and any system $\{b_i\}_{i\in S}\subseteq\mathbb{C}_p$, the condition $$\sum_{i\in S}b_if_i\subseteq p^N\mathcal{O}_p$$ for an integer $N$ implies that $$\sum_{i\in S} b_ia_i\in p^N\mathcal{O}_p.$$

The proof of this can be found in [@Panchishkin pp. 19–20]; it covers $\mathbb{C}_p$-valued measures as well by multiplication of some non-zero constant. An easy example of these criteria is the Fourier coefficients of the Eisenstein series given in the previous section.

\[Sigma\] If $m\in\Omega_{n, k}$, then there exists a $p$-adic distribution $\Sigma_{\sigma, m}$ and measure $\Sigma_{\sigma}$, defined for $\chi\in X_p^{{\operatorname{tors}}}$ by $$\int_{\mathbb{Z}_p^{\times}}\chi x_p^md\Sigma_{\sigma} = \iota_p[\Sigma_{\sigma, m}(\chi)] := \iota_p\left[\prod_{p\neq q\in\mathbf{c}}f_{\sigma, q}(\bar{\chi}(q)q^{\frac{n-\delta}{2}-m})\right].$$

That $\Sigma_{\sigma, m}$ satisfies the compatibility criterion is immediate. Take ${\operatorname{\mathfrak{Re}}}(s)\to\infty$ in the identity (16.46) of [@Shimurabook] and we see that the product of polynomials has no constant term. Take $\chi x_p^m$ as the system $\{f_i\}_{i\in I}$ in the statement of the Kummer congruences. If $\mathcal{X}\subseteq X_p^{{\operatorname{tors}}}$ is a finite subset and $\sum_{\chi\in\mathcal{X}}b_{\chi}\chi x_p^m\subseteq p^N\mathcal{O}_p$, then $$\sum_{\chi\in\mathcal{X}}b_{\chi}\int_{\mathbb{Z}_p^{\times}}\chi x_p^md\Sigma_{\sigma}\in P^N\mathcal{O}_p$$ is immediate, by the crucial fact that the coefficients of $f_{\sigma, q}$ are independent of $\chi$.

If $\nu(\chi x_p^m) = a_m(\chi)$ for some $a_m:\{\text{$\mathbb{T}$-valued characters}\}\to\mathbb{C}_p$ and $\omega$ is a primitive $\mathbb{T}$-valued character whose conductor is prime to $p$, then the twist of $\nu$ by $\omega$, given by $[\nu\otimes\omega](\chi x_p^m) := a_m(\chi\omega)$, is also a $p$-adic measure.

A non-zero Hecke eigenform $f$ is *$p$-ordinary* if $$\left|p^\frac{n(n+1)}{2}\lambda_{p, 1}\cdots\lambda_{p, n}\right|_p =1$$ and assume $p\nmid{\mathfrak}{b}^{-1}\cap{\mathfrak}{c}$. The main theorem is given as follows.

\[main\] Let $k>2n$ and $f\in\mathcal{S}_k(\Gamma[{\mathfrak}{b}^{-1}, {\mathfrak}{bc}], \psi)$ be a $p$-ordinary Hecke eigenform. Assume the existence of $\tau\in {\mathfrak}{b}S_+^{\triangledown}$ such that $c_f(\tau, 1)\neq 0$ and $c(\tau, 1; f_0)\neq 0$. There exist bounded $\mathbb{C}_p$-analytic functions $$\begin{aligned}
    \nu_f^{\pm}:X_p\to\mathbb{C}_p\end{aligned}$$ that are uniquely determined by the following. In both cases $\chi\in X_p^{{\operatorname{tors}}}$ is a primitive Dirichlet character of conductor $p^{\ell_{\chi}}$ with $1\leq \ell_{\chi}\in\mathbb{Z}$, $\eta = \psi\bar{\chi}\rho_{\tau}$, $\mu\in\{0, 1\}$ is chosen so that $(\psi_{\infty}\chi)(-1) = (-1)^{[k]+\mu}$, $\hat{\tau} = N({\mathfrak}{t})(2\tau)^{-1}$, $\Lambda_{\tau}(s) := (\Lambda_{{\mathfrak}{c}_0}/\Lambda_{{\mathfrak}{y}_0})(\frac{2s-n}{4})$ is a finite product of Euler factors over $p\mid{\mathfrak}{t}$, $$\mathcal{G}_{\tau}(s) := \prod_{q\in\mathbf{b}}g_q\left((\psi^{{\mathfrak}{c}_0}\bar{\chi})(q)q^{-s}\right)^{-1},$$ $Z = 1$ and $d= 0$ if $n$ is odd, $Z = e^{\frac{i\pi n}{4}}$ and $d = \frac{n^2}{2}$ if $n$ is even.

(i) For any $m\in \Omega_{n, k}^+{\backslash}\{n+\frac{1}{2}\}$, assume that $m\neq n+\frac{3}{2}$ if $n>1$ and $(\psi^*\bar{\chi})^2 = 1$, and let $c_m := \frac{n}{2}(3k-3n-\mu+m-3+2\delta) + (1-\delta)m$; the measure $\nu_f^+$ is defined by $$\begin{aligned}
        \int_{\mathbb{Z}_p^{\times}}\chi x_p^md\nu_f^{+} &=\iota_p\left[\frac{(-1)^{n[k]}|2\tau|^{\frac{n}{2}+\mu}}{Zi^{d}N({\mathfrak}{tbc})^{n\mu}}\left|-\tfrac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}}{2}\hat{\tau}\right|^{-\frac{k+m-\mu-1-2n}{2}}\frac{p^{n\ell_{\chi}(n+1-k-m)}G_n(\bar{\chi})}{\Lambda_{\tau}(m)\mathcal{G}_{\tau}(m)}\right. \\
        &\hspace{40pt}\times\left.\left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)^{-\ell_{\chi}}\frac{L_{\psi}(m, f, \bar{\chi})}{\pi^{c_m}\langle f, f\rangle}\right].\end{aligned}$$

(ii) For any $m\in\Omega_{n, k}^-$ let $c_m := \frac{n}{2}(3k-5n-\mu+3m-2)$; the measure $\nu_f^-$ is given by $$\begin{aligned}
        \int_{\mathbb{Z}_p^{\times}}\chi x_p^m d\nu_f^{-} &=\iota_p\left[\frac{(-1)^{n[k]}|2\tau|^{\frac{n}{2}+\mu}}{Zi^{d}N({\mathfrak}{tbc})^{n\mu}p^{\ell_{\chi}\frac{n^2}{2}}}\left|-\tfrac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}}{2}\hat{\tau}\right|^{-\frac{k+3m-\mu-2-4n}{2}}\frac{p^{n\ell_{\chi}(n+1-k-m)}G_n(\bar{\chi})}{\Lambda_{\tau}(m)\mathcal{G}_{\tau}(m)}\right. \\
        &\hspace{40pt}\times\left.\left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)^{-\ell_{\chi}} \frac{L_{\psi}(m, f, \bar{\chi})}{\pi^{c_m}\langle f, f\rangle}\right].\end{aligned}$$

The condition that $m\neq n+\frac{1}{2}$ (and $m\neq n+\frac{3}{2}$ when $(\psi^*\bar{\chi})^2 = 1$) arises as a result of complications in the Fourier expansion of the Eisenstein series at this value, as seen in the previous section. It is unique to the half-integral weight case since $n+\frac{1}{2}$ does not belong to the set of special values when $k\in\mathbb{Z}$. Most likely it can still be interpolated since one can use the Mazur measure to interpolate the extra Fourier coefficients arising here, as is done in the integral-weight case of [@Panchishkin pp. 113–114], but it is not necessary in order to give the existence of the measure here.

The $p$-adic Mellin transform of a $p$-adic measure is defined by $$L_{\nu}(x) := \int_{\mathbb{Z}_p^{\times}} xd\nu$$ for any $x\in X_p$.

\[padiclfun\] Let $f\in \mathcal{S}_k(\Gamma, \psi)$ be a $p$-ordinary Hecke eigenform. The $p$-adic $L$-functions of $f$ are defined by: $$\mathcal{L}_p^{\pm}(s, f, \chi) := L_{\nu_f^{\pm}}(\chi x_p^s) = \int_{\mathbb{Z}_p^{\times}}\chi x_p^sd\nu_f^{\pm}.$$

Proof of Theorem \[main\]
-------------------------

The proof of the main theorem now follows along predictable lines: prove the existence of $p$-adic distributions interpolating $L_{\psi}(m, f, \bar{\chi})$ and show that they satisfy the Kummer congruences, thus defining $p$-adic measures. Though it is enough to define the $p$-adic distribution in terms of non-trivial primitive characters $\chi\in X_p^{{\operatorname{tors}}}$, to use the Kummer congruences we need *all* characters in $X_p^{{\operatorname{tors}}}$ and we achieve this by lifting the undesirable primitive characters into desirable imprimitive characters. The definition of the distribution on primitive characters is similar to that seen in Theorem \[main\]. For imprimitive characters, we cannot use the transformation formula of the theta series \[thetatrans\], instead we define it in terms of the Rankin-Selberg Dirichlet series $$D(s, f, g) := \sum_{\sigma\in S_+/GL_n(\mathbb{Z})}\nu_{\sigma}^{-1}c_f(\sigma, 1)c_g(\sigma, 1)|\sigma|^{-s-\frac{k-\ell}{2}},$$ where $f\in\mathcal{M}_k$, $g\in\mathcal{M}_{\ell}$, and $\nu_{\sigma} := \#\{a\in GL_n(\mathbb{Z})\mid a^T\sigma a=\sigma\}$. This Rankin-Selberg Dirichlet series has an integral expression similar to \[RS\] – in fact it is used as an intermediary step in the proof of \[RS\] – and the flexibility in choice of $g$ allows us to pre-empt the right-hand side of the transformation formula \[thetatrans\].

\[distplus\] There exists a complex distribution $\nu_s^+$ on $\mathbb{Z}_p^{\times}$ which is uniquely determined on Dirichlet characters of $p$-power conductors $p^{\ell_{\chi}}$ as follows. If $\chi$ is primitive and $1\leq \ell_{\chi}\in\mathbb{Z}$, then it is defined by $$\begin{aligned}
\begin{split}
    \nu_s^+(\chi) :&=\frac{(-1)^{n[k]}|2\tau|^{\frac{n}{2}+\mu}}{Zi^dN({\mathfrak}{tbc})^{n\mu}}\left|-\tfrac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}}{2}\hat{\tau}\right|^{-\frac{k+s-\mu-1-2n}{2}}\frac{p^{n\ell_{\chi}(n+1-k-s)}G_n(\bar{\chi})}{\Lambda_{\tau}(s)\mathcal{G}_{\tau}(s)} \\
    &\hspace{10pt}\times\left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)^{-\ell_{\chi}}L_{\psi}(s, f, \bar{\chi}),
\end{split}
    \label{plusdistprim}\end{aligned}$$ In general, for any $\ell>\ell_{\chi}\geq 0$, let $\chi_{\ell}$ denote the character modulo $p^{\ell}$ associated to $\chi$ and, for any $r>\ell$, define $$\begin{aligned}
    \begin{split}
    \nu_s^+(\chi) :&= \frac{|\tau|^{\frac{s-n-1+k+\mu}{2}}}{Zc_{f_0}(\tau, 1)}\left|-\tfrac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}}{2}\hat{\tau}\right|^{-\frac{k+s-\mu-1-2n}{2}} \Lambda_{{\mathfrak}{y}_{0}}(\tfrac{2s-n}{4})p^{rn(\frac{3n}{2}+1-k-s)}\\
    &\hspace{10pt}\times\left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)^{-r}D\left(\tfrac{2s-3n-2}{4} , f_0, \theta_{\chi_{\ell}}^{\star}|W(Y_{r})\right),
    \end{split}\label{plusdistimp}\end{aligned}$$ where $\theta_{\chi_{\ell}}^{\star}(z)$ is defined in \[thetastar\].

By the compatibility criterion, Proposition \[compat\], we just need to show that the definition of $\nu_s$ is independent of $\ell$ and $r$. When $\chi$ is primitive, this is immediate. The expression \[plusdistimp\] is evidently independent of $\ell$ since $\ell>\ell_{\chi}$. Now fix $\ell$, to show independence of $r$, let $V(M)$ be the operator associated to $\left(\begin{smallmatrix} MI_n & 0 \\ 0 & M^{-1}I_n\end{smallmatrix}\right)$ and note that $$\theta_{\chi_{\ell}}^{\star}|W(Y_r) = p^{n(\ell-r)(\frac{n}{2}+\mu)}\theta_{\chi_{\ell}}^{\star}|W(Y_{\ell})V(p^{r-\ell}).$$ So the Dirichlet series in the bottom line of the \[plusdistimp\] becomes $$p^{n(\ell-r)(\frac{n}{2}+\mu)}\sum_{\sigma\in S_+/GL_n(\mathbb{Z})}\nu_{\sigma}^{-1}c_{f_0}(p^{2r-2\ell}\sigma, 1)c\left(\sigma, 1; \theta_{\chi_{\ell}}^{\star}|W(Y_{\ell})\right)|p^{2(\ell-r)}\sigma|^{-\frac{s-n-1+k+\mu}{2}},$$ hence the powers of $p^r$ in \[plusdistimp\] cancel.

Since $U_p$ shifts coefficients, Proposition \[pstabprop\] gives $$c_{f_0}(p^{2r-2\ell}\sigma, 1) = \left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)^{r-\ell}c_{f_0}(\tau, 1),$$ which cancels the remaining dependence on $r$ of \[plusdistimp\].

Through the identities [@Shimuraint (5.9b, 8.8)] relating $D(s, f_0, g)$ to $L_{\psi}(s, f_0, \chi)$, Corollary \[pstablfun\], the transformation formula \[thetatrans\] with the fact that $G_n(\chi)^{-1} = \chi(-1)^n p^{-n^2\ell_{\chi}}G_n(\bar{\chi})$, and the manipulations on $D(s, f, g)$ found in the above proof, one can check that the two definitions, \[plusdistprim\] and \[plusdistimp\], coincide if $\chi$ is primitive.

If $k>2n$ then, for any Dirichlet character $\chi$ of $p$-power conductor and $m\in\Omega_{n, k}^+$, we have $$\frac{\nu_m^+(\chi)}{\pi^{c_m}\langle f, f\rangle}\in\overline{\mathbb{Q}}.$$

In Theorem 7.6 of [@Mercuri2] we showed the existence of a non-zero constant $\mu(\Lambda, k, \psi)$ through which the Petersson inner product, and subsequently the $L$-value, satisfied an algebraicity result. Plugging $g = f$ into that theorem gives $\langle f, f\rangle \in\mu(\Lambda, k, \psi)\overline{\mathbb{Q}}$. So whenever $\chi$ is primitive, this is immediate from the main theorem, Theorem 7.8, of that paper.

If $\chi$ is not primitive, then use the unfolded integral expression [@Shimuraint (8.5)] of $D(s, f, g)$ to obtain the expression $$\begin{aligned}
    \begin{split}
        \nu_m^+(\chi) &= \left[2Zc(\tau, 1;f_0)\Gamma_n\left(\tfrac{m-n-1+k+\mu}{2}\right)\right]^{-1}N({\mathfrak}{b})^{n(n+1)}\left|-\tfrac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}}{2}\hat{\tau}\right|^{-\frac{k+m-\mu-1-2n}{2}} \\
    &\hspace{10pt}\times |4\pi\tau|^{\frac{m-n-1+k+\mu}{2}}p^{rn(\frac{3n}{2}+1-k-m)}\left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)^{-r} \\
    &\hspace{10pt}\times\left\langle f_0, \theta_{\chi_{\ell}}^{\star}|W(Y_r)\mathcal{E}(z, s; k-\tfrac{n}{2}-\mu, \bar{\eta}, \Gamma_r)\right\rangle_{{\mathfrak}{y}_r}V_r.
    \end{split}\label{distplusintexp}\end{aligned}$$ Repeating the process of proving the algebraicity of $L$-values in [@Mercuri2] – applying Proposition 7.5 with $g = \theta_{\chi_{\ell}}^{\star}|W(Y_r)$ and Theorem 7.6 found in that paper – proves the proposition in this case too.

By the above proposition we can define a $p$-adic distribution $$\nu_m^{0+} := \iota_p\left[\frac{\nu_m^+}{\pi^{c_m}\langle f, f\rangle}\right],$$ for $m\in\Omega_{n, k}^+$. To make the following expressions more manageable, we collect superfluous terms into $C_r = C_{f, r, n}(m)$ by putting $$\begin{aligned}
    C_{r} :&= (-1)^{n\left(\left[k-\frac{n}{2}-\mu\right]\right)}\left[\pi^{c_m}2c(\tau, 1;f_0)\Gamma_n\left(\tfrac{m+k+\mu-n-1}{2}\right)\right]^{-1}N({\mathfrak}{b})^{n(n+1)} \\
    &\hspace{10pt}\times|4\pi\tau|^{\frac{m+k+\mu-n-1}{2}} p^{rn(2n+1-k-m)}V_r.\end{aligned}$$ The factor of $(-1)$ appears as a result of $\theta_{\chi_{\ell}}^{\star}|W(Y_r)^2$ in the following calculation. Combining the integral expression \[distplusintexp\] above with the case $g = \theta_{\chi_{\ell}}^{\star}|W(Y_r)$, for large enough $r$, of Proposition \[tracereduce\], we get $$\begin{aligned}
    \begin{split}
    \nu_m^{0+}(\chi) &=  \iota_p\left[Z^{-1}C_r\left|-\tfrac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}}{2}\hat{\tau}\right|^{-\frac{k+m-\mu-1-2n}{2}}\right. \\
    &\hspace{45pt}\left.\times\left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)^{-r}\frac{\langle f_0, [\theta_{\chi_{\ell}}^{\star}\mathcal{E}_{\bar{\eta}}^{\star}(\cdot, \tfrac{2m-n}{4})]|U_p^rW(Y_0)\rangle_{{\mathfrak}{y}_0}}{\langle f, f\rangle}\right].
\end{split} \label{distplusreduce}\end{aligned}$$ In light of the Fourier expansion in Proposition \[eisenexp\] we make one final, artificial, adjustment to this expression by inserting a constant $D_k$. Define $$\begin{aligned}
    D_k :&= \frac{i^{n(k-\frac{n}{2}-\mu)}2^{-n(k-\mu+\frac{3}{2})}}{N({\mathfrak}{b}^2{\mathfrak}{y}_r)^{n\left(\frac{3n-2m}{4}-k+\mu\right)}\pi^{\frac{n(m+k-n-\mu)}{2}}}\Gamma_n\left(\tfrac{m+k-n-\mu}{2}\right) , \\
    {\mathfrak}{R}_r^+(\cdot, m; \chi_{\ell}) :&= D_kZ^{-1}\left|-\tfrac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}}{2}\hat{\tau}\right|^{-\frac{k+m-\mu-1-2n}{2}}\mathbf{Pr}\left([\theta_{\chi_{\ell}}^{\star}\mathcal{E}^{\star}(\cdot, \tfrac{2m-n}{4})]|U_p^r\right),\end{aligned}$$ which latter is an element of $\mathcal{M}_k(\Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{y}_{\chi}], \psi)$ (which is defined for all $m\in\Omega_{n, k}$). For the values of $m$ in Theorem \[main\] (i), it has cyclotomic Fourier coefficients that are non-zero only when $\sigma>0$ at which point, by Proposition \[eisenexp\], they are $$\begin{aligned}
    \begin{split}
    c(\sigma, 1; {\mathfrak}{R}_r^+(\cdot, m;\chi_{\ell})) &= \sum_{(\sigma_1, \sigma_2)\in V_{p^r\sigma}}\chi_{\ell}(|\sigma_1|)|\sigma_1|^{\mu}{\mathfrak}{C}_+^{\star}(\sigma_2, m)P\left(\sigma, p^r\sigma; \tfrac{k-m-\mu}{2}\right), \\
    {\mathfrak}{C}_+^{\star}(\sigma_2, m) :&=\left|-\tfrac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}}{2}\hat{\tau}\right|^{-\frac{k+m-\mu-1-2n}{2}}|\sigma_2|^{m-n-\frac{1}{2}}\prod_{q\in\mathbf{c}}f_{\sigma, q}(\bar{\eta}(q)q^{\frac{n-\delta}{2}-m}).
    \end{split}\label{Rplus}\end{aligned}$$ Insertion of $D_k$ to the expression \[distplusreduce\] above leaves us with $$\begin{aligned}
    \nu_m^{0+}(\chi) = \iota_p\left[C_rD_k^{-1}\left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)\frac{\langle f_0, {\mathfrak}{R}_r^+(\cdot, m;\chi_{\ell})|W(Y_0)\rangle_{{\mathfrak}{y}_0}}{\langle f, f\rangle}\right]. \label{artificialplus}\end{aligned}$$

#### The distribution $\nu_s^-$.

To define $\nu_s^-$ we just replace $\left|-2^{-1}N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}\hat{\tau}\right|^{-\frac{k+s-\mu-1-2n}{2}}$ in the definitions \[plusdistprim\] and \[plusdistimp\] with $\left|-2^{-1}N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}\hat{\tau}\right|^{-\frac{k+3s-\mu-2-4n}{2}}$. As before, for $m\in\Omega_{n, k}^-$, normalise this into the following $p$-adic distribution $$\begin{aligned}
    \nu_m^{0-} := \iota_p\left[\frac{\nu_m^-}{\pi^{c_m}\langle f, f\rangle}\right]. \label{Rminus}\end{aligned}$$ Define ${\mathfrak}{R}_r^{-} :=\left|-2^{-1}N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}\hat{\tau}\right|^{-(m-n-\frac{1}{2})}{\mathfrak}{R}_r^+$, this has cyclotomic Fourier coefficients that are non-zero only when $\sigma>0$. In such a case they are given, when $m\in\Omega_{n, k}^-$, by $$\begin{aligned}
    c(\sigma, 1; {\mathfrak}{R}_r^-(\cdot, m; \chi_{\ell})) &= \sum_{(\sigma_1, \sigma_2)\in V_{p^r\sigma}}\chi_{\ell}(|\sigma_1|)|\sigma_1|^{\mu}{\mathfrak}{C}_-^{\star}(\sigma_2, m)P\left(\sigma_2, p^r\sigma; \tfrac{k+m-\mu-1-2n}{2}\right),\end{aligned}$$ where ${\mathfrak}{C}_-^{\star}(\sigma_2, m) = \left|-\tfrac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}}{2}\hat{\tau}\sigma_2\right|^{-(m-n-\frac{1}{2})}{\mathfrak}{C}_+^{\star}(\sigma_2, m)$. We obtain the expression $$\begin{aligned}
    \nu_m^{0-}(\chi) = \iota_p\left[C_rD_k^{-1}\left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)^{-r}\frac{\langle f_0, {\mathfrak}{R}_r^-(\cdot, m;\chi_{\ell})|W(Y_0)\rangle_{{\mathfrak}{y}_0}}{\langle f, f\rangle}\right]. \label{artificialminus}\end{aligned}$$

Define the linear functional $$\begin{aligned}
    \ell_f:\mathcal{M}_k(\Gamma[{\mathfrak}{b}^{-2}, {\mathfrak}{b}^2{\mathfrak}{y}_0], \psi) &\to\overline{\mathbb{Q}} \\
    g&\mapsto \frac{\langle f_0, g|W(Y_0)\rangle}{\langle f, f\rangle},\end{aligned}$$ and we have $\ell_f(g) \in\mathbb{Q}(f, g, \Lambda, \psi)$ where $\Lambda$ are the eigenvalues of $f$. Note $\ell_f = \langle f_0, f_0\rangle\langle f, f\rangle^{-1}\mathcal{L}_f$, where $\mathcal{L}_f$ is defined in (3.51) of [@Panchishkin p. 109], so that by (3.52) of [@Panchishkin p. 109] there exist positive definite $\sigma_1, \dots, \sigma_t\in{\mathfrak}{b}^2S_+^{\triangledown}$ and $\beta_1, \dots, \beta_t\in \mathbb{Q}(f, \Lambda, \psi)$ satisfying $$\begin{aligned}
    \ell_f(g) = \sum_{i=1}^t \beta_ic_g(\sigma_i, 1). \label{functionalsum}\end{aligned}$$

For any subset $\mathcal{X}\subseteq X_p^{{\operatorname{tors}}}$ take the integers $r$ and $\ell$ large enough so that all $\chi\in\mathcal{X}$ are defined modulo $p^{\ell}$ and the expressions \[artificialplus\] and \[artificialminus\] hold for all $\chi\in\mathcal{X}$; this is why the Kummer congruences involved will use imprimitive characters. We have $$\nu_m^{0\pm}(\chi) = \iota_p\left[C_rD_k^{-1}\left(p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}\right)^{-r}\ell_f({\mathfrak}{R}_r^{\pm}(\cdot, m; \chi_{\ell}))\right].$$ By assumption $|p^{\frac{n(n+1)}{2}}\lambda_{p, 1}\cdots \lambda_{p, n}|_p = 1$ and, by definition, $C_r$ and $D_k$ are independent of $\chi$ and have bounded $p$-adic valuation. So whether $\nu_m^{0\pm}(\chi)$ defines a measure or not is directly dependent on the $p$-adic boundedness of $\ell_f({\mathfrak}{R}_r^{\pm})$. By the expressions \[Rplus\] and \[Rminus\], the $\mathbb{Q}_{ab}$-coefficients of ${\mathfrak}{R}_r^{\pm}$ do not depend on the modulus $p^{\ell}$ of $\chi$, and therefore $c_{\sigma, r}^{\pm}(\chi x_p^m) := \iota_p[c(\sigma, 1; {\mathfrak}{R}_r^{\pm}(\cdot, m; \chi_{\ell})]$ for $m$ as in Theorem \[main\] defines a $p$-adic distribution. By \[functionalsum\], we see that whether $\ell_f({\mathfrak}{R}_r^{\pm})$ satisfies the Kummer congruences of Proposition \[kummer\] is directly dependent on whether $c_{\sigma, r}^{\pm}$ satisfy them (and therefore define $p$-adic measure themselves).

Fix $\sigma$ and for any $(\sigma_1, \sigma_2)\in V_{p^r\sigma}$ fix $\sigma_1$; by definition of $c_{\sigma, r}^{\pm}$ we may assume $p\nmid|\sigma_1|$. By Proposition \[eisenexp\] we have, for any $\ell\in\mathbb{Z}$, the congruences $$\begin{aligned}
    |\sigma_2|&\equiv \left|-\tfrac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}}{2}\hat{\tau}\right||\sigma_1|^2\pmod{p^r}, \label{congruence1}\\
    P(\sigma_2, p^r\sigma; \ell) &\equiv |\sigma_2|^{\ell}\equiv \left[\left|-\tfrac{N(\sqrt{{\mathfrak}{t}}{\mathfrak}{bc})^{2}}{2}\hat{\tau}\right||\sigma_1|^2\right]^{\ell}\pmod{p^r}. \label{congruence2}\end{aligned}$$ We have, by \[Rplus\], \[Rminus\], the analogous expressions for $m\in\Omega_{n, k}^-$, and Corollary \[Sigma\], that $$\begin{aligned}
    c_{\sigma, r}^{\pm}(\chi x_p^m) \equiv\sum_{(\sigma_1, \sigma_2)\in V_{p^r\sigma}}\chi_{\ell}(|\sigma_1|)|\sigma_1|^{k+m-1-2n}\left[\Sigma_{\sigma_2}\otimes\omega_{\tau}\right](\chi_{\ell} x_p^m) \pmod{p^r}, \label{congruence3}\end{aligned}$$ where $\omega_{\tau}$ is the primitive character associated to $\bar{\psi}^*\rho_{\tau}$. The conductor of $\omega_{\tau}$ is $t{\mathfrak}{c}$, where $t$ is the conductor of $\rho_{\tau}$, and this is coprime to $p$ by the following argument. Under the assumption $p\nmid{\mathfrak}{b}^{-1}\cap{\mathfrak}{c}$ we just show $(t, p) =1$ and note that $\rho_{\tau}(p) \neq 0$ if and only if $\left(\frac{|2N({\mathfrak}{b})^{-1}\tau|}{p}\right)\neq 0$. We can assume $|\sigma_2|\not\equiv 0\pmod{p^r\mathcal{O}_p}$, since otherwise $c_{\sigma, r}^{\pm}(\chi x_p^m)\equiv 0\pmod{p^r}$ by congruence \[congruence2\]. By definition, followed by the congruence \[congruence1\] above, we have $$\left(\frac{|2N({\mathfrak}{b})^{-1}\tau|}{p}\right)^r = \left(\frac{N({\mathfrak}{b}^{-1}{\mathfrak}{t})}{p}\right)^{rn} \left(\frac{|\hat{\tau}|}{p}\right)^r = \left(\frac{-N({\mathfrak}{bc}^2)/2}{p}\right)^{rn}\left(\frac{|\sigma_2|}{p^r}\right),$$ which is non-zero by assumption and so we have $(p, t) = 1$. Therefore $\Sigma_{\sigma_2}\otimes\omega_{\tau}$ defines a $p$-adic measure.

The Kummer congruences now complete the proof that $c_{\sigma, r}^{\pm}$ defines a measure. Assume $N\geq r$ and that $$\sum_{\chi\in\mathcal{X}}b_{\chi}\chi x_p^m\subseteq p^N\mathcal{O}_p,$$ for some $b_{\chi}\in\mathcal{O}_p$. Then the congruence \[congruence3\] above gives $$\begin{aligned}
\sum_{\chi\in\mathcal{X}}b_{\chi}c_{\sigma, r}^{\pm}(\chi x_p^m)\in\sum_{(\sigma_1, \sigma_2)\in V_{p^r\sigma}}|\sigma_1|^{k-1-2n}\sum_{\chi\in\mathcal{X}}b_{\chi}\int_{\mathbb{Z}_p^{\times}}(\chi_{\ell}x_p^m)(y|\sigma_1|)[d\Sigma_{\sigma_2}\otimes\omega_{\tau}](y) + p^N\mathcal{O}_p\end{aligned}$$ which is clearly in $p^N\mathcal{O}_p$ since we know $\Sigma_{\sigma_2}\otimes \omega_{\tau}$ is a measure satisfying the Kummer congruences.

We have shown that $\nu_m^{0\pm}$ defines a $p$-adic measure. Take $m$ as in the statement of Theorem \[main\]; to the finish the proof of that theorem, we put $\nu_f^{\pm}(\chi x_p^m) = \nu_m^{0\pm}(\chi)$ for primitive characters $\chi$ and use the primitive character definitions, for example \[plusdistprim\], of the original distributions $\nu_s^{\pm}$.

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<span style="font-variant:small-caps;">Department of Mathematical Sciences, Durham University, Stockton Road, Durham DH1 3LE, U.K.</span>

*E-mail*: `salvatore.mercuri@dur.ac.uk`
