3.1 Symplectic groups

We set up some further notation, which will be used this way throughout Section 3, unless it is stated otherwise. Let $(L,(\cdot ,\cdot ))$ be an even lattice, that is a free $\mathbb{Z}$ -module of finite rank $m$ , equipped with a symmetric bilinear form $(\cdot ,\cdot )$ such that $(x,x)\in 2\mathbb{Z}$ for all $x\in L$ . We assume that $m$ is even, $L$ is non-degenerate and denote its signature by $(b^{+},b^{-})$ . We write $V=L\otimes \mathbb{Q}$ and define by

$$\begin{eqnarray}L^{\prime }=\{x\in V\mid (x,y)\in \mathbb{Z}~\text{for all}~y\in L\}\end{eqnarray}$$

the dual lattice of $L$ . Associated to $(\cdot ,\cdot )$ there is a quadratic form

$$\begin{eqnarray}q(x)={\textstyle \frac{1}{2}}(x,x).\end{eqnarray}$$

Since $L$ is even, we have $L\subset L^{\prime }$ . It can be proved that $L^{\prime }/L$ is a finite abelian group. The modulo 1 reduction of $(\cdot ,\cdot )$ yields a $\mathbb{Q}/\mathbb{Z}$ -valued bilinear form $(\cdot ,\cdot )_{L}$ on $L^{\prime }/L$ . The associated quadratic form $q_{L}$ is the modulo 1 reduction of $q$ with values in $\mathbb{Q}/\mathbb{Z}$ . $(L^{\prime }/L,(\cdot ,\cdot )_{L})$ is called a finite quadratic module or a discriminant form.

To the vector space $(V,(\cdot ,\cdot ))$ we associate a quadratic character

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D712}_{V}:\mathbb{A}^{\times }/\mathbb{Q}^{\times }\longrightarrow \mathbb{C},\qquad x\mapsto \unicode[STIX]{x1D712}_{V}(x)=(x,(-1)^{m/2}\det (V))_{\mathbb{ A}},\end{eqnarray}$$

where $(\cdot ,\cdot )_{\mathbb{A}}$ is the global Hilbert symbol and $\det (V)$ is the Gram determinant of  $V$ . Since $V=L\otimes \mathbb{Q}$ , we have $|\det (V)|=|L^{\prime }/L|$ . We denote by $H=\text{O}(V)$ the orthogonal group over $\mathbb{Q}$ attached to $(V,(\cdot ,\cdot ))$ . Moreover, let $W$ be the standard vector space over $\mathbb{Q}$ of dimension $2n$ equipped with the non-degenerate skew-symmetric bilinear form ${<}\cdot ,\cdot >$ defined by

$$\begin{eqnarray}w=\left(\begin{array}{@{}cc@{}}0 & 1_{n}\\ -1_{n} & 0\end{array}\right).\end{eqnarray}$$

Let $G=\operatorname{Sp}(W)\cong \operatorname{Sp}(n,\mathbb{Q})$ be the symplectic group over $\mathbb{Q}$ attached to $(W,<\cdot ,\cdot >)$ .

More generally, we consider the symplectic group $G(\mathbb{A})=\operatorname{Sp}(n,\mathbb{A})$ . Within $G(\mathbb{A})$ , we have the following subgroups:

(3.2) $$\begin{eqnarray}\displaystyle & \displaystyle M(\mathbb{A})=\left\{m(a)=\left.\left(\begin{array}{@{}cc@{}}a & 0\\ 0 & (a^{-1})^{t}\end{array}\right)\right|a\in \operatorname{GL}_{n}(\mathbb{A})\right\}, & \displaystyle\end{eqnarray}$$

(3.3) $$\begin{eqnarray}\displaystyle & \displaystyle N(\mathbb{A})=\left\{n(b)=\left.\left(\begin{array}{@{}cc@{}}1_{n} & b\\ 0 & 1_{n}\end{array}\right)\right|b\in \operatorname{Sym}_{n}(\mathbb{A})\right\}. & \displaystyle\end{eqnarray}$$

These define the Siegel parabolic subgroup $P(\mathbb{A})=N(\mathbb{A})M(\mathbb{A})$ of $G(\mathbb{A})$ , which is part of the Iwasawa decomposition

(3.4) $$\begin{eqnarray}G(\mathbb{A})=N(\mathbb{A})M(\mathbb{A})K\end{eqnarray}$$

of $G(\mathbb{A})$ , where $K=\prod _{v}K_{v}$ is the maximal compact subgroup of $G(\mathbb{A})$ . Here for a non-archimedean place $v=p$ , the group $K_{p}$ is given by $\operatorname{Sp}(n,\mathbb{Z}_{p})$ . For the archimedean place $v=\infty$ , we have

(3.5) $$\begin{eqnarray}K_{\infty }=\left\{k=\left.\left(\begin{array}{@{}cc@{}}a & b\\ -b & a\end{array}\right)\in \operatorname{Sp}(n,\mathbb{R})\right|\mathbf{k}=a+ib\in U(n)\right\},\end{eqnarray}$$

where $U(n)$ means the unitary group.

The following lemma describes the structure of $K_{p}$ and might be known. For $n=1$ , this result can be found in [Reference Bruinier and YangBY, p. 644]. However, I could not find any reference for a general $n\in \mathbb{N}$ . Therefore, we give a proof here.

Lemma 3.1. For each prime number $p$ , the group $\operatorname{Sp}(n,\mathbb{Z}_{p})$ is generated by the set $G_{n}$ of matrices

(3.6) $$\begin{eqnarray}\begin{array}{@{}l@{}}w,\\ m(a),\quad a\in \operatorname{GL}(n,\mathbb{Z}_{p}),\quad \text{and}\\ n(b),\quad b\in \operatorname{Sym}_{n}(\mathbb{Z}_{p}).\end{array}\end{eqnarray}$$

Proof. The proof is an adaptation of the one which proves [Reference ScharlauSc, Theorem 7.5, Chapter 7]. This theorem states the same result for the group $\operatorname{Sp}(n,F)$ , where $F$ is a field. The proof proceeds by induction on $n$ and starts with a matrix $M_{n}=\left(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\right)\in \operatorname{Sp}(n,\mathbb{Z}_{p})$ , which is then multiplied with suitable matrices of set (3.6). It can be assumed that $A\not =0$ since otherwise, we can replace $M_{n}$ with $M_{n}w$ . By [Reference KedlayaKe, Theorem 4.3.4], there exist matrices $U,V\in \operatorname{GL}(n,\mathbb{Z}_{p})$ such that $UAV=D$ , where $D$ is of the form $\left(\!\begin{smallmatrix}S & 0\\ 0 & 0\end{smallmatrix}\!\right)$ and $S\in M_{k,k}(\mathbb{Z}_{p})$ is diagonal with nonzero entries in $\mathbb{Z}_{p}$ . Multiplying $M$ with $\left(\!\begin{smallmatrix}U & 0\\ 0 & U^{-1}\end{smallmatrix}\!\right)$ from the left and $\left(\!\begin{smallmatrix}V & 0\\ 0 & V^{-1}\end{smallmatrix}\!\right)$ from the right yields

(3.7)

where $S,B_{1},C_{1},D_{1}$ are $k\times k$ -matrices as in [Reference ScharlauSc]. Since matrix (3.7) is an element of $\operatorname{Sp}(n,\mathbb{Z}_{p})$ , it follows that $(A^{1})^{t}C^{1}\in \operatorname{Sym}_{n}(\mathbb{Z}_{p})$ , which in turn implies $C_{2}=0$ and $C_{1}\in \operatorname{Sym}_{k}(\mathbb{Z}_{p})$ . One easily checks that multiplication of (3.7) with $\left(\!\begin{smallmatrix}1 & 0\\ T & 1\end{smallmatrix}\!\right)$ on the left produces

(3.8)

Here $T$ is the matrix $\left(\!\begin{smallmatrix}-C_{1} & -C_{3}^{t}\\ C_{3} & 0\end{smallmatrix}\!\right)\in \operatorname{Sym}_{n}(\mathbb{Z}_{p})$ . Again, (3.8) is in $\operatorname{Sp}(n,\mathbb{Z}_{p})$ and the relation $(A^{2})^{t}D^{2}-(C^{2})^{t}B^{2}=1_{n}$ implies $SD_{1}=1_{k}$ , $D_{2}=0,B_{3}=0$ , and $-C_{4}^{t}B_{4}=1_{n-k},$ as is deduced in [Reference ScharlauSc]. Exactly the same conclusions and transformations (out of (3.6)) as in [Reference ScharlauSc] remain valid for the case of $p$ -adic integers. As a consequence, we obtain the matrix

(3.9)

By multiplying (3.9) from the right with

we finally get

(3.10)

which is the same matrix as in [Reference ScharlauSc]. Since (3.10) is in $\operatorname{Sp}(n,\mathbb{Z}_{p})$ , it can be checked that $M_{n-k}=\left(\!\begin{smallmatrix}0 & B_{4}\\ C_{4} & D_{4}\end{smallmatrix}\!\right)$ is in $\operatorname{Sp}(n-k,\mathbb{Z}_{p})$ . By induction, multiplication of $M_{n-k}$ from the left and right with (finitely many) matrices from $G_{n-k}$ gives the identity matrix. Each used matrix corresponds to the multiplication of (3.10) with the same type of matrix in $G_{n}$ and the result for the blocks $0,B_{4},C_{4},D_{4}$ .◻

3.2 The Weil representation

The following subsection gathers some well-known facts on the Weil representation of $G(\mathbb{A})\times H(\mathbb{A})$ . In particular, the Schrödinger model of the Weil representation with explicit formulas is given. We follow [Reference KudlaKu1, Chapter I.1], [Reference GelbartGe, Chapter 2.3], and [Reference AsmuthAs].

Let

(3.11) $$\begin{eqnarray}\unicode[STIX]{x1D713}=\mathop{\prod }_{p\leqslant \infty }\unicode[STIX]{x1D713}_{p}:\mathbb{A}/\mathbb{Q}\rightarrow \mathbb{C}^{\times },\qquad x=(x_{p})\mapsto \unicode[STIX]{x1D713}(x)=e^{2\unicode[STIX]{x1D70B}i(x_{\infty }-\mathop{\sum }_{p<\infty }x_{p}^{\prime })}\end{eqnarray}$$

be the standard non-trivial additive character, where $x_{p}^{\prime }$ is the usual partial fraction of  $x_{p}$ . Attached to this character and the dual pair $(G,H)$ there is a Weil representation  $\unicode[STIX]{x1D714}$ . The Schrödinger model of $\unicode[STIX]{x1D714}$ defines an action of $G(\mathbb{A})\times H(\mathbb{A})$ on the space $S(V(\mathbb{A})^{n})$ of Schwartz functions on

$$\begin{eqnarray}V(\mathbb{A})^{n}=\{v=(v_{1},\ldots ,v_{n})\mid v_{1},\ldots ,v_{n}\in V\otimes \mathbb{A}\}.\end{eqnarray}$$

Note that since $\dim (V)=m$ is assumed to be even, we do not have to work with the metaplectic cover of $G(\mathbb{A})$ . The action of $\unicode[STIX]{x1D714}$ on the generators of $G(\mathbb{A})$ can be described explicitly for $\unicode[STIX]{x1D711}\in S(V(\mathbb{A})^{n})$ by the following formulas

(3.12) $$\begin{eqnarray}\begin{array}{@{}l@{}}\unicode[STIX]{x1D714}(h)\unicode[STIX]{x1D711}(v)=\unicode[STIX]{x1D711}(h^{-1}v),\quad h\in H(\mathbb{A}),\\ \unicode[STIX]{x1D714}(m(a))\unicode[STIX]{x1D711}(v)=\unicode[STIX]{x1D712}_{V}(\det a)|\text{det}\,a|_{\mathbb{A}}^{m/2}\unicode[STIX]{x1D711}(va),\quad m(a)\in M(\mathbb{A}),\\ \unicode[STIX]{x1D714}(n(b))\unicode[STIX]{x1D711}(v)=\unicode[STIX]{x1D713}(\operatorname{tr}bQ[v])\unicode[STIX]{x1D711}(v),\quad n(b)\in N(\mathbb{A}),\\ \displaystyle \unicode[STIX]{x1D714}(w)\unicode[STIX]{x1D711}(v)=\unicode[STIX]{x1D6FE}\int _{V(\mathbb{A})^{n}}\unicode[STIX]{x1D713}(tr(v,w))\unicode[STIX]{x1D711}(w)\,dw,\end{array}\end{eqnarray}$$

where

(3.13) $$\begin{eqnarray}(v,w)=((v_{i},w_{j}))_{i,j}\in \operatorname{Sym}_{n}(\mathbb{A}),\qquad Q[v]={\textstyle \frac{1}{2}}(v,v)\in \operatorname{Sym}_{n}(\mathbb{A})\end{eqnarray}$$

is the moment matrix. Note that if the bilinear form on $L$ is given by a symmetric matrix $S$ , it is easily seen that

$$\begin{eqnarray}(v,w)=v^{t}Sw\qquad \text{and}\qquad Q[v]={\textstyle \frac{1}{2}}v^{t}Sv\end{eqnarray}$$

for $v,w\in V(\mathbb{A})^{n}$ . Moreover, $dw=\prod _{p}\,dw_{p}$ , where $dw_{p}$ is the Haar measure on $V(\mathbb{Q}_{p})^{n}$ for $p<\infty$ , which is self-dual with respect to  $\unicode[STIX]{x1D713}_{p}$ . By $\unicode[STIX]{x1D6FE}$ we mean

(3.14) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}=\mathop{\prod }_{p\leqslant \infty }\unicode[STIX]{x1D6FE}_{p}.\end{eqnarray}$$

Here

(3.15) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6FE}_{p}=\unicode[STIX]{x1D6FE}_{p}(V)^{n},\quad p<\infty ,\end{eqnarray}$$

(3.16) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6FE}_{\infty }=\unicode[STIX]{x1D6FE}_{\infty }(V)^{n},\quad p=\infty ,\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}_{p}(V)$ is the local Weil index and $\unicode[STIX]{x1D6FE}_{\infty }(V)=e((b^{+}-b^{-})/8)$ . Note that by the product formula we have

(3.17) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}_{\infty }(V)\mathop{\prod }_{p<\infty }\unicode[STIX]{x1D6FE}_{p}(V)=1.\end{eqnarray}$$

3.3 The degenerate principal series

Using the Iwasawa decomposition (3.4), each element $g\in G(\mathbb{A})$ can be written as a product $g=n(b)m(a)k$ . For any $g=n(b)m(a)k\in G(\mathbb{A})$ , we set

$$\begin{eqnarray}|a(g)|=|\det (a)|_{\mathbb{A}}.\end{eqnarray}$$

One can prove that this is a well-defined function, which is left- $N(\mathbb{A})M(\mathbb{Q})$ invariant and right- $K$ invariant.

The character $\unicode[STIX]{x1D712}_{V}$ (3.1) is extendable to the group $P(\mathbb{A})$ via $\unicode[STIX]{x1D712}_{V}\left(\!\begin{smallmatrix}a & b\\ 0 & (a^{-1})^{t}\end{smallmatrix}\!\right)=\unicode[STIX]{x1D712}_{V}(\det (a))$ . For $s\in \mathbb{C}$ , we denote by $I_{n}(s,\unicode[STIX]{x1D712}_{V})$ the normalized induced representation from $P(\mathbb{A})$ to $G(\mathbb{A})$ . It is called the degenerate principal series representation. It is realized on the space of all smooth functions $\unicode[STIX]{x1D6F7}(g,s)$ on $G(\mathbb{A})$ satisfying

(3.18) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(n(b)m(a)g,s)=\unicode[STIX]{x1D712}_{V}(\det a)|\det (a)|^{s+\unicode[STIX]{x1D70C}_{n}}\unicode[STIX]{x1D6F7}(g,s)\end{eqnarray}$$

for all $a\in \operatorname{GL}_{n}(\mathbb{A})$ and $b\in \operatorname{Sym}_{n}(\mathbb{A})$ , where

(3.19) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{n}=\frac{n+1}{2}.\end{eqnarray}$$

The action of $G(\mathbb{A})$ is given by right translations. Let $I_{n,p}(s,\unicode[STIX]{x1D712}_{V})$ denote the corresponding local induced representations. According to [Reference KudlaKu1], $I_{n}(s,\unicode[STIX]{x1D712}_{V})$ can be written as a restricted tensor product $I_{n}(s,\unicode[STIX]{x1D712}_{V})=\otimes _{p\leqslant \infty }I_{n,p}(s,\unicode[STIX]{x1D712}_{V})$ . Also, each $\unicode[STIX]{x1D711}\in I_{n}(s,\unicode[STIX]{x1D712}_{V})$ is determined by its restriction to  $K$ .

An element $\unicode[STIX]{x1D6F7}\in I_{n}(s,\unicode[STIX]{x1D712}_{V})$ is called a standard section if its restriction to $K$ is independent of  $s$ . It is called factorizable if it has the form $\unicode[STIX]{x1D6F7}(g,s)=\otimes _{p\leqslant \infty }\unicode[STIX]{x1D6F7}_{p}(g,s)$ with $\unicode[STIX]{x1D6F7}_{p}(g,s)\in I_{n,p}(s,\unicode[STIX]{x1D712}_{V})$ .

For this paper, the normalized standard section plays an important role. The non-archimedian components are constructed by means of the intertwining map

(3.20) $$\begin{eqnarray}\unicode[STIX]{x1D706}:S(V(\mathbb{A})^{n})\longrightarrow I_{n}(s_{0},\unicode[STIX]{x1D712}_{V}),\qquad \unicode[STIX]{x1D711}\mapsto \unicode[STIX]{x1D6F7}(g,s_{0})=(\unicode[STIX]{x1D714}(g)\unicode[STIX]{x1D711})(0),\end{eqnarray}$$

where

Using the Iwasawa decomposition, it can be proved that $\unicode[STIX]{x1D706}(\unicode[STIX]{x1D711})\in I_{n}(s_{0},\unicode[STIX]{x1D712}_{V})$ has a unique extension to a standard section by

(3.21) $$\begin{eqnarray}\unicode[STIX]{x1D706}(\unicode[STIX]{x1D711})(g,s)=|a(g)|^{s-s_{0}}(\unicode[STIX]{x1D714}(g)\unicode[STIX]{x1D711})(0)\in I_{n}(s,\unicode[STIX]{x1D712}_{V}).\end{eqnarray}$$

We now want to define a section $\unicode[STIX]{x1D6F7}^{l}$ , which depends on a lattice  $L$ . To this end, we briefly recall some necessary facts on lattices and discriminant forms. For more details, see for example [Reference Miranda and MorrisonMM].

Remark 3.2. Let $L\subset V$ be a lattice as introduced in Section 3.1. It is well known that we can attach to each prime $p$ a $p$ -adic lattice $L_{p}=L\otimes \mathbb{Z}_{p}$ . The bilinear form $(\cdot ,\cdot )$ and the corresponding quadratic form $q$ can be extended to $L_{p}$ in the usual way. It is then possible to define the dual lattice $L_{p}^{\prime }$ and the discriminant group $L_{p}^{\prime }/L_{p}$ , which can be endowed with a finite quadratic form $q_{L_{p}}$ coming from the quadratic form on $L_{p}$ . The groups $L^{\prime }/L$ and $L_{p}^{\prime }/L_{p}$ and the quadratic forms on them are closely related:

$$\begin{eqnarray}L_{p}^{\prime }/L_{p}\cong (L^{\prime }/L)_{p},\end{eqnarray}$$

where $(L^{\prime }/L)_{p}$ is the $p$ -component of $L^{\prime }/L$ . The quadratic form $q_{L_{p}}$ is just the restriction of $q_{L}$ to the $p$ -component $(L^{\prime }/L)_{p}$ . In particular, for an element $\unicode[STIX]{x1D707}\in L^{\prime }/L$ , the $p$ -part of $\unicode[STIX]{x1D707}$ can be interpreted as an element of $L_{p}^{\prime }/L_{p}$ .

Definition 3.3. For $\unicode[STIX]{x1D707}\in (L^{\prime }/L)^{n}$ , we define $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}\in S(V(\mathbb{A}_{f})^{n})$ by

(3.22) $$\begin{eqnarray}\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D707}+\hat{L}^{n}}=\mathop{\prod }_{p<\infty }\unicode[STIX]{x1D711}_{p}^{(\unicode[STIX]{x1D707})}=\mathop{\prod }_{p<\infty }\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D707}+L_{p}^{n}}.\end{eqnarray}$$

Here $L_{p}=L\otimes \mathbb{Z}_{p}$ , which is the $p$ -part of $\hat{L}=L\otimes \hat{\mathbb{Z}}$ with $\hat{\mathbb{Z}}=\prod _{p<\infty }\mathbb{Z}_{p}$ , and $\unicode[STIX]{x1D712}_{\unicode[STIX]{x1D707}+L_{p}^{n}}$ is the characteristic function of $\unicode[STIX]{x1D707}+L_{p}^{n}$ .

We associate to $(\unicode[STIX]{x1D711}_{p}^{(\unicode[STIX]{x1D707})})_{p<\infty }$ the standard section $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D707}}^{l}(s)=\unicode[STIX]{x1D6F7}_{\infty }^{l}(s)\prod _{p<\infty }\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(s)\in I_{n}(s,\unicode[STIX]{x1D712}_{V})$ with $l\in \mathbb{Z}$ and

(3.23) $$\begin{eqnarray}\begin{array}{@{}l@{}}\unicode[STIX]{x1D6F7}_{\infty }^{l}(k,s)=\det (\mathbf{k})^{l}\quad \text{for all}~k\in K_{\infty },\\ \unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(g,s)=\unicode[STIX]{x1D706}_{p}(\unicode[STIX]{x1D711}_{p}^{(\unicode[STIX]{x1D707})})(g,s),\end{array}\end{eqnarray}$$

where $\mathbf{k}\in U(n)$ corresponds to $k\in K_{\infty }$ .

The lemma below is well known for $n=1$ ; see for example [Reference HörmannHo]. Although it might be known for a general $n$ , I was not able to find it in the literature.

Lemma 3.4. Let $\unicode[STIX]{x1D707}=(\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{n})\in (L^{\prime }/L)^{n}$ and $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D707}}^{l}(s)\in I_{n}(s,\unicode[STIX]{x1D712}_{V})$ be as in Definition 3.3. Then

(3.24) $$\begin{eqnarray}\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(k,s)=1\quad \text{for all}~k\in \operatorname{Sp}(n,\mathbb{Z}_{p})\end{eqnarray}$$

for all primes $p$ coprime to $|L^{\prime }/L|$ . In particular, $\unicode[STIX]{x1D712}_{V,p}=(\cdot ,(-1)^{m/2}\det (V))_{p}$ is unramified for all those primes.

Proof. The latter assumption follows directly from the explicit formulas of the local Hilbert symbol $(\cdot ,(-1)^{m/2}\det (V))_{p}$ . Note that $(-1)^{m/2}\det (V)\equiv 0,1\hspace{0.2em}{\rm mod}\hspace{0.2em}4$ .

By definition, $\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(k)=(\unicode[STIX]{x1D714}(k)\unicode[STIX]{x1D711}_{p}^{(\unicode[STIX]{x1D707})})(0)$ . It suffices to prove (3.24) on the generators of $\operatorname{Sp}(n,\mathbb{Z}_{p})$ . According to Lemma 3.1,  $\operatorname{Sp}(n,\mathbb{Z}_{p})$ is generated by the elements  $w$ , $n(b)$ , and $m(a)$ . Using the formulas (3.12), we obtain the following formulas for the Weil representation acting on $\unicode[STIX]{x1D711}_{p}^{(\unicode[STIX]{x1D707})}$ , which are a straightforward generalization of [Reference Bruinier and YangBY, (2.33)] (see also [Reference NguyenN, Section 3.5]).

(3.25) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(n(b)) & = & \displaystyle (\unicode[STIX]{x1D714}(n(b))\unicode[STIX]{x1D711}_{p}^{(\unicode[STIX]{x1D707})})(0)\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D713}_{p}(\operatorname{tr}(bQ[\unicode[STIX]{x1D707}]))\unicode[STIX]{x1D711}_{p}^{(\unicode[STIX]{x1D707})}(0)\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D713}_{p}(\operatorname{tr}(bQ[\unicode[STIX]{x1D707}]))\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(1)\quad \text{for}~b\in \operatorname{Sym}_{n}(\mathbb{Z}_{p}),\end{eqnarray}$$

(3.26) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(w) & = & \displaystyle (\unicode[STIX]{x1D714}(w)\unicode[STIX]{x1D711}_{p}^{(\unicode[STIX]{x1D707})})(0)\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D6FE}_{p}(V)^{n}|L_{p}^{\prime }/L_{p}|^{-n/2}\mathop{\sum }_{\unicode[STIX]{x1D708}\in (L_{p}^{\prime }/L_{p})^{n}}\unicode[STIX]{x1D713}_{p}(\operatorname{tr}(\unicode[STIX]{x1D708},\unicode[STIX]{x1D707}))\unicode[STIX]{x1D711}_{p}^{(\unicode[STIX]{x1D708})}(0)\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D6FE}_{p}(V)^{n}|L_{p}^{\prime }/L_{p}|^{-n/2}\mathop{\sum }_{\unicode[STIX]{x1D708}\in (L_{p}^{\prime }/L_{p})^{n}}\unicode[STIX]{x1D713}_{p}(\operatorname{tr}(\unicode[STIX]{x1D708},\unicode[STIX]{x1D707}))\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D708})}(1),\end{eqnarray}$$

and

(3.27) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(m(a)) & = & \displaystyle (\unicode[STIX]{x1D714}(m(a))\unicode[STIX]{x1D711}_{p}^{(\unicode[STIX]{x1D707})})(0)\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D712}_{V,p}(\det a)|\text{det}\,a|_{p}^{m/2}\unicode[STIX]{x1D711}_{p}^{(\unicode[STIX]{x1D707}a^{-1})}(0)\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D712}_{V,p}(\det a)|\text{det}\,a|_{p}^{m/2}\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707}a^{-1})}(1)\quad \text{for}~a\in \operatorname{GL}_{n}(\mathbb{Z}_{p}).\end{eqnarray}$$

If $p$ is coprime to $|L^{\prime }/L|$ , we can conclude the following:

1. (1) $L_{p}$ is unimodular since $|L_{p}^{\prime }/L_{p}|=p^{\text{ord}_{p}(|L^{\prime }/L|)}$ ; see for example [Reference KitaokaKi, Section 5.1].

2. (2) $\unicode[STIX]{x1D713}_{p}(\operatorname{tr}bQ[\unicode[STIX]{x1D707}])=\unicode[STIX]{x1D713}_{p}(\operatorname{tr}(\unicode[STIX]{x1D707},\unicode[STIX]{x1D708}))=1$ .

3. (3) $\unicode[STIX]{x1D6FE}(V)=1$ by [Reference WeilWe, Theorem 5].

4. (4) $\unicode[STIX]{x1D712}_{V,p}(\det a)=1$ for all $a\in \operatorname{GL}_{n}(\mathbb{Z}_{p})$ using the definition of the Hilbert symbol.

This finally proves the claim. ◻

As in [Reference Bruinier and YangBY], we consider the $|L^{\prime }/L|^{n}$ -dimensional subspace

(3.28) $$\begin{eqnarray}S_{L}=\bigoplus _{\unicode[STIX]{x1D707}\in (L^{\prime }/L)^{n}}\mathbb{C}\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}\subset S(V(\mathbb{A}_{f})^{n}).\end{eqnarray}$$

From the proof of Lemma 3.4, it is obvious that space (3.28) is stable under the action of the group $\operatorname{Sp}(n,\hat{\mathbb{Z}})$ via the Weil representation. Thereby, a finite dimensional representation $\unicode[STIX]{x1D70C}_{L,n}$ of $\operatorname{Sp}(n,\mathbb{Z})$ on $S_{L}$ is defined by

(3.29) $$\begin{eqnarray}\unicode[STIX]{x1D70C}_{L,n}(g)\unicode[STIX]{x1D711}=\overline{\unicode[STIX]{x1D714}}(g_{f})\unicode[STIX]{x1D711},\end{eqnarray}$$

where $g_{f}\in \operatorname{Sp}(n,\hat{\mathbb{Z}})$ is the projection of $g\in \operatorname{Sp}(n,\mathbb{Z})$ into $\operatorname{Sp}(n,\hat{\mathbb{Z}})$ . Equations (3.25), (3.26), and (3.27) yield explicit formulas for $\unicode[STIX]{x1D70C}_{L,n}$ on the generators of $\operatorname{Sp}(n,\mathbb{Z})$ :

(3.30) $$\begin{eqnarray}\begin{array}{@{}l@{}}\unicode[STIX]{x1D70C}_{L,n}(n(b))\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}=e(\operatorname{tr}(bQ[\unicode[STIX]{x1D707}]))\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}\\ \displaystyle \unicode[STIX]{x1D70C}_{L,n}(w)\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}=\frac{e\left(\frac{n(b^{+}-b^{-})}{8}\right)}{|L^{\prime }/L|^{n/2}}\mathop{\sum }_{\unicode[STIX]{x1D708}\in (L^{\prime }/L)^{n}}e(\operatorname{tr}(\unicode[STIX]{x1D708},\unicode[STIX]{x1D707}))\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D708}}\\ \unicode[STIX]{x1D70C}_{L,n}(m(a))\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}=\unicode[STIX]{x1D712}_{V}(\det a)|\text{det}\,a|^{m/2}\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}a^{-1}}.\end{array}\end{eqnarray}$$

Let $\{\mathfrak{e}_{\unicode[STIX]{x1D707}}\mid \unicode[STIX]{x1D707}\in (L^{\prime }/L)^{n}\}$ be the standard basis of the group ring $\mathbb{C}[(L^{\prime }/L)^{n}]$ . Then the map $\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}\mapsto \mathfrak{e}_{\unicode[STIX]{x1D707}}$ defines an isomorphism between $S_{L}$ and $\mathbb{C}[(L^{\prime }/L)^{n}]$ , and $\unicode[STIX]{x1D70C}_{L,n}$ can be interpreted as a representation acting on the group ring $\mathbb{C}[(L^{\prime }/L)^{n}]$ . The formulas in (3.30) can also be found in [Reference ZhangZh, Section 2.1.3].

For later purposes, we introduce the standard inner product on $\mathbb{C}[(L^{\prime }/L)^{n}]$ by

(3.31) $$\begin{eqnarray}\left\langle \mathop{\sum }_{\unicode[STIX]{x1D706}\in (L^{\prime }/L)^{n}}a_{\unicode[STIX]{x1D706}}\mathfrak{e}_{\unicode[STIX]{x1D706}},\mathop{\sum }_{\unicode[STIX]{x1D707}\in (L^{\prime }/L)^{n}}b_{\unicode[STIX]{x1D707}}\mathfrak{e}_{\unicode[STIX]{x1D707}}\right\rangle =\mathop{\sum }_{\unicode[STIX]{x1D706}\in (L^{\prime }/L)^{n}}a_{\unicode[STIX]{x1D706}}\overline{b_{\unicode[STIX]{x1D706}}}.\end{eqnarray}$$

Since $\unicode[STIX]{x1D70C}_{L,n}$ is a unitary representation with respect to (3.31), its dual representation $\unicode[STIX]{x1D70C}_{L,n}^{\ast }$ is just the complex conjugate of $\unicode[STIX]{x1D70C}_{L,n}$ . That is,

$$\begin{eqnarray}\unicode[STIX]{x1D70C}_{L,n}^{\ast }(g)=\unicode[STIX]{x1D70C}_{L,n}^{-1}(g)=\overline{\unicode[STIX]{x1D70C}_{L,n}(g)}\end{eqnarray}$$

for all $g\in \operatorname{Sp}(n,\mathbb{Z})$ . The explicit formulas (3.30) show immediately that $\unicode[STIX]{x1D70C}_{L,n}^{\ast }$ is equal to $\unicode[STIX]{x1D70C}_{L(-1),n}$ , where $L(-1)$ is the lattice $L$ equipped with the bilinear form $-(\cdot ,\cdot )$ . Thus, without loss of generality, one can work with the dual Weil representation.

3.4 Local Whittaker functions

In this section, we compute a particular $p$ -adic integral. It will turn out that a local Whittaker integral at a ramified prime can be expressed in terms of this integral. We follow mostly [Reference Kudla and YangKY, 4.1], [Reference YangYa, Chapter 2], and [Reference Hironaka and SatoHS]. First, we consider anisotropic discriminant forms, which will be needed in the sequel.

3.4.1 Anisotropic quadratic modules

Definition 3.5. Let $(L^{\prime }/L,(\cdot ,\cdot ))$ be a finite quadratic module with associated quadratic form  $q$ . We call $(L^{\prime }/L,(\cdot ,\cdot ))$ anisotropic if $q(\unicode[STIX]{x1D707})\not =0$ for all $\unicode[STIX]{x1D707}\in L^{\prime }/L\setminus \{0\}$ .

The structure of anisotropic finite quadratic modules of prime-power order can be found in [Reference Bruinier, Ehlen and FreitagBEF, Lemma 4.9].

Remark 3.6. Let $p$ be an odd prime, and let $(L^{\prime }/L,(\cdot ,\cdot ))$ be an anisotropic finite quadratic module whose order is a power of  $p$ . Then we have

(3.32) $$\begin{eqnarray}(L^{\prime }/L,q)\cong \left\{\begin{array}{@{}ll@{}}\displaystyle \left(\mathbb{Z}/p\mathbb{Z},q(x)=\frac{\unicode[STIX]{x1D6FC}_{1}x^{2}}{p}\right)~\text{or}~\left(\mathbb{Z}/p\mathbb{Z},q(x)=\frac{\unicode[STIX]{x1D6FC}_{2}x^{2}}{p}\right)\quad & \\ \quad \displaystyle \oplus \left(\mathbb{Z}/p\mathbb{Z},q(x)=\frac{x^{2}}{p}\right),\quad & p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4,\\ \displaystyle \left(\mathbb{Z}/p\mathbb{Z},q(x)=\frac{\unicode[STIX]{x1D6FD}_{1}x^{2}}{p}\right)~\text{or}~\left(\mathbb{Z}/p\mathbb{Z},q(x)=\frac{\unicode[STIX]{x1D6FD}_{2}x^{2}}{p}\right)\quad & \\ \quad \displaystyle \oplus \left(\mathbb{Z}/p\mathbb{Z},q(x)=\frac{x^{2}}{p}\right),\quad & p\equiv 3\hspace{0.2em}{\rm mod}\hspace{0.2em}4,\end{array}\right.\end{eqnarray}$$

where $(\frac{\unicode[STIX]{x1D6FC}_{1}}{p})=(\frac{\unicode[STIX]{x1D6FD}_{1}}{p})=\pm (\frac{2}{p}),(\frac{\unicode[STIX]{x1D6FD}_{2}}{p})=(\frac{2}{p})$ , and $(\frac{\unicode[STIX]{x1D6FC}_{2}}{p})=-(\frac{2}{p})$ .

These results can be transferred to $p$ -adic lattices.

Remark 3.7.

1. (1) Let $p$ be an odd prime, and let $(L,(\cdot ,\cdot ))$ be a non-degenerate and even $p$ -adic lattice of rank $m$ with quadratic form $q$ (even means that $q(x)\in \mathbb{Z}_{p}$ for all $x\in L$ ). Then $L^{\prime }/L$ is a finite $p$ -group, and $(L^{\prime }/L,(\cdot ,\cdot ))$ can be interpreted as a finite quadratic module.

2. (2) If $(L^{\prime }/L,(\cdot ,\cdot ))$ is anisotropic, the quadratic form on the lattice $L$ is $\mathbb{Z}_{p}$ -equivalent to

   (3.33) $$\begin{eqnarray}\left\{\begin{array}{@{}ll@{}}\displaystyle \mathop{\sum }_{i=1}^{m-1}\frac{x_{i}^{2}}{2}+\frac{p\unicode[STIX]{x1D6FC}_{m}x_{m}^{2}}{2}~\text{or}~\mathop{\sum }_{i=1}^{m-2}\frac{x_{i}^{2}}{2}+\frac{p\unicode[STIX]{x1D6FF}_{m-1}x_{m-1}^{2}}{2}+\frac{px_{m}^{2}}{2},\quad & p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4,\\ \displaystyle \mathop{\sum }_{i=1}^{m-1}\frac{x_{i}^{2}}{2}+\frac{p\unicode[STIX]{x1D6FD}_{m}x_{m}^{2}}{2}~\text{or}~\mathop{\sum }_{i=1}^{m-2}\frac{x_{i}^{2}}{2}+\frac{p\unicode[STIX]{x1D700}_{m-1}x_{m-1}^{2}}{2}+\frac{px_{m}^{2}}{2},\quad & p\equiv 3\hspace{0.2em}{\rm mod}\hspace{0.2em}4,\end{array}\right.\end{eqnarray}$$
   where $(\frac{\unicode[STIX]{x1D6FC}_{m}}{p})=(\frac{\unicode[STIX]{x1D6FD}_{m}}{p})=\pm (\frac{2}{p})$ , $(\frac{\unicode[STIX]{x1D6FF}_{m-1}}{p})=-(\frac{2}{p})$ , and $(\frac{\unicode[STIX]{x1D700}_{m-1}}{p})=(\frac{2}{p})$ (see [Reference Miranda and MorrisonMM, Proposition 2.12], applied to (3.32)).

3.4.2 Local Whittaker integrals

Lemma 3.8. Let $p>2$ be a prime, $l\in \mathbb{Z}$ and let $(L,(\cdot ,\cdot ))$ be an even and non-degenerate $p$ -adic lattice with associated quadratic form $q$ such that the attached discriminant form $L^{\prime }/L$ is anisotropic. Further, let $V=L\otimes \mathbb{Q}_{p}$ , $d_{V}x$ the self-dual Haar measure on  $V$ , $dx$ the standard Haar measure on $\mathbb{Z}_{p}^{m}$ , and $\unicode[STIX]{x1D700}\in \mathbb{Z}_{p}^{\times }$ .

If $L^{\prime }/L\cong \mathbb{Z}/p\mathbb{Z}$ , we have

(3.34) $$\begin{eqnarray}\begin{array}{@{}l@{}}\displaystyle \int _{L}\unicode[STIX]{x1D713}_{p}(\unicode[STIX]{x1D700}p^{-l}q(x))d_{V}x=|L^{\prime }/L|^{-1/2}\left\{\begin{array}{@{}ll@{}}1\quad & \text{if}~l\leqslant 0,\\ \displaystyle \left(\left(\frac{-\unicode[STIX]{x1D700}}{p}\right)\unicode[STIX]{x1D6FF}_{p}\right)^{m-1}p^{-(1/2)(ml-1)}\quad & \text{if}~l\geqslant 1~\text{is odd},\\ \displaystyle \pm \left(\frac{2}{p}\right)\left(\frac{-\unicode[STIX]{x1D700}}{p}\right)\unicode[STIX]{x1D6FF}_{p}p^{-(1/2)(ml-1)}\quad & \text{if}~l>1~\text{is even}\end{array}\right.\\ \text{and}\\ \displaystyle \int _{L\times L}\unicode[STIX]{x1D713}_{p}(\unicode[STIX]{x1D700}p^{-l}(x,y))\,d_{V}x\,d_{V}y=|L^{\prime }/L|^{-1}\cdot \left\{\begin{array}{@{}ll@{}}p^{-lm+1}\quad & \text{if}~l\geqslant 1,\\ 1\quad & \text{if}~l<1.\end{array}\right.\end{array}\end{eqnarray}$$

Similarly, if $L^{\prime }/L\cong (\mathbb{Z}/p\mathbb{Z})^{2}$ ,

(3.35) $$\begin{eqnarray}\begin{array}{@{}l@{}}\displaystyle \int _{L}\unicode[STIX]{x1D713}_{p}(\unicode[STIX]{x1D700}p^{-l}q(x))d_{V}x\\ \displaystyle \quad =|L^{\prime }/L|^{-1/2}\left\{\begin{array}{@{}ll@{}}1\quad & \text{if}~l\leqslant 0,\\ \displaystyle \left(\frac{-1}{p}\right)^{(m-2)/2}p^{-(1/2)(ml-2)}\quad & \text{if}~l\geqslant 1~\text{is odd},\\ \displaystyle \left(\frac{-1}{p}\right)(-1)^{(p-3)/2}\left(\frac{2}{p}\right)p^{-(1/2)(ml-2)}\quad & \text{if}~l>1~\text{is even}\end{array}\right.\\ \text{and}\\ \displaystyle \int _{L\times L}\unicode[STIX]{x1D713}_{p}(\unicode[STIX]{x1D700}p^{-l}(x,y))\,d_{V}x\,d_{V}y=|L^{\prime }/L|^{-1}\cdot \left\{\begin{array}{@{}ll@{}}p^{-lm+2}\quad & \text{if}~l\geqslant 1,\\ 1\quad & \text{if}~l<1.\end{array}\right.\end{array}\end{eqnarray}$$

As usual,

$$\begin{eqnarray}\unicode[STIX]{x1D6FF}_{p}=\left\{\begin{array}{@{}ll@{}}1\quad & \text{if}~p\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}4,\\ i\quad & \text{if}~p\equiv 3\hspace{0.2em}{\rm mod}\hspace{0.2em}4.\end{array}\right.\end{eqnarray}$$

Proof. By [Reference Kudla and YangKY, p. 2287], we have $d_{V}x=|L^{\prime }/L|^{-1/2}\,dx$ . Also, by Remark 3.7,  $q$ is $\mathbb{Z}_{p}$ -equivalent to one of the quadratic forms in (3.33).

The integral $\int _{\mathbb{Z}_{p}^{m}}\unicode[STIX]{x1D713}_{p}(\unicode[STIX]{x1D700}p^{-l}q(x))dx$ is computed explicitly for a general quadratic form in [Reference YangYa, Lemma 2.2]. The first formula of (3.34) and (3.35) respectively follows easily from the general formula applied to the special quadratic form.

The integral $\int _{\mathbb{Z}_{p}^{m}\times \mathbb{Z}_{p}^{m}}\unicode[STIX]{x1D713}_{p}(\unicode[STIX]{x1D700}p^{-l}(x,y))\,dx\,dy$ is calculated in [Reference Hironaka and SatoHS, Proposition 3.2], for a general quadratic form. The second formula of (3.34) and (3.35) respectively follows immediately from the just quoted general one.◻

For the following lemma, we introduce some notation, which is adopted from [Reference Hironaka and SatoHS]. We consider the set

$$\begin{eqnarray}T_{n}(\mathbb{Q}_{p})=\{b\in \operatorname{Sym}_{n}(\mathbb{Q}_{p})\mid \det (b)\not =0\}\end{eqnarray}$$

and the subgroup

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{0}(p)=\{\unicode[STIX]{x1D6FE}=(\unicode[STIX]{x1D6FE}_{ij})\in \operatorname{GL}_{n}(\mathbb{Z}_{p})\mid \unicode[STIX]{x1D6FE}_{ij}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}p~\text{for all}~i>j\}\end{eqnarray}$$

of $\operatorname{GL}_{n}(\mathbb{Z}_{p})$ . It acts on $T_{n}(\mathbb{Q}_{p})$ by $b\mapsto \unicode[STIX]{x1D6FE}\cdot b=\unicode[STIX]{x1D6FE}b\unicode[STIX]{x1D6FE}^{t}$ . For the next lemma, we will need a complete set of representatives of $\unicode[STIX]{x1D6E4}_{0}(p)\backslash T_{n}(\mathbb{Q}_{p})$ with respect to this action, which was determined in [Reference Hironaka and SatoHS, Theorem 2.1]. To describe this result, let $S_{n}$ be the symmetric group acting on $I=\{1,\ldots ,n\}$ , and let $\unicode[STIX]{x1D6FF}\in \mathbb{Z}_{p}^{\times }\setminus (\mathbb{Z}_{p}^{\times })^{2}$ be a $p$ -adic non-square unit. Such a set of representatives is given by

(3.36) $$\begin{eqnarray}\displaystyle & & \displaystyle \{S_{\unicode[STIX]{x1D70E},e,\unicode[STIX]{x1D700}}=(s_{ij})\mid s_{ij}=\unicode[STIX]{x1D700}_{i}p^{e_{i}}\unicode[STIX]{x1D6FF}_{i,\unicode[STIX]{x1D70E}(j)}\nonumber\\ \displaystyle & & \displaystyle \qquad \text{with}~(\unicode[STIX]{x1D70E},e=(e_{1},\ldots ,e_{n}),\unicode[STIX]{x1D700}=(\unicode[STIX]{x1D700}_{1},\ldots ,\unicode[STIX]{x1D700}_{n}))\in S_{n}\times \mathbb{Z}^{n}\times \{1,\unicode[STIX]{x1D6FF}\}^{n}\},\end{eqnarray}$$

where

(3.37) $$\begin{eqnarray}\unicode[STIX]{x1D70E}^{2}=\operatorname{id},\qquad e_{\unicode[STIX]{x1D70E}(i)}=e_{i}\quad \text{for all}~i\in I,\qquad \unicode[STIX]{x1D700}_{i}=1\quad \text{if}~\unicode[STIX]{x1D70E}(i)\not =i.\end{eqnarray}$$

The following constants occur in the formula for local densities in [Reference Hironaka and SatoHS, Theorem 4.1]. For the convenience of the reader, we repeat these quantities in full detail.

$$\begin{eqnarray}\displaystyle & & \displaystyle c_{1}(\unicode[STIX]{x1D70E})=|\{i\in I\mid \unicode[STIX]{x1D70E}(i)=i,\}|\qquad \text{and}\nonumber\\ \displaystyle & & \displaystyle c_{2}(\unicode[STIX]{x1D70E})=|\{i\in I\mid \unicode[STIX]{x1D70E}(i)\not =i,\}|.\nonumber\end{eqnarray}$$

For a partition $I=I_{0}\cup \cdots \cup I_{s}$ into disjoint $\unicode[STIX]{x1D70E}$ -stable subsets, we put

(3.38) $$\begin{eqnarray}\begin{array}{@{}l@{}}n_{l}=|I_{l}|,\\ \displaystyle n(l)=\frac{n^{(l)}(n^{(l)}+1)}{2},\end{array}\end{eqnarray}$$

where $n^{(l)}=n_{l}+n_{l+1}+\cdots +n_{s}$ and $l\geqslant 0$ . We also define

(3.39) $$\begin{eqnarray}\begin{array}{@{}l@{}}\displaystyle t(\unicode[STIX]{x1D70E},\{I_{i}\})=\mathop{\sum }_{l=0}^{s}|\{(i,j)\in I_{l}\times I_{l}\mid i<j<\unicode[STIX]{x1D70E}(i),\unicode[STIX]{x1D70E}(j)<\unicode[STIX]{x1D70E}(i)\}|,\\ \displaystyle \unicode[STIX]{x1D70F}(\{I_{i}\})=\mathop{\sum }_{l=1}^{s}|\{(i,j)\in I_{l}\times (I_{0}\cup \cdots \cup I_{l-1})\mid j<i\}|,\\ c_{1}^{(l)}(\unicode[STIX]{x1D70E})=|\{i\in I_{l}\cup \cdots \cup I_{s}\mid \unicode[STIX]{x1D70E}(i)=i\}|,l\geqslant 0,\\ \text{and}\\ c_{1,l}(\unicode[STIX]{x1D70E})=|\{i\in I_{l}\mid \unicode[STIX]{x1D70E}(i)=i\}|,\\ c_{2,l}(\unicode[STIX]{x1D70E})=|\{i\in I_{l}\mid \unicode[STIX]{x1D70E}(i)>i\}|.\end{array}\end{eqnarray}$$

The terms $c_{1,l}(\unicode[STIX]{x1D70E})$ and $c_{2,l}(\unicode[STIX]{x1D70E})$ occur frequently in subsequent formulas in this paper. In order to make these formulas more readable, we drop the $(\unicode[STIX]{x1D70E})$ -part and write from now on $c_{1,l}$ and $c_{2,l}$ instead.

Let $(L,(\cdot ,\cdot ))$ be a $p$ -adic lattice as in Lemma 3.8, and let $q$ be the attached quadratic form. For $r\in \mathbb{N}$ , we set $L_{r}=L\oplus \mathbb{Z}_{p}^{2r}$ , let $q_{r}$ be the quadratic form

(3.40) $$\begin{eqnarray}q_{r}=q+q^{\prime }\quad \text{with}~q^{\prime }(x)=\frac{1}{2}x^{t}\left(\begin{array}{@{}cc@{}}0 & 1_{r}\\ 1_{r} & 0\end{array}\right)x,\end{eqnarray}$$

and denote the corresponding bilinear form by $(\cdot ,\cdot )_{r}$ . We then define for $\unicode[STIX]{x1D700}\in \mathbb{Z}_{p}^{\times }$ and $l\in \mathbb{Z}$

$$\begin{eqnarray}J_{r}(\unicode[STIX]{x1D700}p^{l})=\int _{L_{r}}\unicode[STIX]{x1D713}_{p}(\unicode[STIX]{x1D700}p^{l}q_{r}(x))\,d_{V}x\qquad \text{and}\qquad I_{r}(\unicode[STIX]{x1D700}p^{l})=\int _{L_{r}\times L_{r}}\unicode[STIX]{x1D713}_{p}(2\unicode[STIX]{x1D700}p^{l}(x,y)_{r})\,d_{V}x\,d_{V}y.\end{eqnarray}$$

Finally, let $k\in \mathbb{N}$ . The following set will be used later on:

(3.41) $$\begin{eqnarray}\{(\unicode[STIX]{x1D708}_{0},\unicode[STIX]{x1D708}_{1},\ldots ,\unicode[STIX]{x1D708}_{k-1})\in \mathbb{Z}\times \mathbb{N}^{k-1}\mid \unicode[STIX]{x1D708}_{0}+\unicode[STIX]{x1D708}_{1}+\cdots +\unicode[STIX]{x1D708}_{l}\leqslant -1,0\leqslant l\leqslant k-1\}.\end{eqnarray}$$

The variable $\unicode[STIX]{x1D708}_{0}$ can be parametrized by $t\in \mathbb{N}$ by setting $\unicode[STIX]{x1D708}_{0}=-t-\sum _{i=1}^{k-1}\unicode[STIX]{x1D708}_{i}$ . Note that $t$ is determined by a tuple $\unicode[STIX]{x1D708}$ by the previously stated equation for  $\unicode[STIX]{x1D708}_{0}$ . Based on this parametrization, (3.41) can be written in the form

(3.42) $$\begin{eqnarray}\{\unicode[STIX]{x1D708}\}_{k}=\left\{\left(\unicode[STIX]{x1D708}_{0}=-\mathop{\sum }_{i=1}^{k-1}\unicode[STIX]{x1D708}_{i}-t,\unicode[STIX]{x1D708}_{1},\ldots ,\unicode[STIX]{x1D708}_{k-1}\right)\in \mathbb{Z}\times \mathbb{N}^{k-1}\mid t\in \mathbb{N}\right\},\end{eqnarray}$$

which is more suitable for later estimates and calculations. To each tuple $\unicode[STIX]{x1D708}=(\unicode[STIX]{x1D708}_{0}=-\sum _{i=1}^{k-1}\unicode[STIX]{x1D708}_{i}-t,\unicode[STIX]{x1D708}_{1},\ldots ,\unicode[STIX]{x1D708}_{k-1})$ , we associate the quantities

(3.43) $$\begin{eqnarray}\begin{array}{@{}l@{}}\displaystyle S_{l}^{(k)}(\unicode[STIX]{x1D708})=\mathop{\sum }_{i=0}^{l}\unicode[STIX]{x1D708}_{i}=-t-\mathop{\sum }_{i=l+1}^{k-1}\unicode[STIX]{x1D708}_{i},\quad 0\leqslant l\leqslant k-1,\\ o(\unicode[STIX]{x1D708})_{k}=\{0\leqslant l\leqslant k-1\mid S_{l}^{(k)}(\unicode[STIX]{x1D708})\equiv 1\hspace{0.2em}{\rm mod}\hspace{0.2em}2\},\\ e(\unicode[STIX]{x1D708})_{k}=\{0\leqslant l\leqslant k-1\mid S_{l}^{(k)}(\unicode[STIX]{x1D708})\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}2\}\end{array}\end{eqnarray}$$

and

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D708},q_{r})=\left\{\begin{array}{@{}ll@{}}0\quad & \text{if}~L^{\prime }/L\cong \mathbb{Z}/p\mathbb{Z},\\ \displaystyle (-1)^{\mathop{\sum }_{l\in e(\unicode[STIX]{x1D708})_{k}}c_{1,l}}\left(\frac{-1}{p}\right)^{(m-2)/2\mathop{\sum }_{l\in o(\unicode[STIX]{x1D708})_{k}}c_{1,l}}\quad \\ \displaystyle \quad \times \left(\frac{2}{p}\right)^{\mathop{\sum }_{l\in e(\unicode[STIX]{x1D708})_{k}}c_{1,l}}p^{(m+2r)\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})((1/2)c_{1,l}+c_{2,l})}\quad & \text{if}~L^{\prime }/L\cong (\mathbb{Z}/p\mathbb{Z})^{2}.\end{array}\right.\nonumber\end{eqnarray}$$

Note that the sets $o(\unicode[STIX]{x1D708})_{k}$ and $e(\unicode[STIX]{x1D708})_{k}$ depend only on the parity of each component  $\unicode[STIX]{x1D708}_{i}$ . In this context, we further introduce the subsets

(3.44) $$\begin{eqnarray}\{{\unicode[STIX]{x1D708}_{i_{1},\ldots ,i_{d}}\}}_{k},\quad 0\leqslant i_{1}<i_{2}<\cdots <i_{d}\leqslant n,\end{eqnarray}$$

of $\{\unicode[STIX]{x1D708}\}_{k}$ , which consist of all tuples $\unicode[STIX]{x1D708}$ , where the components $\unicode[STIX]{x1D708}_{i_{j}}$ , $j=1,\ldots ,d$ , run over all odd integers and the remaining components over all even integers. Note that if the parity of $\unicode[STIX]{x1D708}_{1},\ldots ,\unicode[STIX]{x1D708}_{k-1}$ is fixed, then the parity of $\unicode[STIX]{x1D708}_{0}$ is determined by that of  $t$ . To emphasize the dependency of $o(\unicode[STIX]{x1D708})_{k}$ and $e(\unicode[STIX]{x1D708})_{k}$ on the parity of the components of $\unicode[STIX]{x1D708}$ , we will later write $o(i_{1},\ldots ,i_{d})_{k}$ and $e(i_{1},\ldots ,i_{d})_{k}$ respectively if $\unicode[STIX]{x1D708}\in \{{\unicode[STIX]{x1D708}_{i_{1},\ldots ,i_{d}}\}}_{k}$ .

Lemma 3.9. Let $p>2$ be a prime, $(L,(\cdot ,\cdot ))$ be a $p$ -adic lattice of even rank $m$ with the same properties as in Lemma 3.8, and $\unicode[STIX]{x1D707}\in L_{r}^{\prime }/L_{r}$ for $r\in \mathbb{N}$ .

Then

(3.45) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\operatorname{Sym}_{n}(\mathbb{Q}_{p})}\int _{\unicode[STIX]{x1D707}+L_{r}^{n}}\unicode[STIX]{x1D713}_{p}\left(\frac{1}{2}\operatorname{tr}(bQ^{r}[x])\right)dx\,db\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D707},0}\frac{1}{|L^{\prime }/L|^{n/2}}\mathop{\sum }_{\substack{ \unicode[STIX]{x1D70E}\in S_{n} \\ \unicode[STIX]{x1D70E}^{2}=\operatorname{id}}}2^{-c_{1}(\unicode[STIX]{x1D70E})}(1-p^{-1})^{c_{2}(\unicode[STIX]{x1D70E})+c_{1}(\unicode[STIX]{x1D70E})}p^{-c_{2}(\unicode[STIX]{x1D70E})}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{I=I_{0}\cup \cdots \cup I_{s}}p^{-\unicode[STIX]{x1D70F}(\{I_{i}\})-t(\unicode[STIX]{x1D70E},\{I_{i}\})}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{k=0}^{s+1}\frac{2^{c_{1}^{(k)}(\unicode[STIX]{x1D70E})}2^{\mathop{\sum }_{l=0}^{k-1}c_{1,l}}(1-p^{-1})^{c_{1}^{(k)}(\unicode[STIX]{x1D70E})}p^{-\mathop{\sum }_{l=k+1}^{s}n(l)}p^{\mathop{\sum }_{l=0}^{k-1}(c_{1,l}+2c_{2,l})}}{\mathop{\prod }_{l=k}^{s}(1-p^{-n(l)})}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{\{\unicode[STIX]{x1D708}\}_{k}}p^{\mathop{\sum }_{j=0}^{k-1}\unicode[STIX]{x1D708}_{j}(n(k)-n(j))}\unicode[STIX]{x1D6EF}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D708},q_{r}),\end{eqnarray}$$

where

1. (1) $Q^{r}[x]$ is the moment matrix with respect to $(\cdot ,\cdot )_{r}$ ;

2. (2) $dx$ is the self-dual Haar measure on $L_{r}^{n}\otimes \mathbb{Q}_{p}$ with respect to the character  $\unicode[STIX]{x1D713}_{p}$ ;

3. (3) $db$ is the product measure of the standard Haar measure on  $\mathbb{Q}_{p}$ ;

4. (4) $\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D707},0}$ is the Kronecker delta and $s\leqslant n$ is an integer;

5. (5) $\sum _{I=I_{0}\cup \cdots \cup I_{s}}$ is the sum over all partitions of $I$ into disjoint $\unicode[STIX]{x1D70E}$ -stable subsets;

6. (6) and the summation with respect to $\{\unicode[STIX]{x1D708}\}_{k}$ is taken over the set (3.42); if $k=0$ , we understand this sum to be equal to 1.

Proof. First, note that by an idea of Liu [Reference LiuLi2, Proposition 2.4], for $r>\unicode[STIX]{x1D70C}_{n}$ , in the region of absolute convergence, we have

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\operatorname{Sym}_{n}(\mathbb{Q}_{p})}\int _{\unicode[STIX]{x1D707}+L_{r}^{n}}\unicode[STIX]{x1D713}_{p}\left(\frac{1}{2}\operatorname{tr}(bQ^{r}[x])\right)dx\,db=\int _{\unicode[STIX]{x1D707}+L_{r}^{n}}\int _{\operatorname{Sym}_{n}(\mathbb{Q}_{p})}\unicode[STIX]{x1D713}_{p}\left(\frac{1}{2}\operatorname{tr}(bQ^{r}[x])\right)db\,dx\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{\unicode[STIX]{x1D707}+L_{r}^{n}}\int _{\operatorname{Sym}_{n}(\mathbb{Q}_{p})/\operatorname{Sym}_{n}(\mathbb{Z}_{p})}\unicode[STIX]{x1D713}_{p}\left(\frac{1}{2}\operatorname{tr}(bQ^{r}[x])\right)db\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\int _{b^{\prime }\in \operatorname{Sym}_{n}(\mathbb{Z}_{p})}\unicode[STIX]{x1D713}_{p}\left(\frac{1}{2}\operatorname{tr}(b^{\prime }Q^{r}[x])\right)db^{\prime }\,dx.\nonumber\end{eqnarray}$$

A simple calculation applying standard $p$ -adic integration formulas as in [Reference Fleig, Gustafsson, Kleinschmidt and PerssonFGKP, Example 3.15] allows us to conclude that the integral over $b^{\prime }$ is equal to one if and only if

$$\begin{eqnarray}q_{r}(x_{i})\in \mathbb{Z}_{p}\qquad \text{and}\qquad (x_{i},x_{j})_{r}\in \mathbb{Z}_{p}\end{eqnarray}$$

for all $i,j=1,\ldots ,n$ and zero otherwise. Since $L^{\prime }/L$ is anisotropic, this is the case if and only if $\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{n}\in L$ . The double integral on the left-hand side in (3.45) then simplifies to

(3.46) $$\begin{eqnarray}\int _{\operatorname{Sym}_{n}(\mathbb{Q}_{p})}\int _{L_{r}^{n}}\unicode[STIX]{x1D713}_{p}\left(\frac{1}{2}\operatorname{tr}(bQ^{r}[x])\right)dx\,db.\end{eqnarray}$$

It can be evaluated using the methods in [Reference Hironaka and SatoHS], where the local density $\unicode[STIX]{x1D6FC}_{p}(S,T)$ depending on the matrices $S=\text{diag}(u_{1}p^{\unicode[STIX]{x1D6FC}_{1}},\ldots ,u_{m}p^{\unicode[STIX]{x1D6FC}_{m}})$ and $T=\text{diag}(v_{1}p^{\unicode[STIX]{x1D6FD}_{1}},\ldots ,v_{n}p^{\unicode[STIX]{x1D6FD}_{n}})$ , $u_{i},v_{j}\in \mathbb{Z}_{p}^{\times },\unicode[STIX]{x1D6FC}_{i},\unicode[STIX]{x1D6FD}_{j}\in \mathbb{Z}$ , is calculated for an odd prime $p$ and $n\in \mathbb{N}$ . By comparing (3.46) with the integral in [Reference Hironaka and SatoHS, Lemma 1.1], it becomes apparent that we are interested in the case $T=0$ . It can be verified that all results from [Reference Hironaka and SatoHS] are valid for $T=0$ , that is, for $\unicode[STIX]{x1D6FD}_{1}=\cdots =\unicode[STIX]{x1D6FD}_{n}=\infty$ . In this case, the explicit formula for the integral $\mathscr{G}_{\unicode[STIX]{x1D6E4}_{0}(p)}(S_{\unicode[STIX]{x1D70E},e,\unicode[STIX]{x1D700}},T)$ in [Reference Hironaka and SatoHS, Proposition 3.3] is equal to

$$\begin{eqnarray}(1-p^{-1})^{2c_{2}(\unicode[STIX]{x1D70E})+c_{1}(\unicode[STIX]{x1D70E})}p^{-n(n-1)/2}.\end{eqnarray}$$

In particular, condition (9) in this proposition is always satisfied and $\mathscr{G}_{\unicode[STIX]{x1D6E4}_{0}(p)}(S_{\unicode[STIX]{x1D70E},e,\unicode[STIX]{x1D700}},T)$ is nonzero for all $e\in \mathbb{Z}^{n}$ . Also, the bound $b_{l}(\unicode[STIX]{x1D70E},T)$ in the introduction of [Reference Hironaka and SatoHS] is equal to $\infty$ for all $l\in \mathbb{N}$ and $\unicode[STIX]{x1D70E}\in S_{n}$ . Consequently, the sum $\sum _{i=0}^{l}\unicode[STIX]{x1D708}_{i}$ in the set

(3.47) $$\begin{eqnarray}\{(\unicode[STIX]{x1D708}_{0},\unicode[STIX]{x1D708}_{1},\ldots ,\unicode[STIX]{x1D708}_{k-1})\in \mathbb{Z}\times \mathbb{N}^{k-1}\mid -b_{l}(\unicode[STIX]{x1D70E},T)\leqslant \unicode[STIX]{x1D708}_{0}+\unicode[STIX]{x1D708}_{1}+\cdots +\unicode[STIX]{x1D708}_{l}\leqslant -1,0\leqslant l\leqslant k-1\}\end{eqnarray}$$

appearing in [Reference Hironaka and SatoHS, Theorem 4.1] is not bounded from below for $T=0$ . Note that (without the bound $-b_{l}(\unicode[STIX]{x1D70E},T)$ ) it is equal to $\{\unicode[STIX]{x1D708}\}_{k}$ in (3.42).

Since [Reference Hironaka and SatoHS, Proposition 1.2] can also be carried over to our situation, we obtain

(3.48) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\operatorname{Sym}_{n}(\mathbb{Q}_{p})}\int _{L_{r}^{n}}\unicode[STIX]{x1D713}_{p}\left(\frac{1}{2}\operatorname{tr}(bQ^{r}[x])\right)dx\,db\nonumber\\ \displaystyle & & \displaystyle \quad =p^{-n(n-1)/2}\mathop{\sum }_{b\in \unicode[STIX]{x1D6E4}_{0}(p)\backslash T_{n}(\mathbb{Q}_{p})}\frac{(1-p^{-1})^{2c_{2}(\unicode[STIX]{x1D70E})+c_{1}(\unicode[STIX]{x1D70E})}}{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6E4}_{0}(p),b)}\int _{L_{r}^{n}}\unicode[STIX]{x1D713}_{p}\left(\frac{1}{2}\operatorname{tr}(bQ^{r}[x])\right)dx.\end{eqnarray}$$

The term $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6E4}_{0}(p),b)$ is explicitly calculated in [Reference Hironaka and SatoHS, Theorem 2.2]: for some representative $S_{\unicode[STIX]{x1D70E},e,\unicode[STIX]{x1D700}}$ of $\unicode[STIX]{x1D6E4}_{0}(p)\backslash T_{n}(\mathbb{Q}_{p})$ , the expression $\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6E4}_{0}(p),S_{\unicode[STIX]{x1D70E},e,\unicode[STIX]{x1D700}})$ is equal to

(3.49) $$\begin{eqnarray}\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6E4}_{0}(p),S_{\unicode[STIX]{x1D70E},e,\unicode[STIX]{x1D700}})=2^{c_{1}(\unicode[STIX]{x1D70E})}(1-p^{-1})^{c_{2}(\unicode[STIX]{x1D70E})}p^{c(\unicode[STIX]{x1D70E},e,\unicode[STIX]{x1D700})},\end{eqnarray}$$

where

$$\begin{eqnarray}c(\unicode[STIX]{x1D70E},e,\unicode[STIX]{x1D700})=-\frac{n(n-1)}{2}+\unicode[STIX]{x1D70F}(\{I_{i}\})+t(\unicode[STIX]{x1D70E},\{I_{i}\})+c_{2}(\unicode[STIX]{x1D70E})+\mathop{\sum }_{l=0}^{s}\unicode[STIX]{x1D708}_{l}n(l).\end{eqnarray}$$

The same ideas for calculating the integral on the right-hand side of (3.48), as in [Reference Hironaka and SatoHS, Proposition 3.2], lead to

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{b\in \unicode[STIX]{x1D6E4}_{0}(p)\backslash T_{n}(\mathbb{Q}_{p})}\frac{1}{\unicode[STIX]{x1D6FC}(\unicode[STIX]{x1D6E4}_{0}(p),b)}\int _{L_{r}^{n}}\unicode[STIX]{x1D713}_{p}\left(\frac{1}{2}\operatorname{tr}(bQ^{r}[x])\right)dx\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{(\unicode[STIX]{x1D70E},e,\unicode[STIX]{x1D700})}2^{-c_{1}(\unicode[STIX]{x1D70E})}(1-p^{-1})^{-c_{2}(\unicode[STIX]{x1D70E})}p^{-c(\unicode[STIX]{x1D70E},e,\unicode[STIX]{x1D700})}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \mathop{\prod }_{\substack{ i=1 \\ \unicode[STIX]{x1D70E}(i)=i}}^{n}\int _{L_{r}}\unicode[STIX]{x1D713}_{p}(\unicode[STIX]{x1D700}_{i}p^{e_{i}}q_{r}(x))\,d_{V}x\cdot \mathop{\prod }_{\substack{ j=1 \\ \unicode[STIX]{x1D70E}(j)>j}}^{n}\int _{L_{r}\times L_{r}}\unicode[STIX]{x1D713}_{p}(2\unicode[STIX]{x1D700}_{j}p^{e_{j}}(x,y)_{r})\,d_{V}x\,d_{V}y.\nonumber\end{eqnarray}$$

Following the proof of Theorem 4.1 of [Reference Hironaka and SatoHS] and taking the thoughts with respect to set (3.47) from above into account, we find that the last expression is equal to

$$\begin{eqnarray}\displaystyle & & \displaystyle p^{n(n-1)/2}\mathop{\sum }_{\substack{ \unicode[STIX]{x1D70E}\in S^{n} \\ \unicode[STIX]{x1D70E}^{2}=\operatorname{id}}}2^{-c_{1}(\unicode[STIX]{x1D70E})}(1-p^{-1})^{-c_{2}(\unicode[STIX]{x1D70E})}p^{-c_{2}(\unicode[STIX]{x1D70E})}\mathop{\sum }_{I=I_{0}\cup \cdots \cup I_{s}}p^{-\unicode[STIX]{x1D70F}(\{I_{i}\})-t(\unicode[STIX]{x1D70E},\{I_{i}\})}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{k=0}^{s+1}\frac{2^{c_{1}^{(k)}(\unicode[STIX]{x1D70E})}(1-p^{-1})^{c_{1}^{(k)}(\unicode[STIX]{x1D70E})}p^{-\mathop{\sum }_{l=k+1}^{s}n(l)}}{\mathop{\prod }_{l=k}^{s}(1-p^{-n(l)})}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{\{\unicode[STIX]{x1D708}\}_{k}}p^{\mathop{\sum }_{j=0}^{k-1}\unicode[STIX]{x1D708}_{l}(n(k)-n(j))}\mathop{\prod }_{l=0}^{k-1}\mathop{\prod }_{\substack{ i\in I_{l} \\ \unicode[STIX]{x1D70E}(i)=i}}\left(\mathop{\sum }_{\unicode[STIX]{x1D700}_{i}\in \{1,\unicode[STIX]{x1D6FF}\}}J_{r}(\unicode[STIX]{x1D700}_{i}p^{S_{l}^{(k)}(\unicode[STIX]{x1D708})})\right)\mathop{\prod }_{\substack{ i\in I_{l} \\ \unicode[STIX]{x1D70E}(i)>i}}I_{r}(2p^{S_{l}^{(k)}(\unicode[STIX]{x1D708})}).\nonumber\end{eqnarray}$$

In light of [Reference YangYa, Corollary 2.3 and Notation (1.6)], we obtain

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\prod }_{l=0}^{k-1}\mathop{\prod }_{\substack{ i\in I_{l} \\ \unicode[STIX]{x1D70E}(i)=i}}\left(\mathop{\sum }_{\unicode[STIX]{x1D700}_{i}\in \{1,\unicode[STIX]{x1D6FF}\}}J_{r}(\unicode[STIX]{x1D700}_{i}p^{S_{l}^{(k)}(\unicode[STIX]{x1D708})})\right)\mathop{\prod }_{\substack{ i\in I_{l} \\ \unicode[STIX]{x1D70E}(i)>i}}I_{r}(2p^{S_{l}^{(k)}(\unicode[STIX]{x1D708})})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\prod }_{l=0}^{k-1}\mathop{\prod }_{\substack{ i\in I_{l} \\ \unicode[STIX]{x1D70E}(i)=i}}\left(1+\left(\frac{\unicode[STIX]{x1D6FF}}{p}\right)^{l(S_{l}^{(k)}(\unicode[STIX]{x1D708}),1)}\right)J_{r}(p^{S_{l}^{(k)}(\unicode[STIX]{x1D708})})\mathop{\prod }_{\substack{ i\in I_{l} \\ \unicode[STIX]{x1D70E}(i)>i}}I_{r}(2p^{S_{l}^{(k)}(\unicode[STIX]{x1D708})}).\nonumber\end{eqnarray}$$

Decomposing the product $\prod _{l=0}^{k-1}$ into products over $o(\unicode[STIX]{x1D708})_{k}$ and $e(\unicode[STIX]{x1D708})_{k}$ and using formulas (3.34) and (3.35) allows us to write the last expression in the form

$$\begin{eqnarray}\displaystyle & & \displaystyle |L^{\prime }/L|^{-n/2}\nonumber\\ \displaystyle & & \displaystyle \quad \times \left\{\begin{array}{@{}ll@{}}0\quad & \text{if}~L^{\prime }/L\cong \mathbb{Z}/p\mathbb{Z}\\ \displaystyle \mathop{\prod }_{l\in o(\unicode[STIX]{x1D708})_{k}}\left(2\left(\frac{-1}{p}\right)^{(m-2)/2}p^{(1/2)((m+2r)S_{l}^{(k)}(\unicode[STIX]{x1D708})+2)}\right)^{c_{1,l}}\quad & \\ \quad \displaystyle \times \mathop{\prod }_{l\in e(\unicode[STIX]{x1D708})_{k}}\left(-2\left(\frac{2}{p}\right)p^{(1/2)((m+2r)S_{l}^{(k)}(\unicode[STIX]{x1D708})+2)}\right)^{c_{1,l}}\quad & \\ \quad \displaystyle \times \mathop{\prod }_{l=0}^{k-1}p^{((m+2r)S_{l}^{(k)}(\unicode[STIX]{x1D708})+2)c_{2,l}}\quad & \text{if}~L^{\prime }/L\cong (\mathbb{Z}/p\mathbb{Z})^{2},\end{array}\right.\nonumber\end{eqnarray}$$

which leads to the formula in (3.45).◻

3.5 Eisenstein series

3.5.1 Adelic and vector valued Eisenstein series

To any standard section $\unicode[STIX]{x1D6F7}\in I_{n}(s,\unicode[STIX]{x1D712}_{V})$ and $g\in G(\mathbb{A})$ , we can associate an Eisenstein series

(3.50) $$\begin{eqnarray}E(g,s,\unicode[STIX]{x1D6F7})=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in P(\mathbb{Q})\backslash G(\mathbb{Q})}\unicode[STIX]{x1D6F7}(\unicode[STIX]{x1D6FE}g,s).\end{eqnarray}$$

It converges absolutely for $\operatorname{Re}(s)>\unicode[STIX]{x1D70C}_{n}$ and defines an automorphic form on $G(\mathbb{A})$ . Moreover, as is important for this paper, it can be proved that $E(g,s,\unicode[STIX]{x1D6F7})$ has a meromorphic analytic continuation to the whole complex $s$ -plane and that it satisfies a functional equation

(3.51) $$\begin{eqnarray}E(g,s,\unicode[STIX]{x1D6F7})=E(g,-s,M(s)\unicode[STIX]{x1D6F7}),\end{eqnarray}$$

where

(3.52) $$\begin{eqnarray}\begin{array}{@{}c@{}}\displaystyle M(s)=\mathop{\prod }_{p\leqslant \infty }M_{p}(s):I_{n}(s,\unicode[STIX]{x1D712}_{V})\longrightarrow I_{n}(-s,\unicode[STIX]{x1D712}_{V}),\\ \displaystyle \unicode[STIX]{x1D6F7}(g,s)\mapsto M(s)\unicode[STIX]{x1D6F7}(g,s)=\int _{N(\mathbb{A})}\unicode[STIX]{x1D6F7}(wng,s)\,dn\end{array}\end{eqnarray}$$

is the global intertwining operator. Since $N(\mathbb{A})\cong \operatorname{Sym}_{n}(\mathbb{A})$ , for each $p\leqslant \infty$ , the operator $M_{p}(s)$ can be written as

(3.53) $$\begin{eqnarray}M_{p}(s)\unicode[STIX]{x1D6F7}_{p}(g,s)=\int _{\operatorname{Sym}_{n}(\mathbb{Q}_{p})}\unicode[STIX]{x1D6F7}_{p}(wn(b)g,s)\,db,\end{eqnarray}$$

where $\mathbb{Q}_{\infty }=\mathbb{R}$ and $db$ is the self-dual Haar measure on $\operatorname{Sym}_{n}(\mathbb{Q}_{p})$ with respect to the pairing $(b_{1},b_{2})\mapsto \unicode[STIX]{x1D713}_{p}(\operatorname{tr}(b_{1}b_{2}))$ .

The following result is well known; see [Reference Kudla and RallisKR, Lemma 4.6]. We give an elementary proof, which is in principal also known, but to the best of my knowledge not explicitly written down. So, we add it here for the convenience of the reader.

Theorem 3.10. Let $\unicode[STIX]{x1D6F7}_{\infty }^{l}(s)$ be the archimedean part of the section defined in (3.23), $\unicode[STIX]{x1D6FC}=\frac{1}{2}(s+\unicode[STIX]{x1D70C}_{n}+l)$ , and $\unicode[STIX]{x1D6FD}=\frac{1}{2}(s+\unicode[STIX]{x1D70C}_{n}-l)$ . Then

(3.54) $$\begin{eqnarray}M_{\infty }(s)\unicode[STIX]{x1D6F7}_{\infty }^{l}(g,s)=i^{-ln}2^{(1-s)n}\unicode[STIX]{x1D70B}^{n(n+1)/2}\frac{\unicode[STIX]{x1D6E4}_{n}(s)}{\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FD})}\unicode[STIX]{x1D6F7}_{\infty }^{l}(g,-s)\end{eqnarray}$$

for all $g\in \operatorname{Sp}(n,\mathbb{R})$ .

Proof. The proof is basically a straightforward generalization of the analogous result in [Reference KudlaKu2, Lemma 9.2]. Let $n(x)m(a)k$ be the Iwasawa decomposition of  $g$ . We can assume that $\det (a)>0$ . Lemma 9.2 and the subsequent arguments in [Reference KudlaKu2] carry over to the case of a general  $n$ . We therefore obtain (see also [Reference Bruinier, Funke and KudlaBFK, (1.5)])

(3.55) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\operatorname{Sym}_{n}(\mathbb{R})}\unicode[STIX]{x1D6F7}_{\infty }^{l}(wn(b)n(x)m(a)k,s)\,db\nonumber\\ \displaystyle & & \displaystyle \qquad =\unicode[STIX]{x1D712}_{V}(\det (a))|\det (a)|^{s+\unicode[STIX]{x1D70C}_{n}}\det (\mathbf{k})^{l}\int _{\operatorname{Sym}_{n}(\mathbb{R})}\det (b+iy)^{-\unicode[STIX]{x1D6FC}}\det (b-iy)^{-\unicode[STIX]{x1D6FD}}\,db,\end{eqnarray}$$

where $y=a^{t}a$ . The integral on the right-hand side of (3.55) can be evaluated with the help of [Reference ShimuraSh, (1.31)]. We find for $\operatorname{Re}(s+\unicode[STIX]{x1D70C}_{n})>n+1$

(3.56) $$\begin{eqnarray}\int _{\operatorname{Sym}_{n}(\mathbb{R})}\det (b+iy)^{-\unicode[STIX]{x1D6FC}}\det (b-iy)^{-\unicode[STIX]{x1D6FD}}\,db=i^{-ln}2^{(1-s)n}\unicode[STIX]{x1D70B}^{n(n+1)/2}\det (y)^{-s}\frac{\unicode[STIX]{x1D6E4}_{n}(s)}{\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FD})}.\end{eqnarray}$$

Combining (3.55) and (3.56), we finally obtain

$$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\operatorname{Sym}_{n}(\mathbb{R})}\unicode[STIX]{x1D6F7}_{\infty }^{l}(wn(b)n(b)n(x)m(a)k,s)\,db\nonumber\\ \displaystyle & & \displaystyle \quad =i^{-ln}2^{(1-s)n}\unicode[STIX]{x1D70B}^{n(n+1)/2}\frac{\unicode[STIX]{x1D6E4}_{n}(s)}{\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FD})}|\det (a)|^{-s+\unicode[STIX]{x1D70C}_{n}}\det (\mathbf{k})^{l}.\nonumber\end{eqnarray}$$

The next theorem provides the local analogue of Theorem 3.10 for an odd ramified prime  $p$ . We first need a preparatory lemma.

Lemma 3.11. Let the sets $\{\unicode[STIX]{x1D708}\}_{k}$ and $\{{\unicode[STIX]{x1D708}_{i_{1},\ldots ,i_{d}}\}}_{k}$ be defined as in (3.42) and (3.44) respectively, $n(l)$ be given in (3.38), $X=p^{-s}$ , and

(3.57) $$\begin{eqnarray}v(\unicode[STIX]{x1D708})=-\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})(c_{1,l}+2c_{2,l})\end{eqnarray}$$

with $c_{1,l},c_{2,l}$ specified in (3.39). We further define coefficients depending on the decomposition $I=I_{0}\cup \cdots \cup I_{s}$ of $I$ (indicated by the abbreviation $\mathbf{I}_{\mathbf{l}}$ ) by

(3.58) $$\begin{eqnarray}A_{i}(\mathbf{I}_{\mathbf{l}})=\mathop{\sum }_{j=0}^{i-1}\left(\frac{1}{2}c_{1,j}+c_{2,j}\right),\qquad B_{i}(\mathbf{I}_{\mathbf{l}})=-mA_{i}(\mathbf{I}_{\mathbf{l}})+n(0)-n(i),\,i=1,\ldots ,k.\end{eqnarray}$$

Then the series

(3.59) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\{\unicode[STIX]{x1D708}\}_{k}}(-1)^{\mathop{\sum }_{l\in e(\unicode[STIX]{x1D708})_{k}}c_{1,l}}\left(\frac{-1}{p}\right)^{((m-2)/2)\mathop{\sum }_{l\in o(\unicode[STIX]{x1D708})_{k}}c_{1,l}}\left(\frac{2}{p}\right)^{\mathop{\sum }_{l\in e(\unicode[STIX]{x1D708})_{k}}c_{1,l}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,p^{\mathop{\sum }_{j=0}^{k-1}\unicode[STIX]{x1D708}_{j}(n(k)-n(j))}p^{m\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})((1/2)c_{1,l}+c_{2,l})}X^{v(\unicode[STIX]{x1D708})}\end{eqnarray}$$

is equal to

$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{0\leqslant i_{1}\leqslant \cdots \leqslant i_{d}\leqslant n}(-1)^{\mathop{\sum }_{l\in e(i_{1},\ldots ,i_{d})_{k}}c_{1,l}}\left(\frac{-1}{p}\right)^{((m-2)/2)\mathop{\sum }_{l\in o(i_{1},\ldots ,i_{d})_{k}}c_{1,l}}\left(\frac{2}{p}\right)^{\mathop{\sum }_{l\in e(i_{1},\ldots ,i_{d})_{k}}c_{1,l}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,p^{\mathop{\sum }_{j=1}^{d}2A_{i_{j}}(\mathbf{I}_{\mathbf{l}})s-B_{i_{j}}(\mathbf{I}_{\mathbf{l}})}\mathop{\prod }_{j=1}^{k}(\unicode[STIX]{x1D701}_{p}(4A_{j}(\mathbf{I}_{\mathbf{l}})s-2B_{j}(\mathbf{I}_{\mathbf{l}}))-1),\nonumber\end{eqnarray}$$

where the sum $\sum _{0\leqslant i_{1}\leqslant \cdots \leqslant i_{d}\leqslant n}$ runs over all subsets of $\{0,\ldots ,n\}$ indicating the positions of the odd components of $\unicode[STIX]{x1D708}$ (see (3.44)).

In particular, (3.59) can be meromorphically continued to the whole complex $s$ -plane.

Proof. Series (3.59) is bounded by

$$\begin{eqnarray}\mathop{\sum }_{\{\unicode[STIX]{x1D708}\}_{k}}\left|p^{\mathop{\sum }_{j=0}^{k-1}\unicode[STIX]{x1D708}_{l}(n(k)-n(j))}p^{m\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})((1/2)c_{1,l}+c_{2,l})}X^{v(\unicode[STIX]{x1D708})}\right|.\end{eqnarray}$$

Since the exponents $m\sum _{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})((1/2)c_{1,l}+c_{2,l})$ are negative for all $\unicode[STIX]{x1D708}\in \{\unicode[STIX]{x1D708}\}_{k}$ and $n(k)-n(l)$ is negative for all $1\leqslant l\leqslant k-1$ , the latter series in turn is bounded by $\sum _{\{\unicode[STIX]{x1D708}\}_{k}}p^{S_{0}^{(k)}(\unicode[STIX]{x1D708})(n(k)-n(0))}p^{\operatorname{Re}(s)\sum _{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})}$ , where we have used $\unicode[STIX]{x1D708}_{0}=S_{0}^{(k)}(\unicode[STIX]{x1D708})$ . The identity $\sum _{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})=-kt-\sum _{i=1}^{k-1}i\unicode[STIX]{x1D708}_{i}$ gives

(3.60) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\{\unicode[STIX]{x1D708}\}_{k}}p^{S_{0}^{(k)}(\unicode[STIX]{x1D708})(n(k)-n(0))}p^{\operatorname{Re}(s)\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})}\nonumber\\ \displaystyle & & \displaystyle \qquad =\mathop{\prod }_{i=1}^{k-1}\left(\mathop{\sum }_{\unicode[STIX]{x1D708}_{i}=1}^{\infty }p^{-\unicode[STIX]{x1D708}_{i}(i\operatorname{Re}(s)+(n(k)-n(0)))}\right)\mathop{\sum }_{t=1}^{\infty }p^{-t(k\operatorname{Re}(s)+(n(k)-n(0)))}\nonumber\\ \displaystyle & & \displaystyle \qquad \leqslant \mathop{\prod }_{i=1}^{k-1}\left(\mathop{\sum }_{n=1}^{\infty }n^{-(i\operatorname{Re}(s)+(n(k)-n(0)))}\right)\left(\mathop{\sum }_{n=1}^{\infty }n^{-(k\operatorname{Re}(s)+(n(k)-n(0)))}\right).\end{eqnarray}$$

The right-hand side of (3.60) is uniformly convergent for all $s\in \mathbb{C}$ with $\operatorname{Re}(s)>(1+(n(0)-n(k)))/2$ , which shows that the sum over $\{\unicode[STIX]{x1D708}\}_{k}$ in (3.59) is holomorphic in the same region.

The absolute convergence of the sum $\sum _{\{\unicode[STIX]{x1D708}\}_{k}}$ allows us to change the order of summation. Since $o(\unicode[STIX]{x1D708})_{k}$ and $e(\unicode[STIX]{x1D708})_{k}$ only depend on the parity of the components of $\unicode[STIX]{x1D708}$ , we decompose series (3.59) into partial series in the following way:

(3.61) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{0\leqslant i_{1}\leqslant \cdots \leqslant i_{d}\leqslant n}\mathop{\sum }_{\unicode[STIX]{x1D708}\in \{{\unicode[STIX]{x1D708}_{i_{1},\ldots ,i_{d}}\}}_{k}}(-1)^{\mathop{\sum }_{l\in e(\unicode[STIX]{x1D708})_{k}}c_{1,l}}\left(\frac{-1}{p}\right)^{((m-2)/2)\mathop{\sum }_{l\in o(\unicode[STIX]{x1D708})_{k}}c_{1,l}}\left(\frac{2}{p}\right)^{\mathop{\sum }_{l\in e(\unicode[STIX]{x1D708})_{k}}c_{1,l}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,p^{\mathop{\sum }_{j=0}^{k-1}\unicode[STIX]{x1D708}_{j}(n(k)-n(j))}p^{m\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})((1/2)c_{1,l}+c_{2,l})}X^{v(\unicode[STIX]{x1D708})}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{0\leqslant i_{1}\leqslant \cdots \leqslant i_{d}\leqslant n}(-1)^{\mathop{\sum }_{l\in e(i_{1},\ldots ,i_{d})_{k}}c_{1,l}}\left(\frac{-1}{p}\right)^{((m-2)/2)\mathop{\sum }_{l\in o(i_{1},\ldots ,i_{d})_{k}}c_{1,l}}\left(\frac{2}{p}\right)^{\mathop{\sum }_{l\in e(i_{1},\ldots ,i_{d})_{k}}c_{1,l}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{\unicode[STIX]{x1D708}\in \{{\unicode[STIX]{x1D708}_{i_{1},\ldots ,i_{d}}\}}_{k}}p^{\mathop{\sum }_{j=0}^{k-1}\unicode[STIX]{x1D708}_{j}(n(k)-n(j))}p^{m\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})((1/2)c_{1,l}+c_{2,l})}X^{v(\unicode[STIX]{x1D708})}.\end{eqnarray}$$

A straightforward calculation yields

(3.62) $$\begin{eqnarray}\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})\left(\frac{1}{2}c_{1,l}+c_{2,l}\right)=-t\left(\mathop{\sum }_{l=0}^{k-1}\left(\frac{1}{2}c_{1,l}+c_{2,l}\right)\right)-\mathop{\sum }_{l=1}^{k-1}\left(\mathop{\sum }_{j=0}^{l-1}\left(\frac{1}{2}c_{1,j}+c_{2,l}\right)\right)\unicode[STIX]{x1D708}_{l}.\end{eqnarray}$$

Moreover, by using $\unicode[STIX]{x1D708}_{0}=S_{0}^{(k)}(\unicode[STIX]{x1D708})$ , it is also easily seen that

(3.63) $$\begin{eqnarray}\mathop{\sum }_{l=0}^{k-1}\unicode[STIX]{x1D708}_{l}(n(k)-n(l))=t(n(0)-n(k))+\mathop{\sum }_{l=1}^{k-1}\unicode[STIX]{x1D708}_{l}(n(0)-n(l)).\end{eqnarray}$$

In terms of coefficients (3.58), we can then write

$$\begin{eqnarray}\displaystyle & & \displaystyle m\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})\left(\frac{1}{2}c_{1,l}+c_{2,l}\right)+\mathop{\sum }_{l=0}^{k-1}\unicode[STIX]{x1D708}_{l}(n(k)-n(l))+2s\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})\left(\frac{1}{2}c_{1,l}+c_{2,l}\right)\nonumber\\ \displaystyle & & \displaystyle \quad =tB_{k}(\mathbf{I}_{\mathbf{l}})+\mathop{\sum }_{l=1}^{k-1}\unicode[STIX]{x1D708}_{l}B_{l}(\mathbf{I}_{\mathbf{l}})-2s\left(tA_{k}(\mathbf{I}_{\mathbf{l}})+\mathop{\sum }_{l=1}^{k-1}\unicode[STIX]{x1D708}_{l}A_{l}(\mathbf{I}_{\mathbf{l}})\right).\nonumber\end{eqnarray}$$

Taking the latter equation into account and using the variables $\unicode[STIX]{x1D708}_{i_{j}}=2\unicode[STIX]{x1D707}_{i_{j}}-1$ , $j=1,\ldots ,d$ , $\unicode[STIX]{x1D708}_{j}=2\unicode[STIX]{x1D707}_{j}$ , $j\not =i_{1},\ldots ,i_{d},$ we obtain

(3.64) $$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\unicode[STIX]{x1D708}\in \{{\unicode[STIX]{x1D708}_{i_{1},\ldots ,i_{d}}\}}_{k}}p^{\mathop{\sum }_{j=0}^{k-1}\unicode[STIX]{x1D708}_{j}(n(k)-n(j))}p^{m\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})((1/2)c_{1,l}+c_{2,l})}X^{v(\unicode[STIX]{x1D708})}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\prod }_{j=1}^{d}\left(\mathop{\sum }_{\unicode[STIX]{x1D707}_{i_{j}}=1}^{\infty }p^{(2\unicode[STIX]{x1D707}_{i_{j}}-1)B_{j}(\mathbf{I}_{\mathbf{l}})}p^{-2(2\unicode[STIX]{x1D707}_{i_{j}}-1)A_{j}(\mathbf{I}_{\mathbf{l}})s}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \mathop{\prod }_{\substack{ j=0 \\ j\notin \{i_{1},\ldots ,i_{d}\}}}^{k-1}\left(\mathop{\sum }_{\unicode[STIX]{x1D707}_{j}=1}^{\infty }p^{2\unicode[STIX]{x1D707}_{j}B_{k}(\mathbf{I}_{\mathbf{l}})}p^{-4\unicode[STIX]{x1D707}_{j}A_{k}(\mathbf{I}_{\mathbf{l}})s}\right)\nonumber\\ \displaystyle & & \displaystyle \quad =p^{\mathop{\sum }_{j=1}^{d}2A_{i_{j}}(\mathbf{I}_{\mathbf{l}})s-B_{i_{j}}(\mathbf{I}_{\mathbf{l}})}\mathop{\prod }_{j=1}^{k}(\unicode[STIX]{x1D701}_{p}(4A_{j}(\mathbf{I}_{\mathbf{l}})s-2B_{j}(\mathbf{I}_{\mathbf{l}}))-1).\end{eqnarray}$$

To state the aforementioned theorem, we adopt the notation from Section 3.4. Based on Lemma 3.9, we use the abbreviation

$$\begin{eqnarray}\mathop{\sum }_{(\unicode[STIX]{x1D70E},I,k)}=\mathop{\sum }_{\substack{ \unicode[STIX]{x1D70E}\in S_{n} \\ \unicode[STIX]{x1D70E}^{2}=\operatorname{id}}}\mathop{\sum }_{I=I_{0}\cup \cdots \cup I_{s}}\mathop{\sum }_{k=0}^{s+1},\end{eqnarray}$$

put

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D705}_{p}(\unicode[STIX]{x1D70E},I,k) & = & \displaystyle 2^{-c_{1}(\unicode[STIX]{x1D70E})}(1-p^{-1})^{c_{1}(\unicode[STIX]{x1D70E})+c_{2}(\unicode[STIX]{x1D70E})}p^{-c_{2}(\unicode[STIX]{x1D70E})}p^{-\unicode[STIX]{x1D70F}(\{I_{i}\})-t(\unicode[STIX]{x1D70E},\{I_{i}\})}\nonumber\\ \displaystyle & & \displaystyle \times \,\frac{2^{c_{1}^{(k)}(\unicode[STIX]{x1D70E})}2^{\mathop{\sum }_{l=0}^{k-1}c_{1,l}}(1-p^{-1})^{c_{1}^{(k)(\unicode[STIX]{x1D70E})}}p^{-\mathop{\sum }_{l=k+1}^{s}n(l)}p^{\mathop{\sum }_{l=0}^{k-1}(c_{1,l}+2c_{2,l})}}{\mathop{\prod }_{l=k}^{s}(1-p^{-n(l)})},\nonumber\end{eqnarray}$$

and further introduce

(3.65) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D6EC}_{p}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D708},q)\nonumber\\ \displaystyle & & \displaystyle \quad =\left\{\begin{array}{@{}ll@{}}0\quad & \text{if}~L^{\prime }/L\cong \mathbb{Z}/p\mathbb{Z},\\ \displaystyle (-1)^{\mathop{\sum }_{l\in e(\unicode[STIX]{x1D708})_{k}}c_{1,l}}\left(\frac{-1}{p}\right)^{((m-2)/2)\mathop{\sum }_{l\in o(\unicode[STIX]{x1D708})_{k}}c_{1,l}}\quad & \\ \displaystyle \quad \times \left(\frac{2}{p}\right)^{\mathop{\sum }_{l\in e(\unicode[STIX]{x1D708})_{k}}c_{1,l}}p^{m\mathop{\sum }_{l=0}^{k-1}S_{l}^{(k)}(\unicode[STIX]{x1D708})((1/2)c_{1,l}+c_{2,l})}\quad & \text{if}~L^{\prime }/L\cong (\mathbb{Z}/p\mathbb{Z})^{2},\end{array}\right.\end{eqnarray}$$

(3.66) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E5}_{p}(\unicode[STIX]{x1D70E},q) & = & \displaystyle \mathop{\sum }_{0\leqslant i_{1}\leqslant \ldots \leqslant i_{d}\leqslant n}(-1)^{\mathop{\sum }_{l\in e(i_{1},\ldots ,i_{d})_{k}}c_{1,l}}\nonumber\\ \displaystyle & & \displaystyle \times \left(\frac{-1}{p}\right)^{((m-2)/2)\mathop{\sum }_{l\in o(i_{1},\ldots ,i_{d})_{k}}c_{1,l}}\left(\frac{2}{p}\right)^{\mathop{\sum }_{l\in e(i_{1},\ldots ,i_{d})_{k}}c_{1,l}}\nonumber\\ \displaystyle & & \displaystyle \times \,p^{\mathop{\sum }_{j=1}^{d}2A_{i_{j}}(\mathbf{I}_{\mathbf{l}})s-B_{i_{j}}(\mathbf{I}_{\mathbf{l}})},\end{eqnarray}$$

if $L^{\prime }/L\cong (\mathbb{Z}/p\mathbb{Z})^{2}$ and zero if $L^{\prime }/L\cong \mathbb{Z}/p\mathbb{Z}$ .

Theorem 3.12. Let $(L,(\cdot ,\cdot ))$ be an even lattice of even rank $m$ such that the discriminant form $L^{\prime }/L$ is anisotropic. Moreover, let $p$ be an odd prime dividing $|L^{\prime }/L|$ , $\unicode[STIX]{x1D707}\in (L^{\prime }/L)^{n}$ , and $\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}$ as defined in (3.23). Then

(3.67) $$\begin{eqnarray}\displaystyle M_{p}(s)\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(1,s) & = & \displaystyle \frac{\unicode[STIX]{x1D6FE}_{p}(V)^{n}}{|L_{p}^{\prime }/L_{p}|^{n/2}}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{(\unicode[STIX]{x1D70E},I,k)}\unicode[STIX]{x1D705}_{p}(\unicode[STIX]{x1D70E},I,k)\mathop{\sum }_{\{\unicode[STIX]{x1D708}\}_{k}}p^{\mathop{\sum }_{j=0}^{k-1}\unicode[STIX]{x1D708}_{j}(n(k)-n(j))}\unicode[STIX]{x1D6EC}_{p}(\unicode[STIX]{x1D70E},\unicode[STIX]{x1D708},q)X^{v(\unicode[STIX]{x1D708})}\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(1,-s)\end{eqnarray}$$

for all $s\in \mathbb{C}$ apart from poles on both sides, where $L_{p}=L\otimes \mathbb{Z}_{p}$ and $X=p^{-(s-s_{0})}$ with $s_{0}=(m/2)-\unicode[STIX]{x1D70C}_{n}$ .

Proof. Let $L_{p,r}=L_{p}\oplus \mathbb{Z}_{p}^{2r}$ . Associate to the $p$ -part of $\unicode[STIX]{x1D707}$ the corresponding element in $(L_{p}^{\prime }/L_{p})^{n}$ as indicated in Remark 3.2 (which we also call  $\unicode[STIX]{x1D707}$ ). From Remark 3.2 and [Reference Miranda and MorrisonMM, VI], it also follows that $(L_{p},(\cdot ,\cdot ))$ is non-degenerate and $L_{p}^{\prime }/L_{p}$ is anisotropic.

According to [Reference KudlaKu2, Lemma A.3 and Proposition A.4] and [Reference Kudla and YangKY, Section 4.1], we have for all integers $r\geqslant \unicode[STIX]{x1D70C}_{n}$

(3.68) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\operatorname{Sym}_{n}(\mathbb{Q}_{p})}\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(wn(b),s_{0}+r)\,db\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D6FE}_{p}(V)^{n}\int _{\operatorname{Sym}_{n}(\mathbb{Q}_{p})}\int _{\unicode[STIX]{x1D707}+L_{p,r}^{n}}\unicode[STIX]{x1D713}_{p}\left(\frac{1}{2}\operatorname{tr}(bQ^{r}[x])\right)dx\,db,\end{eqnarray}$$

where $Q^{r}[x]$ is defined by (3.13) based on the quadratic form $q_{r}$ (see (3.40)). The integral on the right-hand side of (3.68) can be evaluated by means of Lemma 3.9. Separating in (3.45) all parts depending on $r$ from the rest, we obtain the formula in (3.67) for $s=r$ , $r>\unicode[STIX]{x1D70C}_{n}$ , using the fact that $\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(1,-s)=\unicode[STIX]{x1D6FF}_{\unicode[STIX]{x1D707},0}$ . Lemma 3.11 guarantees that the right-hand side of (3.67) can be meromorphically continued to the complex plane. More specifically, from the explicit representation as a sum of finite products of local zeta functions, it follows that the right-hand side is a meromorphic function in $p^{-s}$ on the complex plane. By [Reference Kudla and RallisKR, Section 4], the same holds for $M_{p}(s)$ . Combining these facts, we obtain (3.67) by analytic continuation (see also [Reference LiuLi2, p. 950]).◻

We now make some special choices for $g_{0}=(g_{\unicode[STIX]{x1D70F}},g_{f})\in G(\mathbb{A})$ in order to define a vector valued Eisenstein series. To specify $g_{\unicode[STIX]{x1D70F}}$ , let $\unicode[STIX]{x1D70F}=u+iv\in \mathbb{H}_{n}$ . Associated to $\unicode[STIX]{x1D70F}$ , we set

(3.69) $$\begin{eqnarray}g_{\unicode[STIX]{x1D70F}}=n(u)m(a)\in \operatorname{Sp}(n,\mathbb{R})\end{eqnarray}$$

with $^{t}aa=v$ . For all the finite places, we put $g_{f}=(1,1,\ldots ,)$ . Moreover, let $L$ be a lattice as introduced in Section 3.1.

Definition 3.13. Let $l\in 2\mathbb{Z}$ satisfy the condition $2l-b^{-}+b^{+}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}4$ , $\unicode[STIX]{x1D707}\in (L^{\prime }/L)^{n}$ and $\mathfrak{e}_{0}\in \mathbb{C}[(L^{\prime }/L)^{n}]$ .

1. (1) For $g_{0}=(g_{\unicode[STIX]{x1D70F}},g_{f})\in G(\mathbb{A})$ as above and $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D707}}^{l}(s)$ the standard section in Definition 3.3, we define an $S_{L}$ -valued Eisenstein series by

   (3.70) $$\begin{eqnarray}E_{L}(\unicode[STIX]{x1D70F},s,l)=\det (\operatorname{Im}(\unicode[STIX]{x1D70F}))^{-l/2}\mathop{\sum }_{\unicode[STIX]{x1D707}\in (L^{\prime }/L)^{n}}E(g_{0},s,\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D707}}^{l})\unicode[STIX]{x1D711}_{\unicode[STIX]{x1D707}}.\end{eqnarray}$$

2. (2) Further, using the notation $\unicode[STIX]{x1D6E4}_{\infty }=\left\{\left(\!\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\!\right)\in \operatorname{Sp}(n,\mathbb{Z})\mid c=0\right\}$ and $J\left(\left(\!\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\!\right),\unicode[STIX]{x1D70F}\right)=c\unicode[STIX]{x1D70F}+d$ , we define by

   (3.71) $$\begin{eqnarray}E_{l,0}^{n}(\unicode[STIX]{x1D70F},s)=\det (\operatorname{Im}(\unicode[STIX]{x1D70F}))^{s}\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{\infty }\setminus \operatorname{Sp}(n,\mathbb{Z})}|\det (J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F}))|^{-2s}\det (J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F}))^{-l}\unicode[STIX]{x1D70C}_{L,n}^{-1}(\unicode[STIX]{x1D6FE})\mathfrak{e}_{0}\end{eqnarray}$$
   a vector valued non-holomorphic Siegel Eisenstein series $E_{l,0}^{n}$ of weight $l\in 2\mathbb{Z}$ transforming with the “finite” Weil representation  $\unicode[STIX]{x1D70C}_{L,n}$ . We denote by $E_{l,0}^{n\ast }(\unicode[STIX]{x1D70F},s)$ the Eisenstein series in (3.71) with respect to the dual Weil representation $\unicode[STIX]{x1D70C}_{L,n}^{\ast }$ analogous to the one defined in [Reference Bruinier and KühnBK, Section 3].

The next lemma states that there is a close relation between both Eisenstein series in Definition 3.13.

Lemma 3.14. Let $E_{L}(\unicode[STIX]{x1D70F},s,l)$ be the Eisenstein series in (3.70) and $E_{l,0}^{n}(\unicode[STIX]{x1D70F},s)$ be the vector valued Eisenstein series in (3.71). Then the relation

(3.72) $$\begin{eqnarray}E_{L}(\unicode[STIX]{x1D70F},s,l)=E_{l,0}^{n}(\unicode[STIX]{x1D70F},(s+\unicode[STIX]{x1D70C}_{n}-l)/2)\end{eqnarray}$$

holds.

Proof. The proof in [Reference Bruinier and YangBY, Section 2.2], can be generalized to our situation. The computation for the archimedean part of $\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D707}}^{l}$ is the same as in [Reference KudlaKu1, Section IV.2].◻

Remark 3.15. Using the scaled lattice $L(-1)$ instead of $L$ , identity (3.72) reads as follows:

(3.73) $$\begin{eqnarray}E_{L(-1)}(\unicode[STIX]{x1D70F},s,l)=E_{l,0}^{n\ast }(\unicode[STIX]{x1D70F},(s+\unicode[STIX]{x1D70C}_{n}-l)/2).\end{eqnarray}$$

The next theorem provides an explicit functional equation for the vector valued Eisenstein series $E_{L}(\unicode[STIX]{x1D70F},s,l)$ . To this end, we define for a prime  $p$

(3.74) $$\begin{eqnarray}\begin{array}{@{}l@{}}\displaystyle a_{n,p}(s)=L_{p}(s+\unicode[STIX]{x1D70C}_{n}-n,\unicode[STIX]{x1D712}_{V,p})\mathop{\prod }_{k=1}^{[n/2]}\unicode[STIX]{x1D701}_{p}(2s-n+2k),\\ \displaystyle b_{n,p}(s)=L_{p}(s+\unicode[STIX]{x1D70C}_{n},\unicode[STIX]{x1D712}_{V,p})\mathop{\prod }_{k=1}^{[n/2]}\unicode[STIX]{x1D701}_{p}(2s+n-2k+1),\\ \displaystyle D_{p,j}(s)=\unicode[STIX]{x1D701}_{p}(4A_{j}(\mathbf{I}_{\mathbf{l}})(s-s_{0})-2B_{j}(\mathbf{I}_{\mathbf{l}}))\\ \qquad (\text{see Lemma 3.11 for the notation}~A_{i}(\mathbf{I}_{\mathbf{l}}),B_{i}(\mathbf{I}_{\mathbf{l}}))\end{array}\end{eqnarray}$$

and associate to the set $P=\{q~\text{prime}\mid q\nmid |L^{\prime }/L|\}$

(3.75) $$\begin{eqnarray}a_{n}^{P}(s)=\mathop{\prod }_{p\in P}a_{n,p}(s),\qquad b_{n}^{P}(s)=\mathop{\prod }_{p\in P}b_{n,p}(s).\end{eqnarray}$$

We finally put

(3.76) $$\begin{eqnarray}\unicode[STIX]{x1D709}(s,n,l,P)=\frac{(-1)^{l/2}2^{(1-s)n}\unicode[STIX]{x1D70B}^{n(n+1)/2}}{|L^{\prime }/L|^{n/2}}\frac{\unicode[STIX]{x1D6E4}_{n}(s)}{\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FD})}\frac{a_{n}^{P}(s)}{b_{n}^{P}(s)}.\end{eqnarray}$$

Theorem 3.16. Let $E_{L}(\unicode[STIX]{x1D70F},s,l)$ be the Eisenstein series in Definition 3.13. Then $E_{L}(\unicode[STIX]{x1D70F},s,l)$ can be continued meromorphically to the whole complex $s$ -plane. If additionally $L^{\prime }/L$ is anisotropic and $|L^{\prime }/L|$ is odd, it satisfies the functional equation

(3.77) $$\begin{eqnarray}E_{L}(\unicode[STIX]{x1D70F},s,l)=\unicode[STIX]{x1D709}(s,n,l,P)\!\mathop{\prod }_{p\mid |L^{\prime }/L|}\,\mathop{\sum }_{(\unicode[STIX]{x1D70E},I,k)}\!\unicode[STIX]{x1D705}_{p}(\unicode[STIX]{x1D70E},I,k)\unicode[STIX]{x1D6E5}_{p}(\unicode[STIX]{x1D70E},q)\mathop{\prod }_{j=1}^{k}(D_{p,j}(s)-1)E_{L}(\unicode[STIX]{x1D70F},-s,l).\!\end{eqnarray}$$

Proof. The proof follows the one of [Reference Bruinier and YangBY, Proposition 2.5].

Clearly, since each scalar valued Eisenstein series $E(g_{0},s,\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D707}}^{l})$ can be meromorphically continued to the whole complex $s$ -plane, the same holds for $E_{L}(\unicode[STIX]{x1D70F},s,l)$ .

It suffices to prove that the functional equation holds for each $E(g_{0},s,\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D707}}^{l})$ . By (3.51), we have

$$\begin{eqnarray}E(g_{0},s,\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D707}}^{l})=E(g_{0},-s,M(s)\unicode[STIX]{x1D6F7}_{\unicode[STIX]{x1D707}}^{l}).\end{eqnarray}$$

If $p=\infty$ , Theorem 3.10 gives

(3.78) $$\begin{eqnarray}\displaystyle M_{\infty }(s)\unicode[STIX]{x1D6F7}_{\infty }^{l}(\unicode[STIX]{x1D6FE}g_{\unicode[STIX]{x1D70F}},s) & = & \displaystyle i^{-ln}2^{(1-s)n}\unicode[STIX]{x1D70B}^{n(n+1)/2}\frac{\unicode[STIX]{x1D6E4}_{n}(s)}{\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FD})}\unicode[STIX]{x1D6F7}_{\infty }^{l}(\unicode[STIX]{x1D6FE}g_{\unicode[STIX]{x1D70F}},-s)\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D6FE}_{\infty }(V)^{n}(-1)^{l/2}2^{(1-s)n}\unicode[STIX]{x1D70B}^{n(n+1)/2}\frac{\unicode[STIX]{x1D6E4}_{n}(s)}{\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FC})\unicode[STIX]{x1D6E4}_{n}(\unicode[STIX]{x1D6FD})}\unicode[STIX]{x1D6F7}_{\infty }^{l}(\unicode[STIX]{x1D6FE}g_{\unicode[STIX]{x1D70F}},-s),\end{eqnarray}$$

where the last equation comes from $2l-b^{-}+b^{+}\equiv 0\hspace{0.2em}{\rm mod}\hspace{0.2em}4$ and the fact that $l\in 2\mathbb{Z}$ .

If $p$ is a prime not dividing $|L^{\prime }/L|$ , $\unicode[STIX]{x1D712}_{V,p}$ is unramified and $\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}$ is the spherical standard section by Lemma 3.4. By [Reference Kudla and RallisKR, Lemma 4.1], we then have

(3.79) $$\begin{eqnarray}M_{p}(s)\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(g,s)=\frac{a_{n,p}(s)}{b_{n,p}(s)}\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(g,-s)\end{eqnarray}$$

for all $g\in \operatorname{Sp}(n,\mathbb{Q}_{p})$ .

Now we consider the case of an odd prime $p$ dividing $|L^{\prime }/L|$ . The local form (3.21) of $\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}$ and formulas (3.25), (3.26), and (3.27) yield

(3.80) $$\begin{eqnarray}\begin{array}{@{}l@{}}\displaystyle \unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(gn(b))=\unicode[STIX]{x1D713}_{p}(\operatorname{tr}(bQ[\unicode[STIX]{x1D707}]))\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(g),\quad b\in \operatorname{Sym}_{n}(\mathbb{Z}_{p})\\ \displaystyle \unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(gw)=\unicode[STIX]{x1D6FE}_{p}(V)^{n}|L_{p}^{\prime }/L_{p}|^{-n/2}\mathop{\sum }_{\unicode[STIX]{x1D708}\in (L_{p}^{\prime }/L_{p})^{n}}\unicode[STIX]{x1D713}_{p}(\operatorname{tr}(\unicode[STIX]{x1D708},\unicode[STIX]{x1D707}))\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D708})}(g)\\ \displaystyle \unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(gm(a))=\unicode[STIX]{x1D712}_{V,p}(\det a)|\text{det}\,a|_{p}^{m/2}\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707}a^{-1})}(g),\quad a\in \operatorname{GL}_{n}(\mathbb{Z}_{p})\end{array}\end{eqnarray}$$

for any $g\in \operatorname{Sp}(n,\mathbb{Q}_{p})$ . The definition of the intertwining operator then immediately implies that $M_{p}(s)$ satisfies the same equations as (3.25), (3.26), and (3.27). So, by Lemma 3.1, we arrive at

(3.81) $$\begin{eqnarray}M_{p}(s)\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(k,s)=C_{p}\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(k,-s)\end{eqnarray}$$

for all $k\in \operatorname{Sp}(n,\mathbb{Z}_{p})$ , where $C_{p}$ is given by the right-hand side of (3.67). Clearly, since $M_{p}(s)\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}\in I_{n}(-s,\unicode[STIX]{x1D712}_{V,p})$ , the same equation holds for all $g\in \operatorname{Sp}(n,\mathbb{Q}_{p})$ . Taking Theorem 3.12 and notation (3.66), (3.58), and (3.64) respectively into account, we finally get

(3.82) $$\begin{eqnarray}M_{p}(s)\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(g,s)=\frac{\unicode[STIX]{x1D6FE}_{p}(V)^{n}}{|L_{p}^{\prime }/L_{p}|^{n/2}}\mathop{\sum }_{(\unicode[STIX]{x1D70E},I,k)}\unicode[STIX]{x1D705}_{p}(\unicode[STIX]{x1D70E},I,k)\unicode[STIX]{x1D6E5}_{p}(\unicode[STIX]{x1D70E},q)\mathop{\prod }_{j=1}^{k}(D_{p,j}(s)-1)\unicode[STIX]{x1D6F7}_{p}^{(\unicode[STIX]{x1D707})}(g,-s)\end{eqnarray}$$

for all $g\in \operatorname{Sp}(n,\mathbb{Q}_{p})$ .

Combining (3.78), (3.79), and (3.82) together with the identities

$$\begin{eqnarray}\mathop{\prod }_{p\mid |L^{\prime }/L|}|L_{p}^{\prime }/L_{p}|=|L^{\prime }/L|,\qquad \unicode[STIX]{x1D6FE}_{\infty }(V)^{n}\mathop{\prod }_{p<\infty }\unicode[STIX]{x1D6FE}_{p}(V)^{n}=1,\end{eqnarray}$$

the claimed functional equation follows. ◻

With the help of identity (3.72), we can immediately deduce that $E_{l,0}^{n}(\unicode[STIX]{x1D70F},s)$ possesses the same analytic properties as $E_{L}(\unicode[STIX]{x1D70F},s,l)$ . In particular, we can deduce a functional equation for the Eisenstein series $E_{l,0}^{n}(\unicode[STIX]{x1D70F},s)$ .

Corollary 3.17. Let $E_{l,0}^{n}(\unicode[STIX]{x1D70F},s)$ be the Eisenstein series as in Definition 3.13 with respect to  $\unicode[STIX]{x1D70C}_{L,n}$ . Then $E_{l,0}^{n}(\unicode[STIX]{x1D70F},s)$ has a meromorphic continuation in $s$ to the whole complex plane. If $L^{\prime }/L$ is anisotropic and $|L^{\prime }/L|$ is odd, it satisfies the functional equation

(3.83) $$\begin{eqnarray}\displaystyle E_{l,0}^{n}\left(\unicode[STIX]{x1D70F},s-\frac{l}{2}\right) & = & \displaystyle \unicode[STIX]{x1D709}(2s-\unicode[STIX]{x1D70C}_{n},n,l,P)\mathop{\prod }_{p\mid |L^{\prime }/L|}\mathop{\sum }_{(\unicode[STIX]{x1D70E},I,k)}\unicode[STIX]{x1D705}_{p}(\unicode[STIX]{x1D70E},I,k)\unicode[STIX]{x1D6E5}_{p}(\unicode[STIX]{x1D70E},q)\nonumber\\ \displaystyle & & \displaystyle \times \mathop{\prod }_{j=1}^{k}(D_{p,j}(2s-\unicode[STIX]{x1D70C}_{n})-1)E_{l,0}^{n}\left(\unicode[STIX]{x1D70F},\unicode[STIX]{x1D70C}_{n}-s-\frac{l}{2}\right).\end{eqnarray}$$

Proof. The assertion results immediately from the functional equation of the adelic Eisenstein series $E_{L}(\unicode[STIX]{x1D70F},s,l)$ and relation (3.72).◻

Remark 3.18. Let $L(-1)$ be the lattice as defined at the end of Section 3.3. It is easily checked that the integrals in Lemma 3.8 do not change by passing from $q$ to $-q$ if $L^{\prime }/L$ is isomorphic to $(\mathbb{Z}/p\mathbb{Z})^{2}$ . Hence, the same holds for the formula in Lemma 3.9, which in turn implies that $E_{L(-1)}(\unicode[STIX]{x1D70F},s,l)$ satisfies the same functional equation as $E_{L}(\unicode[STIX]{x1D70F},s,l)$ . Taking additionally (3.73) into account, we can deduce that $E_{l,0}^{n\ast }(\unicode[STIX]{x1D70F},s)$ obeys the same functional equation as $E_{l,0}^{n}(\unicode[STIX]{x1D70F},s)$ .

4.1 Jacobi Eisenstein series

In order to define a Jacobi Eisenstein series, we need to introduce some notation and some basic facts about Jacobi forms, which we take from [Reference ZieglerZi, Section 1] and [Reference ArakawaAr2, Section 1].

The Jacobi group $G_{n,m}$ is the semidirect product of the Heisenberg group

$$\begin{eqnarray}H_{n,m}=\{[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\mid \unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in M_{m,n},\unicode[STIX]{x1D705}\in M_{m,m}~\text{where}~(\unicode[STIX]{x1D705}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D706}^{t})\in \operatorname{Sym}_{m}\}\end{eqnarray}$$

and $\operatorname{Sp}(n)$ , that is,

$$\begin{eqnarray}G_{n,m}=H_{n,m}\rtimes \operatorname{Sp}(n).\end{eqnarray}$$

Note that the group law on $H_{n,m}$ is given by

$$\begin{eqnarray}[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\cdot [(\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D707}^{\prime }),\unicode[STIX]{x1D705}^{\prime }]=[(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D707}+\unicode[STIX]{x1D707}^{\prime }),\unicode[STIX]{x1D705}+\unicode[STIX]{x1D705}^{\prime }+\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}^{\prime t}-\unicode[STIX]{x1D707}\unicode[STIX]{x1D706}^{\prime t}]\end{eqnarray}$$

and $\operatorname{Sp}(n)$ acts on the Heisenberg group by multiplication from the left

$$\begin{eqnarray}[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\circ M=[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})M,\unicode[STIX]{x1D705}].\end{eqnarray}$$

The product of two elements $([(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}],M),([(\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D708}^{\prime }),\unicode[STIX]{x1D705}^{\prime }],M^{\prime })\in G_{n,m}$ reads accordingly as

$$\begin{eqnarray}([(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}],M)\cdot ([(\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D708}^{\prime }),\unicode[STIX]{x1D705}^{\prime }],M^{\prime })=([(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})M^{\prime },\unicode[STIX]{x1D705}]\cdot [(\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D707}^{\prime }),\unicode[STIX]{x1D705}^{\prime }],MM^{\prime }).\end{eqnarray}$$

Note that $G_{n,m}$ can be interpreted as a subgroup of $\operatorname{Sp}(n+m)$ using suitable embeddings. Details can be found in [Reference ZieglerZi] and [Reference ArakawaAr2].

In accordance with (3.71), we set for $M=\left(\!\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\!\right)\in \operatorname{Sp}(n,\mathbb{R})$ and $\unicode[STIX]{x1D70F}\in \mathbb{H}_{n}$

$$\begin{eqnarray}M\langle \unicode[STIX]{x1D70F}\rangle =(a\unicode[STIX]{x1D70F}+b)(c\unicode[STIX]{x1D70F}+d)^{-1}\qquad \text{and}\qquad J(M,\unicode[STIX]{x1D70F})=c\unicode[STIX]{x1D70F}+d.\end{eqnarray}$$

In terms of this notation, we can define an action of the real Jacobi group $G_{n,m}(\mathbb{R})$ on

$$\begin{eqnarray}{\mathcal{H}}_{n,m}=\mathbb{H}_{n}\times M_{m,n}(\mathbb{C})\end{eqnarray}$$

by

(4.1) $$\begin{eqnarray}g(\unicode[STIX]{x1D70F},z):=(M\langle \unicode[STIX]{x1D70F}\rangle ,zJ(M,\unicode[STIX]{x1D70F})^{-1}+\unicode[STIX]{x1D706}M\langle \unicode[STIX]{x1D70F}\rangle +\unicode[STIX]{x1D707}),\end{eqnarray}$$

where $g=([(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}],M)\in G_{n,m}(\mathbb{R})$ and $(\unicode[STIX]{x1D70F},z)\in {\mathcal{H}}_{n,m}$ . Let $S=(s_{ij})_{i,j}\in \operatorname{Sym}_{m}(\mathbb{Q})$ be a positive definite half integral matrix, which means that $2s_{ij},s_{ii}\in \mathbb{Z}$ . Attached to $S$ we have a symmetric non-degenerate bilinear form on the lattice  $\mathbb{Z}^{m}$ :

(4.2) $$\begin{eqnarray}(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})=2S(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707})\end{eqnarray}$$

with the corresponding quadratic form

$$\begin{eqnarray}q(\unicode[STIX]{x1D706})=S[\unicode[STIX]{x1D706}]\end{eqnarray}$$

for $\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in \mathbb{Z}^{m}$ ; see also (2.1).

Associated to $S$ and a positive integer $l$ , we specify a factor of automorphy

(4.3) $$\begin{eqnarray}\displaystyle & & \displaystyle J_{S,l}(g,(\unicode[STIX]{x1D70F},z))\nonumber\\ \displaystyle & & \displaystyle \quad =\det (J(M,\unicode[STIX]{x1D70F}))^{l}e(-\operatorname{tr}(S\unicode[STIX]{x1D705})-\operatorname{tr}(S[\unicode[STIX]{x1D706}]M\langle \unicode[STIX]{x1D70F}\rangle +2S(\unicode[STIX]{x1D706},zJ(M,\unicode[STIX]{x1D70F})^{-1})-S[z]J(M,\unicode[STIX]{x1D70F})^{-1}c))\end{eqnarray}$$

and a slash operator in terms of $J_{S,l}$ for a function $f:{\mathcal{H}}_{n,m}\rightarrow \mathbb{C}$ and $g\in G_{m,n}(\mathbb{R})$ by

$$\begin{eqnarray}(f\mid _{l,S}g)(\unicode[STIX]{x1D70F},z)=J_{S,l}(g,(\unicode[STIX]{x1D70F},z))^{-1}f(g(\unicode[STIX]{x1D70F},z)).\end{eqnarray}$$

For later purposes, we introduce the concept of a Jacobi form of higher degree. Let

$$\begin{eqnarray}H_{n,m}(\mathbb{Z})=\{[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\mid \unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in M_{m,n}(\mathbb{Z}),\unicode[STIX]{x1D705}\in M_{m,m}(\mathbb{Z})~\text{where}~(\unicode[STIX]{x1D705}+\unicode[STIX]{x1D707}\unicode[STIX]{x1D706}^{t})\in \operatorname{Sym}_{m}(\mathbb{Z})\}\end{eqnarray}$$

and

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{n,m}=H_{n,m}(\mathbb{Z})\rtimes \operatorname{Sp}(n,\mathbb{Z}).\end{eqnarray}$$

We call a holomorphic function $f:{\mathcal{H}}_{n,m}\rightarrow \mathbb{C}$ a Jacobi form of weight  $l$ and index  $S$ if $f$ fulfills the following conditions:

1. (i) $(f\mid _{l,S}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D70F},z)=f(\unicode[STIX]{x1D70F},z)$ for all $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{n,m}$ and $(\unicode[STIX]{x1D70F},z)\in {\mathcal{H}}_{n,m}$ .

2. (ii) $f(\unicode[STIX]{x1D70F},z)$ has a Fourier–Jacobi expansion of the form

   (4.4) $$\begin{eqnarray}f(\unicode[STIX]{x1D70F},z)=\mathop{\sum }_{\substack{ N\in \operatorname{Sym}_{n}^{\ast }(\mathbb{Z}),r\in M_{m,n}(\mathbb{Z}) \\ N-(1/4)r^{t}S^{-1}r\geqslant 0}}c(N,r)e(\operatorname{tr}(N\unicode[STIX]{x1D70F}+r^{t}z)),\end{eqnarray}$$
   where $\operatorname{Sym}_{n}^{\ast }(\mathbb{Z})$ is the set of all half integral matrices in $\operatorname{Sym}_{n}(\mathbb{R})$ .

The condition on the Fourier expansion of $f$ is only necessary in the case of $n=1$ . We denote by $J_{l,S}(\unicode[STIX]{x1D6E4}_{n,m})$ the space of all Jacobi forms of weight  $l$ and index  $S$ . As usual, $f\in J_{l,S}(\unicode[STIX]{x1D6E4}_{n,m})$ is a cusp form if it possesses a Fourier expansion where the condition $N-(1/4)r^{t}S^{-1}r\geqslant 0$ can be replaced by $N-(1/4)r^{t}S^{-1}r>0$ . By $J_{l,S}^{\text{cusp}}(\unicode[STIX]{x1D6E4}_{m,n})$ we mean the subspace of all cusp forms.

Next, denote by $\unicode[STIX]{x1D6E4}_{n,m}^{\infty }$ the subgroup of $\unicode[STIX]{x1D6E4}_{n,m}$ given by

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{n,m}^{\infty }=\left\{\left([(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}],\left(\begin{array}{@{}cc@{}}a & b\\ c & d\end{array}\right)\right)\in \unicode[STIX]{x1D6E4}_{n,m}\mid c=0,\unicode[STIX]{x1D706}=0\right\}.\end{eqnarray}$$

For an even integer $l\in \mathbb{N}$ and $(\unicode[STIX]{x1D70F},z)\in {\mathcal{H}}_{n,m}$ , we define the real analytic Jacobi Eisenstein series

(4.5) $$\begin{eqnarray}E_{l,S}^{(n)}(\unicode[STIX]{x1D70F},z;s)=\det (\operatorname{Im}(\unicode[STIX]{x1D70F}))^{s}\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{n,m}^{\infty }\backslash \unicode[STIX]{x1D6E4}_{n,m}}J_{S,l}(\unicode[STIX]{x1D6FE},(\unicode[STIX]{x1D70F},z))^{-1}|\det (J(M,\unicode[STIX]{x1D70F}))|^{-2s},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}=([(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}],M)$ . It can be proved that $E_{l,S}^{(n)}$ converges absolutely for $\operatorname{Re}(2s+l)>n+m+1$ ; see [Reference ZieglerZi, Theorem 2.1]. A complete set of coset representatives of $\unicode[STIX]{x1D6E4}_{n,m}^{\infty }\backslash \unicode[STIX]{x1D6E4}_{n,m}$ consists of

(4.6) $$\begin{eqnarray}\left\{\left.\left([(\unicode[STIX]{x1D706}a,\unicode[STIX]{x1D706}b),0],\left(\begin{array}{@{}cc@{}}a & b\\ c & d\end{array}\right)\right)\right|\left(\begin{array}{@{}cc@{}}a & b\\ c & d\end{array}\right)\in \unicode[STIX]{x1D6E4}_{\infty }\backslash \operatorname{Sp}(n,\mathbb{Z})~\text{and}~\unicode[STIX]{x1D706}\in M_{m,n}(\mathbb{Z})\right\};\end{eqnarray}$$

see [Reference ZieglerZi, Section 2].

In order to state the above-mentioned relation between both Eisenstein series, we need to introduce some theta series. For $\unicode[STIX]{x1D707}\in D_{m}(S)=((2S)^{-1}\mathbb{Z}^{m}/\mathbb{Z}^{m})^{n}$ ( $\cong (2S)^{-1}M_{m,n}(\mathbb{Z})/M_{m,n}(\mathbb{Z})$ ), the series

(4.7) $$\begin{eqnarray}\unicode[STIX]{x1D703}_{S,\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F},z)=\mathop{\sum }_{\unicode[STIX]{x1D706}\in M_{m,n}(\mathbb{Z})}e(\operatorname{tr}(S[\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}]\unicode[STIX]{x1D70F}+2S(\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707},z)))\end{eqnarray}$$

is analogous to the theta series as it is known for the classical Jacobi forms and which can be found for example in [Reference ArakawaAr1, Section 1]. It converges normally on ${\mathcal{H}}_{n,m}$ and depends only on $\unicode[STIX]{x1D707}\hspace{0.2em}{\rm mod}\hspace{0.2em}M_{m,n}(\mathbb{Z})$ . The transformation behavior of $\unicode[STIX]{x1D6E9}_{S}(\unicode[STIX]{x1D70F},z)=\sum _{\unicode[STIX]{x1D707}\in D_{m}(s)}\unicode[STIX]{x1D703}_{S,\unicode[STIX]{x1D707}}(\unicode[STIX]{x1D70F},z)\mathfrak{e}_{\unicode[STIX]{x1D707}}$ under the action of the Jacobi group is well known. We have (see [Reference ZieglerZi, Lemma 3.2] or [Reference RungeRu, p. 168])

(4.8) $$\begin{eqnarray}\unicode[STIX]{x1D6E9}_{S}(M(\unicode[STIX]{x1D70F},z))=e(\operatorname{tr}(S[z]J(M,\unicode[STIX]{x1D70F})^{-1}c))\det (J(M,\unicode[STIX]{x1D70F}))^{m/2}\unicode[STIX]{x1D70C}_{L,n}(M)\unicode[STIX]{x1D6E9}_{S}(\unicode[STIX]{x1D70F},z),\end{eqnarray}$$

where $M=\left(\!\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\!\right)$ , $M(\unicode[STIX]{x1D70F},z)=(M\langle \unicode[STIX]{x1D70F}\rangle ,zJ(M,\unicode[STIX]{x1D70F})^{-1})$ (apply (4.1) to $([(0,0),0],M)$ ) and $\unicode[STIX]{x1D70C}_{L,n}$ is the Weil representation in (3.30) for the lattice $L=\mathbb{Z}^{m}$ equipped with the bilinear form (4.2).

Proposition 4.1. Let $E_{l,S}^{(n)}(\unicode[STIX]{x1D70F},z,s)$ be the real analytic Jacobi Eisenstein series, $E_{l,0}^{n\ast }(\unicode[STIX]{x1D70F},s)$ the real analytic vector valued Eisenstein series (3.71) with respect to $\unicode[STIX]{x1D70C}_{L,n}^{\ast }$ , and $\unicode[STIX]{x1D6E9}_{S}(\unicode[STIX]{x1D70F},s)$ the vector valued theta series as introduced before. Then the following relation between both Eisenstein series holds:

(4.9) $$\begin{eqnarray}E_{l,S}^{(n)}(\unicode[STIX]{x1D70F},z;s)=\langle E_{l-(m/2),0}^{n\ast }(\unicode[STIX]{x1D70F},s),\overline{\unicode[STIX]{x1D6E9}_{S}(\unicode[STIX]{x1D70F},z)}\rangle ,\end{eqnarray}$$

where $\langle \cdot ,\cdot \rangle$ is the standard inner product (3.31).

Proof. The proof is a straightforward generalization of the one of [Reference ArakawaAr1, Proposition 3.1]. The Jacobi Eisenstein series $E_{l,S}^{(n)}(\unicode[STIX]{x1D70F},z;s)$ can be written in a more explicit way by making use of the set of coset representatives (4.6). As in [Reference ZieglerZi, Section 2], we obtain for $\unicode[STIX]{x1D6FE}=\left(\!\begin{smallmatrix}a & b\\ c & d\end{smallmatrix}\!\right)$

(4.10) $$\begin{eqnarray}\displaystyle & & \displaystyle E_{l,S}^{(n)}(\unicode[STIX]{x1D70F},z;s)\nonumber\\ \displaystyle & & \displaystyle \quad =\det (\operatorname{Im}(\unicode[STIX]{x1D70F}))^{s}\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{\infty }\backslash \operatorname{Sp}(n,\mathbb{Z})}|\det (J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F}))|^{-2s}\det (J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F}))^{-l}e(-\operatorname{tr}(S[z]J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F})^{-1}c))\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{\unicode[STIX]{x1D706}\in M_{m,n}(\mathbb{Z})}e(\operatorname{tr}(S[\unicode[STIX]{x1D706}]\unicode[STIX]{x1D6FE}\langle \unicode[STIX]{x1D70F}\rangle +2S(\unicode[STIX]{x1D706},zJ(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F})^{-1})))\nonumber\\ \displaystyle & & \displaystyle \quad =\det (\operatorname{Im}(\unicode[STIX]{x1D70F}))^{s}\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{\infty }\backslash \operatorname{Sp}(n,\mathbb{Z})}|\det (J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F}))|^{-2s}\det (J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F}))^{-l}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,e(-\operatorname{tr}(S[z]J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F})^{-1}c))\unicode[STIX]{x1D703}_{S,0}(\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F},z))\nonumber\\ \displaystyle & & \displaystyle \quad =\det (\operatorname{Im}(\unicode[STIX]{x1D70F}))^{s}\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{\infty }\backslash \operatorname{Sp}(n,\mathbb{Z})}|\det (J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F}))|^{-2s}\det (J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F}))^{-l}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,e(-\operatorname{tr}(S[z]J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F})^{-1}c))\langle \mathfrak{e}_{0},\overline{\unicode[STIX]{x1D6E9}_{S}(\unicode[STIX]{x1D6FE}(\unicode[STIX]{x1D70F},z))}\rangle .\end{eqnarray}$$

Employing the transformation formula (4.8) and taking into account the identity $\overline{\unicode[STIX]{x1D70C}_{L,n}}=\unicode[STIX]{x1D70C}_{L,n}^{\ast }$ and that $\unicode[STIX]{x1D70C}_{L,n}^{\ast }$ is unitary with respect to $\langle \cdot ,\cdot \rangle$ , we obtain for the right-hand side of (4.10)

$$\begin{eqnarray}\displaystyle & & \displaystyle \det (\operatorname{Im}(\unicode[STIX]{x1D70F}))^{s}\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{\infty }\backslash \operatorname{Sp}(n,\mathbb{Z})}|\det (J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F}))|^{-2s}\det (J(\unicode[STIX]{x1D6FE},\unicode[STIX]{x1D70F}))^{-l+(m/2)}\langle \unicode[STIX]{x1D70C}_{L,n}(\unicode[STIX]{x1D6FE})\mathfrak{e}_{0},\overline{\unicode[STIX]{x1D6E9}_{S}(\unicode[STIX]{x1D70F},z)}\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =\langle E_{l-(m/2),0}^{n\ast }(\unicode[STIX]{x1D70F},s),\overline{\unicode[STIX]{x1D6E9}_{S}(\unicode[STIX]{x1D70F},z)}\rangle .\nonumber\end{eqnarray}$$

Relation (4.9) allows us to state the following analytic properties of $E_{l,S}^{(n)}(\unicode[STIX]{x1D70F},z;s)$ (which are partly already known) with respect to the variable  $s$ , provided that $\det (2S)$ is odd. We use the notation of Corollary 3.17.

Theorem 4.2. Let $m\in \mathbb{N}$ with $4\mid m$ , and let $S\in \operatorname{Sym}_{m}(\mathbb{Q})$ be positive definite and half integral. Then the real analytic Jacobi Eisenstein series can be meromorphically continued to the whole $s$ -plane. If in addition, the quadratic module $(2S)^{-1}\mathbb{Z}^{m}/\mathbb{Z}^{m}$ is anisotropic and $\det (2S)$ odd, $E_{l,S}^{(n)}(\unicode[STIX]{x1D70F},z;s)$ satisfies the functional equation

(4.11) $$\begin{eqnarray}\displaystyle E_{l,S}^{(n)}\left(\unicode[STIX]{x1D70F},z,s-\frac{l}{2}+\frac{m}{4}\right) & = & \displaystyle \unicode[STIX]{x1D709}\left(2s-\unicode[STIX]{x1D70C}_{n},n,l-\frac{m}{2},P\right)\mathop{\prod }_{p\mid |L^{\prime }/L|}\mathop{\sum }_{(\unicode[STIX]{x1D70E},I,k)}\unicode[STIX]{x1D705}_{p}(\unicode[STIX]{x1D70E},I,k)\unicode[STIX]{x1D6E5}_{p}(\unicode[STIX]{x1D70E},q)\nonumber\\ \displaystyle & & \displaystyle \times \mathop{\prod }_{j=1}^{k}(D_{p,j}(2s-\unicode[STIX]{x1D70C}_{n})-1)E_{l,S}^{(n)}\left(\unicode[STIX]{x1D70F},z,\unicode[STIX]{x1D70C}_{n}-s-\frac{l}{2}+\frac{m}{4}\right).\end{eqnarray}$$

4.2 Analytic properties of zeta functions of common Jacobi eigenforms

In this subsection, we prove analytic properties of a zeta function $Z_{n}(s,f)$ associated to a Hecke Jacobi eigenform  $f$ . To this end, we briefly recall the necessary background following [Reference ArakawaAr2, Section 2] and [Reference BöchererBo, Section 3]. See also the references in [Reference ArakawaAr2] for further details.

Let $\unicode[STIX]{x1D6FC}\in \operatorname{Sp}(n,\mathbb{Q})$ . The double coset $\unicode[STIX]{x1D6E4}_{m,n}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6E4}_{m,n}$ can be decomposed into a disjoint finite union of left cosets

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{n,m}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6E4}_{n,m}=\mathop{\bigcup }_{i=1}^{r}\unicode[STIX]{x1D6E4}_{n,m}\unicode[STIX]{x1D6FD}_{i}.\end{eqnarray}$$

It can be shown that by means of this decomposition, an action of double cosets $\unicode[STIX]{x1D6E4}_{m,n}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6E4}_{m,n}$ on $J_{l,S}(\unicode[STIX]{x1D6E4}_{n,m})$ is given by

(4.12) $$\begin{eqnarray}(\unicode[STIX]{x1D6E4}_{n,m}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6E4}_{n,m},f)\mapsto T(\unicode[STIX]{x1D6FC})f=f|\unicode[STIX]{x1D6E4}_{n,m}\unicode[STIX]{x1D6FC}\unicode[STIX]{x1D6E4}_{n,m}=\mathop{\sum }_{i=1}^{r}f\mid _{l,S}\unicode[STIX]{x1D6FD}_{i},\end{eqnarray}$$

which stabilizes cusp forms. We now define a zeta function associated to a Hecke eigenform.

Definition 4.3. Let $D=\text{diag}(d_{1},d_{2},\ldots ,d_{n})$ be an elementary divisor matrix, $d_{n}(D)=\left(\!\begin{smallmatrix}D & 0\\ 0 & (D^{-1})^{t}\end{smallmatrix}\!\right)\in \operatorname{Sp}(n,\mathbb{Q})$ , and $f\in J_{l,S}^{\text{cusp}}(\unicode[STIX]{x1D6E4}_{n,m})$ be a common Hecke eigenform of all Hecke operators $T(d_{n}(D))$ , that is,

$$\begin{eqnarray}f|\unicode[STIX]{x1D6E4}_{n,m}d_{n}(D)\unicode[STIX]{x1D6E4}_{n,m}=\unicode[STIX]{x1D706}(f,D)f\end{eqnarray}$$

for all $D$ . Then we can assign to $f$ the Dirichlet series

(4.13) $$\begin{eqnarray}Z_{n}(s,f)=\mathop{\sum }_{D\in D_{n}(\mathbb{Z})}\unicode[STIX]{x1D706}(f,D)\det (D)^{-s},\end{eqnarray}$$

where $D_{n}(\mathbb{Z})$ denotes the set of elementary divisor matrices of rank  $n$ .

It can be shown that $Z_{n}(s,f)$ converges absolutely for $\operatorname{Re}(s)>2n+m+1$ [Reference ArakawaAr2, Chapter 2]. Later, we will show that it can be continued meromorphically to the whole $s$ -plane.

We need two further fundamental concepts to state the next result, which gives a representation of $Z_{n}(s,f)$ as a Rankin–Selberg convolution integral.

In the theory of the classical Siegel modular forms, the notion of an Eisenstein series is extended to the Klingen Eisenstein series attached to a cusp form. This construction can be generalized to Jacobi forms of higher degree.

Definition 4.4. Let $l\in \mathbb{N}$ be even, and $r\in \mathbb{Z}$ with $0\leqslant r\leqslant n$ . For $(\unicode[STIX]{x1D70F},z)\in {\mathcal{H}}_{n,m}$ , set

$$\begin{eqnarray}(\unicode[STIX]{x1D70F}^{\ast },z^{\ast })=\left(\unicode[STIX]{x1D70F}\left[\left(\begin{array}{@{}c@{}}0_{n-r}\\ 1_{r}\end{array}\right)\right],z\left(\begin{array}{@{}c@{}}0_{n-r}\\ 1_{r}\end{array}\right)\right).\end{eqnarray}$$

Moreover, depending on $f\in J_{l,S}^{\text{cusp}}(\unicode[STIX]{x1D6E4}_{r,m})$ , we define the function $\tilde{f}(\unicode[STIX]{x1D70F},z)=f(\unicode[STIX]{x1D70F}^{\ast },z^{\ast })$ on ${\mathcal{H}}_{n,m}$ and the Klingen Eisenstein series

(4.14) $$\begin{eqnarray}E_{l,S}^{n,r}(f,(\unicode[STIX]{x1D70F},z);s)=\mathop{\sum }_{\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{n,m}^{r}\setminus \unicode[STIX]{x1D6E4}_{n,m}}(\tilde{f}\mid _{l,S}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D70F},z)\left(\frac{\det (\operatorname{Im}(M\langle \unicode[STIX]{x1D70F}\rangle ))}{\det (\operatorname{Im}(M\langle \unicode[STIX]{x1D70F}\rangle ^{\ast }))}\right)^{s},\end{eqnarray}$$

where $\unicode[STIX]{x1D6FE}=([(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}],M)$ for each $\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{n,m}$ and

$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{n,m}^{r} & = & \displaystyle \left\{([((0,\unicode[STIX]{x1D706}_{1}),\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}],M)\mid \right.\nonumber\\ \displaystyle & & \displaystyle \left.M\in \unicode[STIX]{x1D6E4}_{r,n},\unicode[STIX]{x1D706}_{1}\in M_{m,r}(\mathbb{Z}),\unicode[STIX]{x1D707}\in M_{m,n}(\mathbb{Z}),\unicode[STIX]{x1D705}\in M_{m,m}(\mathbb{Z})~\text{where}~(\unicode[STIX]{x1D705}+\unicode[STIX]{x1D707}(0,\unicode[STIX]{x1D706}_{1})^{t})\in \operatorname{Sym}_{m}(\mathbb{Z})\right\}\nonumber\end{eqnarray}$$

with

$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{r,n}=\left\{\left.\left(\begin{array}{@{}cc@{}}a & b\\ c & d\end{array}\right)\in \operatorname{Sp}(n,\mathbb{Z})\right|c=\left(\begin{array}{@{}cc@{}}0 & 0\\ 0 & c_{4}\end{array}\right),\,d=\left(\begin{array}{@{}cc@{}}d_{1} & 0\\ d_{3} & d_{4}\end{array}\right)~\text{with}~c_{4},d_{4}\in M_{r,r}(\mathbb{Z})\right\}.\end{eqnarray}$$

Ziegler showed that $E_{l,S}^{n,r}$ is absolutely convergent for $l+2\operatorname{Re}(s)>n+m+r+1$ ; see [Reference ZieglerZi, Theorem 2.5].

The definition of a Petersson scalar product on $J_{l,S}^{\text{cusp}}$ can be found in [Reference ArakawaAr2] and [Reference ZieglerZi]. Let $w=(\unicode[STIX]{x1D70F},z)\in {\mathcal{H}}_{n,m}$ . We decompose $\unicode[STIX]{x1D70F}$ and $z$ into their real and imaginary parts, $\unicode[STIX]{x1D70F}=x+iy$ , $z=u+iv$ with $x,y\in \operatorname{Sym}_{n}(\mathbb{R})$ and $u,v\in M_{m,n}(\mathbb{R})$ , and define $dw=d(\unicode[STIX]{x1D70F},z)=\det (y)^{-(m+n+1)}\,dx\,dy\,du\,dv$ . Here $dx\,dy=\prod \,dx_{ij}\prod \,dy_{ij}$ , and $du$ and $dv$ are the standard Haar measures on $M_{m,n}(\mathbb{R})$ . Further, let

$$\begin{eqnarray}C_{S,l}(\unicode[STIX]{x1D70F},z)=\det (y)^{l}\exp (-4\unicode[STIX]{x1D70B}\operatorname{tr}(S[v]y^{-1})).\end{eqnarray}$$

For $f,g\in J_{l,S}(\unicode[STIX]{x1D6E4}_{n,m})$ , one of which is a cusp form, the Petersson scalar product of $f$ and $g$ is given by

(4.15) $$\begin{eqnarray}(f,g)=\int _{\unicode[STIX]{x1D6E4}_{n,m}\backslash {\mathcal{H}}_{n,m}}f(w)\overline{g(w)}C_{S,l}(w)\,dw.\end{eqnarray}$$

We now describe the zeta function $Z_{n}(s,f)$ as a Rankin–Selberg convolution of a Jacobi Eisenstein series and the Jacobi cusp form  $f$ . To this end, let

$$\begin{eqnarray}\unicode[STIX]{x1D707}(l,S,n,s)=(-1)^{nl/2}2^{n(n+1)-2ns-nl}\det (2S)^{-n}I_{n}\left(s+l-\frac{m}{2}-n-1\right)\end{eqnarray}$$

with

$$\begin{eqnarray}I_{n}(s)=\unicode[STIX]{x1D70B}^{n(n+1)/2}2^{-n(n-1)/2}\frac{\unicode[STIX]{x1D6E4}_{n}\left(s+1+\frac{n-1}{2}\right)}{\unicode[STIX]{x1D6E4}_{n}(s+n+1)}.\end{eqnarray}$$

The following theorem is taken from [Reference ArakawaAr2, Section 2].

Theorem 4.5. Let $p,n\in \mathbb{N}$ with $p\geqslant n$ , $l\in \mathbb{N}$ even with $l>p+n+m+1$ , and $f\in J_{l,S}^{\text{cusp}}(\unicode[STIX]{x1D6E4}_{n,m})$ . If $l+2\operatorname{Re}(s)>p+n+m+1$ , then for any $(\unicode[STIX]{x1D70F},z)\in {\mathcal{H}}_{p,m}$ ,

(4.16) $$\begin{eqnarray}\displaystyle & & \displaystyle \int _{\unicode[STIX]{x1D6E4}_{n,m}\backslash {\mathcal{H}}_{n,m}}f(\unicode[STIX]{x1D701},w)\overline{E_{l,S}^{(p+n)}\left(\left(\begin{array}{@{}cc@{}}-\overline{\unicode[STIX]{x1D70F}} & 0\\ 0 & \unicode[STIX]{x1D701}\end{array}\right),(\overline{z},w),\overline{s}\right)}C_{S,l}(\unicode[STIX]{x1D701},w)d(\unicode[STIX]{x1D701},w)\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D707}(l,S,n,s)\mathop{\sum }_{D\in D_{n}(\mathbb{Z})}(\det (D))^{-2s-l}E_{l,S}^{p,n}(f\mid \unicode[STIX]{x1D6E4}_{n,m}d_{n}(D)\unicode[STIX]{x1D6E4}_{n,m},(\unicode[STIX]{x1D70F},z);s),\end{eqnarray}$$

where the integral on the left-hand side and the series on the right-hand side are absolutely convergent. Moreover, if $f$ is a common eigenform, then the right-hand side of (4.16) coincides with

$$\begin{eqnarray}\unicode[STIX]{x1D707}(l,S,n,s)Z_{n}(2s+l,f)E_{l,S}^{p,n}(f,(\unicode[STIX]{x1D70F},z);s).\end{eqnarray}$$

In particular, if $p=n$ , this quantity is equal to

(4.17) $$\begin{eqnarray}\unicode[STIX]{x1D707}(l,S,n,s)Z_{n}(2s+l,f)f(\unicode[STIX]{x1D70F},z).\end{eqnarray}$$

We obtain as a corollary from Theorem 4.5 and identity (4.11) a functional equation for $Z_{n}(s,f)$ .

Corollary 4.6. Let $l>2n+m+1$ be an even positive integer, $m\in \mathbb{N}$ with $4|m$ , and $f\in J_{l,S}^{\text{cusp}}(\unicode[STIX]{x1D6E4}_{n,m})$ be a common eigenform of all Hecke operators $T(d_{n}(D))$ . Then the Dirichlet series $Z_{n}(s,f)$ can be continued meromorphically to the whole $s$ -plane. If $(2S)^{-1}\mathbb{Z}^{m}/\mathbb{Z}^{m}$ is anisotropic and $\det (2S)$ is odd, then

$$\begin{eqnarray}\mathscr{Z}_{n}(s,f)=\unicode[STIX]{x1D707}(l,S,n,s)Z_{n}(2s+l,f)\end{eqnarray}$$

satisfies the following functional equation,

(4.18) $$\begin{eqnarray}\displaystyle \mathscr{Z}_{n}\left(s-\frac{l}{2}+\frac{m}{4},f\right) & = & \displaystyle \unicode[STIX]{x1D709}\left(2s-\unicode[STIX]{x1D70C}_{2n},2n,l-\frac{m}{2},P\right)\mathop{\prod }_{p\mid \det (2S)}\mathop{\sum }_{(\unicode[STIX]{x1D70E},I,k)}\unicode[STIX]{x1D705}_{p}(\unicode[STIX]{x1D70E},I,k)\unicode[STIX]{x1D6E5}_{p}(\unicode[STIX]{x1D70E},q)\nonumber\\ \displaystyle & & \displaystyle \times \mathop{\prod }_{j=1}^{k}(D_{p,j}(2s-\unicode[STIX]{x1D70C}_{2n})-1)\mathscr{Z}_{n}\left(\unicode[STIX]{x1D70C}_{2n}-s-\frac{l}{2}+\frac{m}{4},f\right),\end{eqnarray}$$

where the quantities $\unicode[STIX]{x1D709}(2s-\unicode[STIX]{x1D70C}_{2n},2n,l-(m/2),P),\unicode[STIX]{x1D705}_{p}(\unicode[STIX]{x1D70E},I,k)$ , $\unicode[STIX]{x1D6E5}_{p}(\unicode[STIX]{x1D70E},q)$ , and $D_{p,j}(2s-\unicode[STIX]{x1D70C}_{2n})$ are taken from Corollary 3.17.

Proof. Note that $\overline{E_{l-(m/2),0}^{2n\ast }\left(\left(\!\begin{smallmatrix}-\overline{\unicode[STIX]{x1D70F}} & 0\\ 0 & \unicode[STIX]{x1D701}\end{smallmatrix}\!\right),\overline{s}\right)}=E_{l-(m/2),0}^{2n}\left(\left(\!\begin{smallmatrix}-\overline{\unicode[STIX]{x1D70F}} & 0\\ 0 & \unicode[STIX]{x1D701}\end{smallmatrix}\!\right),s\right)$ and therefore

(4.19) $$\begin{eqnarray}\overline{E_{l,S}^{(2n)}\left(\left(\begin{array}{@{}cc@{}}-\overline{\unicode[STIX]{x1D70F}} & 0\\ 0 & \unicode[STIX]{x1D701}\end{array}\right),(\overline{z},w),\overline{s}\right)}=\left\langle E_{l-(m/2),0}^{2n}\left(\left(\begin{array}{@{}cc@{}}-\overline{\unicode[STIX]{x1D70F}} & 0\\ 0 & \unicode[STIX]{x1D701}\end{array}\right),s\right),\unicode[STIX]{x1D6E9}_{S}\left(\left(\begin{array}{@{}cc@{}}-\overline{\unicode[STIX]{x1D70F}} & 0\\ 0 & \unicode[STIX]{x1D701}\end{array}\right),(\overline{z},w)\right)\right\rangle .\end{eqnarray}$$

If we replace $\overline{E_{l,S}^{(2n)}\left(\left(\!\begin{smallmatrix}-\overline{\unicode[STIX]{x1D70F}} & 0\\ 0 & \unicode[STIX]{x1D701}\end{smallmatrix}\!\right),(\overline{z},w),\overline{s}\right)}$ in (4.16) by the right-hand side of (4.19), we find that $Z_{n}(s,f)$ is a meromorphic function in $s$ whenever $l+2\operatorname{Re}(s)>2n+m+1$ . Moreover, since $E_{l-(m/2),0}^{2n}$ can be continued to the whole $s$ -plane, $Z_{n}(s,f)$ shares the same analytic property. The functional equation results from Corollary 3.17 in the same way.◻

4.3 Analytic properties of the standard $L$ -function

Let $f\in J_{l,S}^{\text{cusp}}$ be a common eigenform of all operators $T(d_{n}(\mathbb{Z}))$ . We now briefly recall the theory necessary to define the standard $L$ -function $L(s,f)$ . More details can be found in [Reference ArakawaAr2, Reference Bouganis and MarzecBM], in particular in [Reference MuraseMu1] and [Reference MuraseMu2]. All of the following facts are based on an assumption imposed on the lattice $L=\mathbb{Z}_{p}^{m}$ with respect to the positive definite and half integral matrix $S\in \operatorname{Sym}_{m}(\mathbb{Q})$ for any prime  $p$ . It is denoted by $M_{p}^{+}$ in accordance with the notation in [Reference MuraseMu1] and [Reference MuraseMu2]:

1. (1) $L$ is a $\mathbb{Z}_{p}$ -maximal lattice with respect to $2S$ .

2. (2) $L^{\ast }=L$ , where $L^{\ast }=\{x\in L^{\prime }=(2S)^{-1}L\mid S[x]\in (1/p)\mathbb{Z}_{p}\}$ .

Note that if $L$ is maximal, one can prove that $\text{ord}_{p}(\det (2S))\in \{0,1,2\}$ ; see [Reference MuraseMu1, Lemma 4.1(i)]. This implies that $|L^{\prime }/L|\in \{1,p,p^{2}\}$ . Also, condition (2) guarantees that $S[x]\notin (1/p)\mathbb{Z}_{p}$ if $x\in L^{\prime }\setminus L$ . It follows that $L^{\prime }/L$ is anisotropic (provided it is non-trivial). Therefore, our assumption that $L^{\prime }/L$ is anisotropic is no further restriction compared to the condition  $M_{p}^{+}$ . As before, we consider only odd primes  $p$ .

Under the assumption $M_{p}^{+}$ , the Hecke algebra ${\mathcal{H}}(G_{n,m}(\mathbb{Q}_{p}),G_{n,m}(\mathbb{Z}_{p}),\unicode[STIX]{x1D713}_{S,p})$ has a nice structure, as is proved in [Reference MuraseMu1, Section 4]. It is associated to a character $\unicode[STIX]{x1D713}_{S,p}:\operatorname{Sym}_{m}(\mathbb{Q}_{p})\rightarrow \mathbb{C}^{\times }$ , $\unicode[STIX]{x1D713}_{S,p}(\unicode[STIX]{x1D705})=\unicode[STIX]{x1D713}_{p}(\operatorname{tr}(S\unicode[STIX]{x1D705}))$ , where $\unicode[STIX]{x1D713}_{p}$ is the $p$ -part of a character $\unicode[STIX]{x1D713}:\mathbb{A}/\mathbb{Q}\rightarrow \mathbb{C}^{\times }$ with $\unicode[STIX]{x1D713}_{\infty }(x)=e(x)$ for $x\in \mathbb{R}$ . ${\mathcal{H}}(G_{n,m}(\mathbb{Q}_{p}),G_{n,m}(\mathbb{Z}_{p}),\unicode[STIX]{x1D713}_{S,p})$ consists of functions $\unicode[STIX]{x1D719}:G_{n,m}(\mathbb{Q}_{p})\rightarrow \mathbb{C}$ , which fulfill the conditions:

1. (1)

   $$\begin{eqnarray}\unicode[STIX]{x1D719}([(0,0),\unicode[STIX]{x1D705}]kgk^{\prime })=\unicode[STIX]{x1D713}_{S,p}(\unicode[STIX]{x1D705})\unicode[STIX]{x1D719}(g),\quad k,k^{\prime }\in G_{n,m}(\mathbb{Z}_{p}),\,g\in G_{n,m}(\mathbb{Q}_{p})~\text{and}~\unicode[STIX]{x1D705}\in \operatorname{Sym}_{m}(\mathbb{Q}_{p}),\end{eqnarray}$$

2. (2) $\unicode[STIX]{x1D719}$ is compactly supported modulo

   $$\begin{eqnarray}Z_{n,m}(\mathbb{Q}_{p})=\{(0,0,\unicode[STIX]{x1D705})\mid \unicode[STIX]{x1D705}\in \operatorname{Sym}_{n}(\mathbb{Q}_{p})\}.\end{eqnarray}$$

We will use the abbreviation ${\mathcal{H}}_{S,p}$ instead of ${\mathcal{H}}(G_{n,m}(\mathbb{Q}_{p}),G_{n,m}(\mathbb{Z}_{p}),\unicode[STIX]{x1D713}_{S,p})$ . It can be shown that ${\mathcal{H}}_{S,p}$ forms a $\mathbb{C}$ -algebra with respect to the convolution product

(4.20) $$\begin{eqnarray}(\unicode[STIX]{x1D719}_{1}\ast \unicode[STIX]{x1D719}_{2})(g)=\int _{Z_{n,m}(\mathbb{Q}_{p})\backslash G_{n,m}(\mathbb{Q}_{p})}\unicode[STIX]{x1D719}_{1}(gx^{-1})\unicode[STIX]{x1D719}_{2}(x)\,dx,\quad \unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}\in {\mathcal{H}}_{S,p},\end{eqnarray}$$

where $dx$ is the Haar measure on $Z_{n,m}(\mathbb{Q}_{p})\backslash G_{n,m}(\mathbb{Q}_{p})$ normalized by $\int _{Z_{n,m}(\mathbb{Q}_{p})\backslash Z_{n,m}(\mathbb{Q}_{p})G_{n,m}(\mathbb{Z}_{p})}\,dx=1$ .

In order to define the standard $L$ -function of a Hecke eigenform, we need to introduce zonal spherical functions  $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}$ , where $\unicode[STIX]{x1D712}$ is an unramified character of

$$\begin{eqnarray}T(\mathbb{Q}_{p})=\{d_{m}(\text{diag}(d_{1},\ldots ,d_{m}))\mid d_{i}\in \mathbb{Q}_{p}^{\times }\},\end{eqnarray}$$

that is, it is an element of

$$\begin{eqnarray}X(T(\mathbb{Q}_{p}))=\{\unicode[STIX]{x1D712}\in \text{Hom}(T(\mathbb{Q}_{p}),\mathbb{C}^{\times })\mid \unicode[STIX]{x1D712}~\text{is trivial on}~T(\mathbb{Z}_{p})\}.\end{eqnarray}$$

Now, $\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}:G_{n,m}(\mathbb{Q}_{p})\rightarrow \mathbb{C}$ is for $\unicode[STIX]{x1D712}=(\unicode[STIX]{x1D712}_{1},\ldots ,\unicode[STIX]{x1D712}_{n})\in X(T(\mathbb{Q}_{p}))$ defined by

$$\begin{eqnarray}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}(g)=\int _{G_{n,m}(\mathbb{Z}_{p})}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D712}}(kg)\,dk.\end{eqnarray}$$

Here $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D712}}$ is another function on $G_{n,m}(\mathbb{Q}_{p})$ given by

$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D712}}\left(\left([(0,\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}],\left(\begin{array}{@{}cc@{}}1_{n} & x\\ 0 & 1_{n}\end{array}\right)d_{n}(\text{diag}(d_{1},\ldots ,d_{n}))\right)([(\unicode[STIX]{x1D706},0),0],k)\right)\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D713}_{S,p}(\unicode[STIX]{x1D705})\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D712}_{i}(d_{i})|d_{i}|_{p}^{2n+m+2-2i}\times \left\{\begin{array}{@{}ll@{}}1\quad & \text{if}~\unicode[STIX]{x1D706}\in M_{m,n}(\mathbb{Z}_{p}),\\ 0\quad & \text{otherwise}.\end{array}\right.\nonumber\end{eqnarray}$$

Here $\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in M_{m,n}(\mathbb{Q}_{p}),\unicode[STIX]{x1D705}\in \operatorname{Sym}_{m}(\mathbb{Q}_{p}),x\in \operatorname{Sym}_{n}(\mathbb{Q}_{p}),d_{i}\in \mathbb{Q}_{p}^{\times }$ , and $k\in G_{n,m}(\mathbb{Z}_{p})$ . The following observation is crucial for the definition of $L(s,f)$ : it is known that the map

(4.21) $$\begin{eqnarray}\unicode[STIX]{x1D719}\mapsto \widehat{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}}(\unicode[STIX]{x1D719})=\int _{Z_{n,m}(\mathbb{Q}_{p})\backslash G_{n,m}(\mathbb{Q}_{p})}\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}}(g)\unicode[STIX]{x1D719}(g^{-1})\,dg\end{eqnarray}$$

gives rise to a $\mathbb{C}$ -algebra homomorphism of ${\mathcal{H}}_{S,p}$ to  $\mathbb{C}$ . Moreover, every such homomorphism is of the form (4.21) for some uniquely determined $\unicode[STIX]{x1D712}\in X(T(\mathbb{Q}_{p}))$ up to some action on $X(T(\mathbb{Q}_{p}))$ ; see [Reference MuraseMu1].

We finally specify the standard $L$ -function attached to a cuspidal Jacobi form of weight $l$ and index  $S$ . To this end, we choose for each $(\unicode[STIX]{x1D70F},z)\in {\mathcal{H}}_{n,m}$ an element $g_{\infty }\in G_{n,m}(\mathbb{R})$ such that $g_{\infty }(i1_{n},0)=(\unicode[STIX]{x1D70F},z)$ and denote by $\mathscr{S}(l,S)$ the set of all bounded functions $f:G_{n,m}(\mathbb{A})\rightarrow \mathbb{C}$ satisfying

1. (i)

   (4.22) $$\begin{eqnarray}f([(0,0),\unicode[STIX]{x1D705}]\unicode[STIX]{x1D6FE}gk_{\infty }k_{f})=\det (J(k_{\infty }i1_{n}))^{-l}\unicode[STIX]{x1D713}_{S}(\unicode[STIX]{x1D705})f(g)\end{eqnarray}$$
   for $\unicode[STIX]{x1D705}\in \operatorname{Sym}_{m}(\mathbb{A})$ , $\unicode[STIX]{x1D6FE}\in G_{n,m}(\mathbb{Q}),g\in G_{n,m}(\mathbb{A}),k_{\infty }\in K_{\infty }$ , and $k_{f}\in K_{f}$ , where $K_{\infty }$ is given by (3.5) and $K_{f}$ by (4.23);

2. (ii)

   $$\begin{eqnarray}F_{f}(\unicode[STIX]{x1D70F},z)=f(g_{\infty })J_{S,l}(g_{\infty },(i1_{n},0))\end{eqnarray}$$
   is holomorphic on ${\mathcal{H}}_{n,m}$ .

One can check that $F_{f}$ does not depend on the choice of $g_{\infty }$ and that for each $f\in \mathscr{S}(l,S)$ , the associated function $F_{f}$ is an element of $J_{k,S}^{\text{cusp}}(\unicode[STIX]{x1D6E4}_{n,m})$ . On the other hand, by means of the strong approximation theorem for $G_{n,m}$ ,

(4.23) $$\begin{eqnarray}G_{n,m}(\mathbb{A})=G_{n,m}(\mathbb{Q})G_{n,m}(\mathbb{R})K_{f},\quad K_{f}=\mathop{\prod }_{p<\infty }G_{n,m}(\mathbb{Z}_{p}),\end{eqnarray}$$

a cusp form $F\in J_{l,S}^{\text{cusp}}(\unicode[STIX]{x1D6E4}_{n,m})$ can be related to $f\in \mathscr{S}(l,S)$ by

(4.24) $$\begin{eqnarray}f(\unicode[STIX]{x1D6FE}g_{\infty }k_{f})=F(g_{\infty }\langle (i1_{n},0)\rangle )J_{l,S}(g_{\infty },(i1_{n},0))^{-1}.\end{eqnarray}$$

More precisely, the following statement holds ([Reference MuraseMu1], [Reference MuraseMu2], and [Reference ArakawaAr2]).

For each prime $p$ , the Hecke algebra ${\mathcal{H}}_{S,p}$ acts on $\mathscr{S}(l,S)$ via convolution

(4.25) $$\begin{eqnarray}(f\ast \unicode[STIX]{x1D719})(g)=\int _{Z_{n,m}(\mathbb{Q}_{p})\backslash G_{n,m}(\mathbb{Q}_{p})}f(gx_{p}^{-1})\unicode[STIX]{x1D719}(x_{p})dx_{p},\quad f\in \mathscr{S}(l,S),\,\unicode[STIX]{x1D719}\in {\mathcal{H}}_{S,p}.\end{eqnarray}$$

We call $f\in \mathscr{S}(l,S)$ a Hecke eigenform if for any prime $p$ and any $\unicode[STIX]{x1D719}\in {\mathcal{H}}_{S,p}$ , the equation

$$\begin{eqnarray}f\ast \unicode[STIX]{x1D719}=\unicode[STIX]{x1D706}_{f,p}(\unicode[STIX]{x1D719})f\quad \text{with}~\unicode[STIX]{x1D706}_{f,p}(\unicode[STIX]{x1D719})\in \mathbb{C}\end{eqnarray}$$

holds.

A Hecke eigenform $f$ defines for each prime $p$ a $\mathbb{C}$ -algebra homomorphism $\unicode[STIX]{x1D706}_{f,p}:{\mathcal{H}}_{S,p}\rightarrow \mathbb{C}$ , $\unicode[STIX]{x1D719}\mapsto \unicode[STIX]{x1D706}_{f,p}(\unicode[STIX]{x1D719})$ , and thereby determines a character $\unicode[STIX]{x1D712}_{f,p}=(\unicode[STIX]{x1D712}_{f,p}^{(1)},\ldots ,\unicode[STIX]{x1D712}_{f,p}^{(n)})\in X(T(\mathbb{Q}_{p}))$ with

$$\begin{eqnarray}\unicode[STIX]{x1D706}_{f,p}(\unicode[STIX]{x1D719})=\widehat{\unicode[STIX]{x1D714}_{\unicode[STIX]{x1D712}_{f,p}}}(\unicode[STIX]{x1D719}),\quad \unicode[STIX]{x1D719}\in {\mathcal{H}}_{S,p}.\end{eqnarray}$$

We are now able to define the standard $L$ -function $L(s,f)$ attached to $f$ by $L(s,f)=\prod _{p<\infty }L_{p}(s,f)$ with

(4.26) $$\begin{eqnarray}L_{p}(s,f)=\mathop{\prod }_{i=1}^{n}[(1-\unicode[STIX]{x1D712}_{f,p}^{(i)}(p)p^{-s})(1-\unicode[STIX]{x1D712}_{f,p}^{(i)}(p)^{-1}p^{-s})]^{-1}.\end{eqnarray}$$

There is a close relation between the action of ${\mathcal{H}}_{S,p}$ on $\mathscr{S}(l,S)$ and action (4.12) of a double coset $\unicode[STIX]{x1D6E4}_{n,m}d_{n}(D)\unicode[STIX]{x1D6E4}_{n,m}$ on $J_{l,S}^{\text{cusp}}(\unicode[STIX]{x1D6E4}_{n,m})$ (see [Reference ArakawaAr2]):

Let $\unicode[STIX]{x1D6FC}=(d_{1},\ldots ,d_{n})\in \mathbb{Z}^{n}$ with $d_{1}\geqslant d_{2}\geqslant \cdots \geqslant d_{n}\geqslant 0$ and set $\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D6FC}}=\text{diag}(p^{d_{1}},\ldots ,p^{d_{n}})$ . Furthermore, we define an element $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}}\in {\mathcal{H}}_{S,p}$ for $g=[(0,0),\unicode[STIX]{x1D705}]kgk^{\prime }\in G_{n,m}(\mathbb{Q}_{p})$ by

$$\begin{eqnarray}\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}}(g)=\left\{\begin{array}{@{}ll@{}}\unicode[STIX]{x1D713}_{S,p}(\unicode[STIX]{x1D705})\quad & \text{if}~g\in Z_{n,m}(\mathbb{Q}_{p})G_{n,m}(\mathbb{Z}_{p})d_{n}(\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D6FC}})G_{n,m}(\mathbb{Z}_{p})~\text{and}~g=[(0,0),\unicode[STIX]{x1D705}]kd_{n}(\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D6FC}})k^{\prime },\\ 0\quad & \text{if}~g\notin Z_{n,m}(\mathbb{Q}_{p})G_{n,m}(\mathbb{Z}_{p})d_{n}(\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D6FC}})G_{n,m}(\mathbb{Z}_{p}).\end{array}\right.\end{eqnarray}$$

Arakawa points out that if $g\in Z_{n,m}(\mathbb{Q}_{p})\unicode[STIX]{x1D6E4}_{n,m}d_{n}(\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D6FC}})\unicode[STIX]{x1D6E4}_{n,m}$ , the value $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}}(g)$ does not depend on $k,k^{\prime }$ , and  $\unicode[STIX]{x1D705}$ . Moreover, it can be easily confirmed that $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}}$ indeed belongs to ${\mathcal{H}}_{S,p}$ . In terms of $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}}$ , we can state the following proposition.

Proposition 4.8. [Reference ArakawaAr2]

Let $F\in J_{k,S}^{\text{cusp}}$ be a cusp form that corresponds to $f\in \mathscr{S}(l,S)$ ; see (4.24). Also, let $\unicode[STIX]{x1D6FC}\in \mathbb{Z}^{n}$ be as above. Then for each $\unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}}\in {\mathcal{H}}_{S,p}$ , we have

$$\begin{eqnarray}F_{f\ast \unicode[STIX]{x1D719}_{\unicode[STIX]{x1D6FC}}}=F\mid \unicode[STIX]{x1D6E4}_{n,m}d_{n}(\unicode[STIX]{x1D70B}_{\unicode[STIX]{x1D6FC}})\unicode[STIX]{x1D6E4}_{n,m}.\end{eqnarray}$$

From Proposition 4.8, it follows immediately that $f\in \mathscr{S}(l,S)$ is a Hecke eigenform if and only $F_{f}\in J_{l,S}^{\text{cusp}}$ is a common eigenform. Therefore, we define the standard $L$ -function $L(s,F_{f})$ of $F_{f}$ to be $L(s,f)$ .

We now want to transfer the analytic properties of the Dirichlet series $Z_{n}(s,f)$ to $L(s,f)$ . Apart from Corollary 4.6, the result of [Reference Bouganis and MarzecBM, Theorem 7.1] is crucial for our proof. It establishes a relation between $Z_{n}(s,f)$ and $L(s,f)$ for a more general situation than that considered in this paper. The factor $G_{p}(s)$ occurring in Theorem 7.1 is closely related to the Siegel series $b(S,s)$ . In [Reference KatsuradaKa, Theorem 4.4], one can find an explicit formula for the Siegel series (note that $S\in \operatorname{Sym}_{m}(\mathbb{Q})$ is non-degenerate and half integral),

(4.27) $$\begin{eqnarray}b(S,s)=\left(\unicode[STIX]{x1D701}(s)\mathop{\prod }_{i=1}^{[m/2]}\unicode[STIX]{x1D701}(2s-2i)\right)^{-1}L\left(s-\frac{m}{2},\unicode[STIX]{x1D712}_{S}\right)\mathop{\prod }_{p\mid \det (2S)}g_{p}(S,p^{-s}),\end{eqnarray}$$

where $L(s,\unicode[STIX]{x1D712}_{S})$ is the $L$ -function associated with the Hecke character corresponding to the extension $\mathbb{Q}(((-1)^{m/2}\det (S))^{1/2})/\mathbb{Q}$ . The quantity $g_{p}(S,p^{-s})$ is a polynomial in $p^{-s}$ , which is explicitly specified in [Reference KatsuradaKa, Theorem 4.3]. Now, the factor $G_{p}(s)$ is according to [Reference Bouganis and MarzecBM] given by

(4.28) $$\begin{eqnarray}G_{p}(s)=\frac{g_{p}(S,p^{-(s+(m/2)+n)})}{g_{p}(S,p^{-(s+(m/2))})}.\end{eqnarray}$$

Tailored to our setting, Theorem 7.1 reads as follows.

Theorem 4.9. [Reference Bouganis and MarzecBM]

Let $0\not =f\in J_{l,S}^{\text{cusp}}(\unicode[STIX]{x1D6E4}_{n,m})$ be a Hecke eigenform of all Hecke operators $T(d_{n}(D))$ . Assume further that the lattice $\mathbb{Z}_{p}^{m}$ with respect to the half integral matrix $S\in \operatorname{Sym}_{m}(\mathbb{Q}_{p})$ satisfies the condition $M_{p}^{+}$ for every prime  $p$ . Then

(4.29) $$\begin{eqnarray}\mathfrak{L}(s)Z_{n}\left(s+\frac{m}{2}+n,f\right)=L(s,f),\end{eqnarray}$$

where $\mathfrak{L}(s)=\prod _{p<\infty }\mathfrak{L}_{p}(s)$ with

(4.30) $$\begin{eqnarray}\mathfrak{L}_{p}(s)=G_{p}(s)\mathop{\prod }_{i=1}^{n}\unicode[STIX]{x1D701}_{p}(2s+2n-2i).\end{eqnarray}$$

In particular, $L(s,f)$ is absolutely convergent for $\operatorname{Re}(s)>2n+m+1$ .

Combining Theorem 4.9 and Corollary 4.6 yields the desired analytic properties of $L(s,f)$ summarized in the next theorem.

Theorem 4.10. Let $l>2n+m+1$ be an even positive integer, $m\in \mathbb{N}$ with $4\mid m$ , and $S\in \operatorname{Sym}_{m}(\mathbb{Q})$ be half integral for which $\mathbb{Z}_{p}^{m}$ satisfies the condition $M_{p}^{+}$ for every  $p$ . Moreover, let $f\in J_{l,S}^{\text{cusp}}(\unicode[STIX]{x1D6E4}_{n,m})$ be a common Hecke eigenform of all Hecke operators $T(d_{n}(D))$ . Then the standard $L$ -function $L(s,f)$ can be continued meromorphically to the whole $s$ -plane. If additionally, $\det (2S)$ is odd, then

$$\begin{eqnarray}\unicode[STIX]{x1D6F9}(s)=\unicode[STIX]{x1D707}\left(l,S,n,\frac{s+n}{2}-\frac{l}{2}+\frac{m}{4}\right)\mathfrak{L}^{-1}(s)L(s,f)\end{eqnarray}$$

satisfies the functional equation

(4.31) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}(s) & = & \displaystyle \unicode[STIX]{x1D709}\left(s-\frac{1}{2},2n,l-\frac{m}{2},P\right)\mathop{\prod }_{p\mid \det (2S)}\mathop{\sum }_{(\unicode[STIX]{x1D70E},I,k)}\unicode[STIX]{x1D705}_{p}(\unicode[STIX]{x1D70E},I,k)\unicode[STIX]{x1D6E5}_{p}(\unicode[STIX]{x1D70E},q)\nonumber\\ \displaystyle & & \displaystyle \times \mathop{\prod }_{j=1}^{k}\left(D_{p,j}\left(s-\frac{1}{2}\right)-1\right)\unicode[STIX]{x1D6F9}(1-s).\end{eqnarray}$$

Proof. Because of (4.29) we have

$$\begin{eqnarray}\mathscr{Z}_{n}(s,f)=\unicode[STIX]{x1D707}(l,S,n,s)\mathfrak{L}^{-1}\left(2s-\frac{m}{2}-n+l\right)L\left(2s-\frac{m}{2}-n+l,f\right).\end{eqnarray}$$

Then the functional equation (4.18) for $\mathscr{Z}_{n}$ for $s/2+n/2$ results in the functional equation

(4.32) $$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D707}\left(l,S,n,\frac{s}{2}+\frac{n}{2}-\frac{l}{2}+\frac{m}{4}\right)\mathfrak{L}^{-1}(s)L(s,f)\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D709}\left(s-\frac{1}{2},2n,l-\frac{m}{2},P\right)\mathop{\prod }_{p\mid \det (2S)}\mathop{\sum }_{(\unicode[STIX]{x1D70E},I,k)}\unicode[STIX]{x1D705}_{p}(\unicode[STIX]{x1D70E},I,k)\unicode[STIX]{x1D6E5}_{p}(\unicode[STIX]{x1D70E},q)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \mathop{\prod }_{j=1}^{k}\left(D_{p,j}\left(s-\frac{1}{2}\right)-1\right)\unicode[STIX]{x1D707}\left(l,S,n,\frac{1-s}{2}+\frac{n}{2}-\frac{l}{2}+\frac{m}{4}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathfrak{L}^{-1}(1-s)L(1-s,f).\end{eqnarray}$$