1 Introduction
The purpose of this paper is to show a certain formula for Dirichlet series of Rankin–Selberg type of certain Siegel cusp forms of half-integral weight.
Let
$2n$
and
$k$
be positive even integers such that
$k>2n+1$
. Let
$g$
be a cusp form of weight
$k-n+\frac{1}{2}$
in the Kohnen plus-space. Let
$f$
be a normalized Hecke eigenform of elliptic cusp form of weight
$2k-2n$
which corresponds to
$g$
by the Shimura correspondence. Let
$F$
be the Siegel cusp form of weight
$k$
of degree
$2n$
which is the Ikeda lift (or Duke–Imamoğlu–Ikeda lift) of
$g$
. In [Reference Katsurada and Kawamura13] Katsurada and Kawamura obtained the identity
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D701}(2s-2k+4n)\mathop{\sum }_{N=1}^{\infty }\frac{\langle \unicode[STIX]{x1D713}_{N},\unicode[STIX]{x1D713}_{N}\rangle }{N^{s}}=\langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{1}\rangle \unicode[STIX]{x1D701}(s-k+1)\unicode[STIX]{x1D701}(s-k+2n)L(s,f), & & \displaystyle \nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D713}_{N}$
denotes the
$N$
th Fourier–Jacobi coefficient of
$F$
and
$\langle \unicode[STIX]{x1D713}_{N},\unicode[STIX]{x1D713}_{N}\rangle$
denotes the Petersson inner product of
$\unicode[STIX]{x1D713}_{N}$
. Here
$\unicode[STIX]{x1D701}(s)$
denotes the Riemann zeta function and
$L(s,f)$
denotes the usual
$L$
-function of
$f$
. In the case of
$n=1$
the above formula was obtained by Kohnen and Skoruppa in [Reference Kohnen and Skoruppa16].
Let
$G$
be the Siegel cusp form of weight
$k-\frac{1}{2}$
of degree
$2n-1$
which is obtained from the first Fourier–Jacobi coefficient
$\unicode[STIX]{x1D713}_{1}$
by the Eichler–Zagier–Ibukiyama correspondence. Here the Eichler–Zagier–Ibukiyama correspondence is the linear isomorphism between Jacobi forms of index 1 and the generalized plus-space of Siegel modular forms (see [Reference Ibukiyama10]). The form
$G$
belongs to the generalized plus-space. We remark that if
$n=1$
, then
$G=g$
. Let
$\unicode[STIX]{x1D719}_{m}$
be the
$m$
th Fourier–Jacobi coefficient of
$G$
for any natural number
$m$
(see §4). We remark that
$\unicode[STIX]{x1D719}_{m}$
is a Jacobi cusp form of weight
$k-\frac{1}{2}$
of index
$m$
of degree
$2n-2$
and
$\unicode[STIX]{x1D719}_{m}$
is identically
$0$
unless
$-m\equiv 0,1\;\text{mod}\;4$
. We denote by
$\langle \unicode[STIX]{x1D719}_{m},\unicode[STIX]{x1D719}_{m}\rangle$
the Petersson inner product of
$\unicode[STIX]{x1D719}_{m}$
(see §2.6 for the definition). If
$n=1$
, then
$\unicode[STIX]{x1D719}_{m}$
is the
$m$
th Fourier coefficient of
$g$
and we put
$\langle \unicode[STIX]{x1D719}_{m},\unicode[STIX]{x1D719}_{m}\rangle :=|\unicode[STIX]{x1D719}_{m}|^{2}$
in this case.
The aim of this paper is to show the following theorem.
Theorem 1.1. Let
$f$
be a normalized Hecke eigenform of elliptic cusp form of weight
$2k-2n$
. Let the symbols be as above. We have
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{\substack{ m\in \mathbb{Z}_{{>}0} \\ -m\equiv 0,1\;\text{mod}\;4}}\frac{\langle \unicode[STIX]{x1D719}_{m},\unicode[STIX]{x1D719}_{m}\rangle }{m^{s+k-n-1/2}}\nonumber\\ \displaystyle & & \displaystyle \qquad =\unicode[STIX]{x1D701}(2s-2n+2)\unicode[STIX]{x1D701}(4s)^{-1}L(2s,f,Ad)\mathop{\sum }_{D_{0}}\frac{\langle \unicode[STIX]{x1D719}_{|D_{0}|},\unicode[STIX]{x1D719}_{|D_{0}|}\rangle }{{|D_{0}|}^{s+k-n-1/2}}\nonumber\\ \displaystyle & & \displaystyle \qquad \quad \times \,\mathop{\prod }_{p\nmid |D_{0}|}\bigg\{1+p^{-2s-1}-\,\frac{({\textstyle \frac{D_{0}}{p}})(1+p^{2n-2})p^{-k+1}a_{f}(p)}{1+p^{2s}}\bigg\}\nonumber\end{eqnarray}$$
for sufficiently large
$\text{Re}(s)$
, where
$D_{0}<0$
runs over all fundamental discriminants and
$a_{f}(p)$
denotes the
$p$
th Fourier coefficient of
$f$
. Here
$L(s,f,Ad)$
denotes the adjoint L-function of
$f$
,
$$\begin{eqnarray}\displaystyle L(s,f,Ad):=\mathop{\prod }_{p}\{(1-p^{-s})(1-\unicode[STIX]{x1D6FC}_{p}^{2}p^{-s})(1-\unicode[STIX]{x1D6FC}_{p}^{-2}p^{-s})\}^{-1}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$\{\unicode[STIX]{x1D6FC}_{p}^{\pm }\}$
are complex values determined by the identity
$$\begin{eqnarray}\displaystyle a_{f}(p)=(\unicode[STIX]{x1D6FC}_{p}+\unicode[STIX]{x1D6FC}_{p}^{-1})p^{k-n-1/2}. & & \displaystyle \nonumber\end{eqnarray}$$
In the case of
$n=1$
the above formula coincides with the formula in [Reference Kohnen and Zagier17, p. 182, line 5], with the modification that the function
$\unicode[STIX]{x1D701}(2s)\unicode[STIX]{x1D701}(4s)^{-1}L(2s,f,Ad)$
should be multiplied on the left in [Reference Kohnen and Zagier17, p. 182, line 5].
To obtain main theorem, a generalization of the Maass relation (Proposition 4.3) plays an important role. This generalization of the Maass relation was essentially shown in [Reference Hayashida7, Theorem 8.2]. We also need calculations of adjoint maps of index-shift maps.
We remark that we have analytic properties of the above Dirichlet series by using the Rankin–Selberg method for the generalized plus-space which is shown in [Reference Hayashida9, Corollary 3.4]. We put
$$\begin{eqnarray}\displaystyle {\mathcal{R}}_{1}(G;s) & := & \displaystyle \unicode[STIX]{x1D70B}^{-2s}\unicode[STIX]{x1D6E4}\biggl(s+k-n-\frac{1}{2}\biggr)\unicode[STIX]{x1D6E4}(s+n-1)\unicode[STIX]{x1D701}(2s+2n-2)\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{\substack{ m\in \mathbb{Z}_{{>}0} \\ -m\equiv 0,1\;\text{mod}\;4}}\frac{\langle \unicode[STIX]{x1D719}_{m},\unicode[STIX]{x1D719}_{m}\rangle }{m^{s+k-n-1/2}}.\nonumber\end{eqnarray}$$
Then
${\mathcal{R}}_{1}(G;s)$
has meromorphic continuation to the whole complex plane and satisfies the functional equation
$$\begin{eqnarray}\displaystyle {\mathcal{R}}_{1}(G;s)={\mathcal{R}}_{1}(G;1-s). & & \displaystyle \nonumber\end{eqnarray}$$
The function
${\mathcal{R}}_{1}(G;s)$
is entire except for
$s=n$
and
$s=1-n$
. The residue at
$s=n$
is
$$\begin{eqnarray}\displaystyle \text{Res}_{s=n}{\mathcal{R}}_{1}(G;s)=(1+\unicode[STIX]{x1D6FF}_{n,1})^{-1}2^{2k-1}\unicode[STIX]{x1D70B}^{k-3/2}\langle G,G\rangle , & & \displaystyle \nonumber\end{eqnarray}$$
where we put
$\unicode[STIX]{x1D6FF}_{n,1}:=1$
if
$n=1$
and
$\unicode[STIX]{x1D6FF}_{n,1}:=0$
if
$n>1$
, and
$\langle G,G\rangle$
denotes the Petersson inner product of
$G$
. Moreover, an explicit formula for
$\langle G,G\rangle$
is shown in [Reference Katsurada and Kawamura14]. To describe the value
$\langle G,G\rangle$
we prepare some notation.
We put
$\unicode[STIX]{x1D6E4}_{\mathbb{C}}(s):=2(2\unicode[STIX]{x1D70B})^{-s}\unicode[STIX]{x1D6E4}(s)$
and
$\tilde{\unicode[STIX]{x1D709}}(s):=\unicode[STIX]{x1D6E4}_{\mathbb{C}}(s)\unicode[STIX]{x1D701}(s)$
. We set
$$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D6EC}}(s,f,Ad):=\unicode[STIX]{x1D6E4}_{\mathbb{C}}(s)\unicode[STIX]{x1D6E4}_{\mathbb{C}}(s+2k-2n-1)L(s,f,Ad). & & \displaystyle \nonumber\end{eqnarray}$$
Then it is known from [Reference Katsurada and Kawamura14] that
$$\begin{eqnarray}\displaystyle \langle G,G\rangle =(1+\unicode[STIX]{x1D6FF}_{n,1})2^{-6k(n-1)+n(2n-3)}\langle g,g\rangle \mathop{\prod }_{i=1}^{n-1}\tilde{\unicode[STIX]{x1D709}}(2i)\tilde{\unicode[STIX]{x1D6EC}}(2i+1,f,Ad). & & \displaystyle \nonumber\end{eqnarray}$$
Therefore the residue of
${\mathcal{R}}_{1}(G;s)$
at
$s=n$
is
$$\begin{eqnarray}\displaystyle & & \displaystyle \text{Res}_{s=n}{\mathcal{R}}_{1}(G;s)\nonumber\\ \displaystyle & & \displaystyle \quad =2^{-2k(3n-4)+n(2n-3)-1}\unicode[STIX]{x1D70B}^{k-3/2}\langle g,g\rangle \mathop{\prod }_{i=1}^{n-1}\tilde{\unicode[STIX]{x1D709}}(2i)\tilde{\unicode[STIX]{x1D6EC}}(2i+1,f,Ad).\nonumber\end{eqnarray}$$
Remark also that the infinite product
$$\begin{eqnarray}\displaystyle \mathop{\prod }_{p\nmid |D_{0}|}\bigg\{1+p^{-2s-1}-\frac{(\frac{D_{0}}{p})(1+p^{2n-2})p^{-k+1}a_{f}(p)}{1+p^{2s}}\bigg\} & & \displaystyle \nonumber\end{eqnarray}$$
in Theorem 1.1 appears in a formula for a certain two-variable Dirichlet series
$\mathscr{L}_{-1}(f;\unicode[STIX]{x1D706},s)$
associated to
$f$
(cf. [Reference Ibukiyama and Katsurada11, p. 225]).
Finally, it is desirable to obtain an explicit formula for
$\langle \unicode[STIX]{x1D719}_{|D_{0}|},\unicode[STIX]{x1D719}_{|D_{0}|}\rangle$
with fundamental discriminants
$D_{0}$
. Then we will obtain a more precise formula for a generalization of the formula in [Reference Kohnen and Zagier17, p. 182, line 5]. An explicit formula for
$\langle \unicode[STIX]{x1D719}_{|D_{0}|},\unicode[STIX]{x1D719}_{|D_{0}|}\rangle$
itself can also be considered as a generalization of the well-known formula from [Reference Kohnen and Zagier17, Theorem 1] for the
$L$
-values and Fourier coefficients of modular forms of half-integral weight. However, it seems difficult to calculate
$\langle \unicode[STIX]{x1D719}_{|D_{0}|},\unicode[STIX]{x1D719}_{|D_{0}|}\rangle$
at the moment.
This article is organized as follows. In §2 we prepare some notation. We also recall definitions of Jacobi forms and the index-shift maps of Jacobi forms. In §3 we recall Ikeda lifts and construct Siegel modular forms of half-integral weight. In §4 we review a generalization of the Maass relation for Siegel modular forms of half-integral weight. We also introduce an index-shift map
$D_{2n-2}(N^{2})$
for Jacobi forms of half-integral weight. In §5 we review a linear isomorphism between Jacobi forms of integral weight and half-integral weight. In §6 we give a formula for the adjoint map
$D_{n^{\prime }}^{\ast }(N^{2})$
of
$D_{n^{\prime }}(N^{2})$
with respect to the Petersson inner product. In §7 we review the Fourier–Jacobi coefficients of the generalized Cohen–Eisenstein series and Jacobi–Eisenstein series. We also calculate their image by an adjoint map
$U_{N}^{\ast }$
of a certain index-shift map
$U_{N}$
. Finally, in §8 we give a proof of Theorem 1.1.
2 Notation and definitions
We denote by
$\mathbb{Z}_{{>}0}$
(respectively,
$\mathbb{R}_{{>}0}$
) the set of all positive integers (respectively, positive real numbers). The symbol
$R^{(n,m)}$
denotes the set of
$n\times m$
matrices with entries in a ring
$R$
. The symbol
$L_{n}^{\ast }$
denotes the set of all semi-positive-definite, half-integral symmetric matrices of size
$n$
, and the symbol
$L_{n}^{+}$
denotes the set of all positive definite, half-integral symmetric matrices of size
$n$
. The transpose of a matrix
$B$
is denoted by
$\text{}^{t\!}B$
. We write
$A[B]:=\text{}^{t\!}BAB$
for two matrices
$A\in R^{(n,n)}$
and
$B\in R^{(n,m)}$
. We write the identity matrix (respectively, zero matrix) of size
$n$
by
$1_{n}$
(respectively,
$0_{n}$
). We denote by
$\text{tr}(S)$
the trace of a square matrix
$S$
and we write
$e(S):=\text{e}^{2\unicode[STIX]{x1D70B}\sqrt{-1}\,\text{tr}(S)}$
for a square matrix
$S$
. For square matrices
$a_{1},\ldots ,a_{n}$
, we denote by
$\text{diag}(a_{1},\ldots ,a_{n})$
the diagonal matrix
$\biggl(\!\begin{smallmatrix}a_{1} & & \\ & \ddots & \\ & & a_{n}\end{smallmatrix}\!\biggr)$
. The symbol
$p$
is reserved for prime numbers. For any odd prime
$p$
the symbol
$(\frac{\ast }{p})$
denotes the Legendre symbol. If
$p=2$
, then we denote
$(\frac{d}{2})=1$
,
$-1$
or
$0$
for
$d\equiv \pm 1\;\text{mod}\;8$
,
$d\equiv \pm 3\;\text{mod}\;8$
or
$d\equiv 0\;\text{mod}\;2$
, respectively.
We denote by
$\mathfrak{H}_{n}$
the Siegel upper half space of degree
$n$
and denote by
$\text{Sp}_{n}(\mathbb{R})$
the real symplectic group of size
$2n$
. We set
$\unicode[STIX]{x1D6E4}_{n}:=\text{Sp}_{n}(\mathbb{Z})$
. We put
$\unicode[STIX]{x1D6E4}_{0}^{(n)}(4):=\big\{\!\big(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\big)\!\in \unicode[STIX]{x1D6E4}_{n}\,\big|\,C\in 4\mathbb{Z}^{(n,n)}\!\big\}$
and put
$\unicode[STIX]{x1D6E4}_{\infty }^{(n)}:=\big\{\!\big(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\big)\in \unicode[STIX]{x1D6E4}_{n}\,\big|\,C=0_{n}\!\big\}$
. We denote by
$M_{k-1/2}^{(n)}$
the vector space of Siegel modular forms of weight
$k-\frac{1}{2}$
of
$\unicode[STIX]{x1D6E4}_{0}^{(n)}(4)$
. We write
$M_{k-1/2}^{+(n)}$
for the plus-space of weight
$k-\frac{1}{2}$
of degree
$n$
, which is a certain subspace of
$M_{k-1/2}^{(n)}$
and it is a generalization of the Kohnen plus-space for general degree (cf. Ibukiyama [Reference Ibukiyama10]; see also §5). The symbol
$S_{k-1/2}^{+(n)}$
denotes the vector space of all Siegel cusp forms in
$M_{k-1/2}^{+(n)}$
.
In the following we quote some notation and definitions from [Reference Hayashida7].
2.1 Jacobi group
For a positive integer
$n$
we define the group
$$\begin{eqnarray}\displaystyle \text{GSp}_{n}^{+}(\mathbb{R}) & := & \displaystyle \left\{g\in \mathbb{R}^{(2n,2n)}~\biggl|~g\left(\begin{array}{@{}cc@{}}0_{n} & -1_{n}\\ 1_{n} & 0_{n}\end{array}\right)\,\text{}^{t\!}g=n(g)\left(\begin{array}{@{}cc@{}}0_{n} & -1_{n}\\ 1_{n} & 0_{n}\end{array}\right)\right.\nonumber\\ \displaystyle & & \displaystyle \left.\vphantom{g\in \mathbb{R}^{(2n,2n)^{2}}\left(\begin{array}{@{}cc@{}}0_{n} & -1_{n}\\ 1_{n} & 0_{n}\end{array}\right)}\text{ for some }n(g)\in \mathbb{R}_{{>}0}\right\}.\nonumber\end{eqnarray}$$
For a matrix
$g\in \text{GSp}_{n}^{+}(\mathbb{R})$
, the number
$n(g)$
in the above definition of
$\text{GSp}_{n}^{+}(\mathbb{R})$
is called the similitude of the matrix
$g$
.
For positive integers
$n$
and
$r$
, we define a subgroup
$G_{n,r}^{J}\subset \text{GSp}_{n+r}^{+}(\mathbb{R})$
by
$$\begin{eqnarray}\displaystyle G_{n,r}^{J} & := & \displaystyle \left\{\left.\left(\begin{array}{@{}cccc@{}}A & & B & \\ & U & & \\ C & & D & \\ & & & V\end{array}\right)\left(\begin{array}{@{}cccc@{}}1_{n} & & & \unicode[STIX]{x1D707}\\ \text{}^{t\!}\unicode[STIX]{x1D706} & 1_{r} & \text{}^{t\!}\unicode[STIX]{x1D707} & \text{}^{t\!}\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}+\unicode[STIX]{x1D705}\\ & & 1_{n} & -\unicode[STIX]{x1D706}\\ & & & 1_{r}\end{array}\right)\right.\right.\nonumber\\ \displaystyle & & \displaystyle \!\quad \in \left.\left.\text{GSp}_{n+r}^{+}(\mathbb{R})\right|\begin{array}{@{}c@{}}A,B,C,D,\\ U,V,\\ \unicode[STIX]{x1D706},\unicode[STIX]{x1D707},\unicode[STIX]{x1D705}\end{array}\right\},\nonumber\end{eqnarray}$$
where
$\big(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\big)\in \text{GSp}_{n}^{+}(\mathbb{R})$
,
$\big(\!\begin{smallmatrix}U & 0\\ 0 & V\end{smallmatrix}\!\big)\in \text{GSp}_{r}^{+}(\mathbb{R})$
,
$\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in \mathbb{R}^{(n,r)}$
and
$\unicode[STIX]{x1D705}=\text{}^{t\!}\unicode[STIX]{x1D705}\in \mathbb{R}^{(r,r)}$
. We remark that two matrices
$\big(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\big)$
and
$\big(\!\begin{smallmatrix}U & 0\\ 0 & V\end{smallmatrix}\!\big)$
in the above notation have the same similitude. We abbreviate an element
$\bigg(\!\begin{smallmatrix}A & & B & \\ & U & & \\ C & & D & \\ & & & V\end{smallmatrix}\!\bigg)\bigg(\!\begin{smallmatrix}1_{n} & & & \unicode[STIX]{x1D707}\\ \text{}^{t\!}\unicode[STIX]{x1D706} & 1_{r} & \text{}^{t\!}\unicode[STIX]{x1D707} & \text{}^{t\!}\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}+\unicode[STIX]{x1D705}\\ & & 1_{n} & -\unicode[STIX]{x1D706}\\ & & & 1_{r}\end{smallmatrix}\!\bigg)$
as
$$\begin{eqnarray}\left(\left(\begin{array}{@{}cc@{}}A & B\\ C & D\end{array}\right)\times \left(\begin{array}{@{}cc@{}}U & 0\\ 0 & V\end{array}\right),[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\right).\end{eqnarray}$$
We will often write
$$\begin{eqnarray}\displaystyle \left(\left(\begin{array}{@{}cc@{}}A & B\\ C & D\end{array}\right),[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\right) & & \displaystyle \nonumber\end{eqnarray}$$
instead of
$\big(\big(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\big)\times 1_{2r},[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\big)$
for simplicity. The element
$\big(\big(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\big),[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\big)$
belongs to
$\text{Sp}_{n+r}(\mathbb{R})$
. Similarly, we abbreviate an element
$$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cccc@{}}1_{n} & & & \unicode[STIX]{x1D707}\\ \text{}^{t\!}\unicode[STIX]{x1D706} & 1_{r} & \text{}^{t\!}\unicode[STIX]{x1D707} & \text{}^{t\!}\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}+\unicode[STIX]{x1D705}\\ & & 1_{n} & -\unicode[STIX]{x1D706}\\ & & & 1_{r}\end{array}\right)\left(\begin{array}{@{}cccc@{}}A & & B & \\ & U & & \\ C & & D & \\ & & & V\end{array}\right) & & \displaystyle \nonumber\end{eqnarray}$$
as
$$\begin{eqnarray}\displaystyle \left([(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}],\left(\begin{array}{@{}cc@{}}A & B\\ C & D\end{array}\right)\times \left(\begin{array}{@{}cc@{}}U & 0\\ 0 & V\end{array}\right)\right), & & \displaystyle \nonumber\end{eqnarray}$$
and as
$\big([(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}],\big(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\big)\big)$
for the case
$U=V=1_{r}$
.
If there is no possibility of confusion, we write
$$\begin{eqnarray}[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\end{eqnarray}$$
for the element
$(1_{2n},[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}])$
for simplicity.
We denote a subgroup of
$G_{n,r}^{J}$
by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{n,r}^{J}:=\{(M,[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}])\in G_{n,r}^{J}\mid M\in \unicode[STIX]{x1D6E4}_{n},\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in \mathbb{Z}^{(n,r)},\unicode[STIX]{x1D705}\in \mathbb{Z}^{(r,r)}\}. & & \displaystyle \nonumber\end{eqnarray}$$
2.2 Groups
$\widetilde{\text{GSp}_{n}^{+}(\mathbb{R})}$
and
$\widetilde{G_{n,1}^{J}}$
The symbol
$\widetilde{\text{GSp}_{n}^{+}(\mathbb{R})}$
denotes the group which consists of pairs
$(M,\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F}))$
, where
$M$
is a matrix
$M=\big(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\big)\in \text{GSp}_{n}^{+}(\mathbb{R})$
, and where
$\unicode[STIX]{x1D711}$
is a holomorphic function on
$\mathfrak{H}_{n}$
which satisfies
$|\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F})|^{2}=\det (M)^{-1/2}|\text{det}(C\unicode[STIX]{x1D70F}+D)|$
. The group operation on
$\widetilde{\text{GSp}_{n}^{+}(\mathbb{R})}$
is given by
$(M,\unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F}))(M^{\prime },\unicode[STIX]{x1D711}^{\prime }(\unicode[STIX]{x1D70F})):=(MM^{\prime },\unicode[STIX]{x1D711}(M^{\prime }\unicode[STIX]{x1D70F})\unicode[STIX]{x1D711}^{\prime }(\unicode[STIX]{x1D70F}))$
for
$(M,\unicode[STIX]{x1D711})$
,
$(M^{\prime },\unicode[STIX]{x1D711}^{\prime })\in \widetilde{\text{GSp}_{n}^{+}(\mathbb{R})}$
.
We denote the theta constant
$\unicode[STIX]{x1D703}^{(n)}(\unicode[STIX]{x1D70F}):=\sum _{p\in \mathbb{Z}^{(n,1)}}e(\unicode[STIX]{x1D70F}[p])$
for
$\unicode[STIX]{x1D70F}\in \mathfrak{H}_{n}$
. We embed
$\unicode[STIX]{x1D6E4}_{0}^{(n)}(4)$
into the group
$\widetilde{\text{GSp}_{n}^{+}(\mathbb{R})}$
via
$M\mapsto (M,\unicode[STIX]{x1D703}^{(n)}(M\unicode[STIX]{x1D70F})\unicode[STIX]{x1D703}^{(n)}(\unicode[STIX]{x1D70F})^{-1})$
.
We denote by
$\unicode[STIX]{x1D6E4}_{0}^{(n)}(4)^{\ast }$
the image of
$\unicode[STIX]{x1D6E4}_{0}^{(n)}(4)$
in
$\widetilde{\text{GSp}_{n}^{+}(\mathbb{R})}$
by this embedding.
We define the group
$$\begin{eqnarray}\displaystyle H_{n,1}(\mathbb{R}):=\{[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\in \text{Sp}_{n+1}(\mathbb{R})\mid \unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in \mathbb{R}^{(n,1)},\unicode[STIX]{x1D705}\in \mathbb{R}\} & & \displaystyle \nonumber\end{eqnarray}$$
and define the group
$$\begin{eqnarray}\displaystyle \widetilde{G_{n,1}^{J}} & := & \displaystyle \widetilde{\text{GSp}_{n}^{+}(\mathbb{R})}\ltimes H_{n,1}(\mathbb{R})\nonumber\\ \displaystyle & = & \displaystyle \{(\tilde{M},[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}])\mid \tilde{M}\in \widetilde{\text{GSp}_{n}^{+}(\mathbb{R})},[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\in H_{n,1}(\mathbb{R})\}\nonumber\end{eqnarray}$$
with the group operation
$$\begin{eqnarray}\displaystyle (\tilde{M_{1}},[(\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D707}_{1}),\unicode[STIX]{x1D705}_{1}])\cdot (\tilde{M_{2}},[(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D707}_{2}),\unicode[STIX]{x1D705}_{2}]):=(\tilde{M_{1}}\tilde{M_{2}},[(\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D707}^{\prime }),\unicode[STIX]{x1D705}^{\prime }]) & & \displaystyle \nonumber\end{eqnarray}$$
for
$(\tilde{M_{i}},[(\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D707}_{i}),\unicode[STIX]{x1D705}_{i}])\in \widetilde{G_{n,1}^{J}}\;(i=1,2)$
, and where
$[(\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D707}^{\prime }),\unicode[STIX]{x1D705}^{\prime }]\in H_{n,1}(\mathbb{R})$
is the matrix determined via the identity
$$\begin{eqnarray}\displaystyle & & \displaystyle \left(M_{1}\times \left(\begin{array}{@{}cc@{}}n(M_{1}) & 0\\ 0 & 1\end{array}\right),[(\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D707}_{1}),\unicode[STIX]{x1D705}_{1}]\right)\left(M_{2}\times \left(\begin{array}{@{}cc@{}}n(M_{2}) & 0\\ 0 & 1\end{array}\right),[(\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D707}_{2}),\unicode[STIX]{x1D705}_{2}]\right)\nonumber\\ \displaystyle & & \displaystyle \quad =\left(M_{1}M_{2}\times \left(\begin{array}{@{}cc@{}}n(M_{1})n(M_{2}) & 0\\ 0 & 1\end{array}\right),[(\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D707}^{\prime }),\unicode[STIX]{x1D705}^{\prime }]\right)\nonumber\end{eqnarray}$$
in
$G_{n,1}^{J}$
. Here
$n(M_{i})$
is the similitude of
$M_{i}$
.
2.3 Action of the Jacobi group
The group
$G_{n,r}^{J}$
acts on
$\mathfrak{H}_{n}\times \mathbb{C}^{(n,r)}$
by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FE}\cdot (\unicode[STIX]{x1D70F},z):=\left(\left(\begin{array}{@{}cc@{}}A & B\\ C & D\end{array}\right)\cdot \unicode[STIX]{x1D70F},\text{}^{t\!}(C\unicode[STIX]{x1D70F}+D)^{-1}(z+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707})\text{}^{t\!}U\right) & & \displaystyle \nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D6FE}=\big(\big(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\big)\times \big(\!\begin{smallmatrix}U & 0\\ 0 & V\end{smallmatrix}\!\big),[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\big)\in G_{n,r}^{J}$
and for any
$(\unicode[STIX]{x1D70F},z)\in \mathfrak{H}_{n}\times \mathbb{C}^{(n,r)}$
. Here
$\big(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\big)\cdot \unicode[STIX]{x1D70F}:=(A\unicode[STIX]{x1D70F}+B)(C\unicode[STIX]{x1D70F}+D)^{-1}$
is the usual transformation.
The group
$\widetilde{G_{n,1}^{J}}$
acts on
$\mathfrak{H}_{n}\times \mathbb{C}^{(n,1)}$
through the projection
$\widetilde{G_{n,1}^{J}}\rightarrow G_{n,1}^{J}$
. This means that
$\widetilde{G_{n,1}^{J}}$
acts on
$\mathfrak{H}_{n}\times \mathbb{C}^{(n,1)}$
by
$$\begin{eqnarray}\displaystyle \tilde{\unicode[STIX]{x1D6FE}}\cdot (\unicode[STIX]{x1D70F},z):=\left(M\times \left(\begin{array}{@{}cc@{}}n(M) & 0\\ 0 & 1\end{array}\right),[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\right)\cdot (\unicode[STIX]{x1D70F},z) & & \displaystyle \nonumber\end{eqnarray}$$
for
$\tilde{\unicode[STIX]{x1D6FE}}=((M,\unicode[STIX]{x1D711}),[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}])\in \widetilde{G_{n,1}^{J}}$
and for
$(\unicode[STIX]{x1D70F},z)\in \mathfrak{H}_{n}\times \mathbb{C}^{(n,1)}$
. Here
$n(M)$
is the similitude of
$M\in \text{GSp}_{n}^{+}(\mathbb{R})$
.
2.4 Factors of automorphy
Let
$k$
be an integer and let
${\mathcal{M}}$
be a symmetric matrix of size
$r$
with entries in
$\mathbb{R}$
. For any
$\unicode[STIX]{x1D6FE}=\big(\big(\!\begin{smallmatrix}A & B\\ C & D\end{smallmatrix}\!\big)\times \big(\!\begin{smallmatrix}U & 0\\ 0 & V\end{smallmatrix}\!\big),[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}]\big)\in G_{n,r}^{J}$
we define the factor of automorphy
$$\begin{eqnarray}\displaystyle & & \displaystyle J_{k,{\mathcal{M}}}(\unicode[STIX]{x1D6FE},(\unicode[STIX]{x1D70F},z))\nonumber\\ \displaystyle & & \displaystyle \quad :=\det (V)^{k}\det (C\unicode[STIX]{x1D70F}+D)^{k}e(V^{-1}{\mathcal{M}}U(((C\unicode[STIX]{x1D70F}+D)^{-1}C)[z+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}]))\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,e(-V^{-1}{\mathcal{M}}U(\text{}^{t\!}\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}\unicode[STIX]{x1D706}+\text{}^{t\!}z\unicode[STIX]{x1D706}+\text{}^{t\!}\unicode[STIX]{x1D706}z+\text{}^{t\!}\unicode[STIX]{x1D707}\unicode[STIX]{x1D706}+\text{}^{t\!}\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}+\unicode[STIX]{x1D705})).\nonumber\end{eqnarray}$$
We define the slash operator
$|_{k,{\mathcal{M}}}$
by
$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D713}|_{k,{\mathcal{M}}}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D70F},z):=J_{k,{\mathcal{M}}}(\unicode[STIX]{x1D6FE},(\unicode[STIX]{x1D70F},z))^{-1}\unicode[STIX]{x1D713}(\unicode[STIX]{x1D6FE}\cdot (\unicode[STIX]{x1D70F},z)) & & \displaystyle \nonumber\end{eqnarray}$$
for any function
$\unicode[STIX]{x1D713}$
on
$\mathfrak{H}_{n}\times \mathbb{C}^{(n,r)}$
and for any
$\unicode[STIX]{x1D6FE}\in G_{n,r}^{J}$
. We remark that
$$\begin{eqnarray}\displaystyle & \displaystyle J_{k,{\mathcal{M}}}(\unicode[STIX]{x1D6FE}_{1}\unicode[STIX]{x1D6FE}_{2},(\unicode[STIX]{x1D70F},z))=J_{k,{\mathcal{M}}}(\unicode[STIX]{x1D6FE}_{1},\unicode[STIX]{x1D6FE}_{2}\cdot (\unicode[STIX]{x1D70F},z))J_{k,V_{1}^{-1}{\mathcal{M}}U_{1}}(\unicode[STIX]{x1D6FE}_{2},(\unicode[STIX]{x1D70F},z)), & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D713}|_{k,{\mathcal{M}}}\unicode[STIX]{x1D6FE}_{1}\unicode[STIX]{x1D6FE}_{2}=(\unicode[STIX]{x1D713}|_{k,{\mathcal{M}}}\unicode[STIX]{x1D6FE}_{1})|_{k,V_{1}^{-1}{\mathcal{M}}U_{1}}\unicode[STIX]{x1D6FE}_{2} & \displaystyle \nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D6FE}_{i}=(M_{i}\times \big(\!\begin{smallmatrix}U_{i} & 0\\ 0 & V_{i}\end{smallmatrix}\!\big),[(\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D707}_{i}),\unicode[STIX]{x1D705}_{i}])\in G_{n,r}^{J}\;(i=1,2)$
.
Let
$k$
and
$m$
be integers. For any
$\tilde{\unicode[STIX]{x1D6FE}}=((M,\unicode[STIX]{x1D711}),[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}])\in \widetilde{G_{n,1}^{J}}$
we define the factor of automorphy
$$\begin{eqnarray}\displaystyle J_{k-1/2,m}(\tilde{\unicode[STIX]{x1D6FE}},(\unicode[STIX]{x1D70F},z)) & := & \displaystyle \unicode[STIX]{x1D711}(\unicode[STIX]{x1D70F})^{2k-1}e(n(M)m(((C\unicode[STIX]{x1D70F}+D)^{-1}C)[z+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D706}+\unicode[STIX]{x1D707}]))\nonumber\\ \displaystyle & & \displaystyle \times \,e(-n(M)m(\text{}^{t\!}\unicode[STIX]{x1D706}\unicode[STIX]{x1D70F}\unicode[STIX]{x1D706}+\text{}^{t\!}z\unicode[STIX]{x1D706}+\text{}^{t\!}\unicode[STIX]{x1D706}z+\text{}^{t\!}\unicode[STIX]{x1D707}\unicode[STIX]{x1D706}+\text{}^{t\!}\unicode[STIX]{x1D706}\unicode[STIX]{x1D707}+\unicode[STIX]{x1D705})),\nonumber\end{eqnarray}$$
where
$n(M)$
is the similitude of
$M$
. We define the slash operator
$|_{k-1/2,m}$
by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}|_{k-1/2,m}\tilde{\unicode[STIX]{x1D6FE}}:=J_{k-1/2,m}(\tilde{\unicode[STIX]{x1D6FE}},(\unicode[STIX]{x1D70F},z))^{-1}\unicode[STIX]{x1D719}(\tilde{\unicode[STIX]{x1D6FE}}\cdot (\unicode[STIX]{x1D70F},z)) & & \displaystyle \nonumber\end{eqnarray}$$
for any function
$\unicode[STIX]{x1D719}$
on
$\mathfrak{H}_{n}\times \mathbb{C}^{(n,1)}$
. We remark that
$$\begin{eqnarray}\displaystyle & \displaystyle J_{k-1/2,m}(\tilde{\unicode[STIX]{x1D6FE}_{1}}\tilde{\unicode[STIX]{x1D6FE}_{2}},(\unicode[STIX]{x1D70F},z))=J_{k-1/2,m}(\tilde{\unicode[STIX]{x1D6FE}_{1}},\tilde{\unicode[STIX]{x1D6FE}_{2}}\cdot (\unicode[STIX]{x1D70F},z))J_{k-1/2,n(M_{1})m}(\tilde{\unicode[STIX]{x1D6FE}_{2}},(\unicode[STIX]{x1D70F},z)) & \displaystyle \nonumber\\ \displaystyle & \displaystyle \unicode[STIX]{x1D719}|_{k-1/2,m}\tilde{\unicode[STIX]{x1D6FE}_{1}}\tilde{\unicode[STIX]{x1D6FE}_{2}}=(\unicode[STIX]{x1D719}|_{k-1/2,m}\tilde{\unicode[STIX]{x1D6FE}_{1}})|_{k-1/2,n(M_{1})m}\tilde{\unicode[STIX]{x1D6FE}_{2}} & \displaystyle \nonumber\end{eqnarray}$$
for any
$\tilde{\unicode[STIX]{x1D6FE}_{i}}=((M_{i},\unicode[STIX]{x1D711}_{i}),[(\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D707}_{i}),\unicode[STIX]{x1D705}_{i}])\in \widetilde{G_{n,1}^{J}}\;(i=1,2)$
.
2.5 Jacobi forms of matrix index
We quote the definition of Jacobi forms of matrix index from [Reference Ziegler18].
Definition 2.1. For an integer
$k$
and for a matrix
${\mathcal{M}}\in L_{r}^{+}$
, a
$\mathbb{C}$
-valued holomorphic function
$\unicode[STIX]{x1D713}$
on
$\mathfrak{H}_{n}\times \mathbb{C}^{(n,r)}$
is called a Jacobi form of weight
$k$
of index
${\mathcal{M}}$
of degree
$n$
, if
$\unicode[STIX]{x1D713}$
satisfies the following two conditions:
(1) the transformation formula
$\unicode[STIX]{x1D713}|_{k,{\mathcal{M}}}\unicode[STIX]{x1D6FE}=\unicode[STIX]{x1D713}$
for any
$\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{n,r}^{J}$
;(2)
$\unicode[STIX]{x1D713}$
has the Fourier expansion $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}(\unicode[STIX]{x1D70F},z)=\mathop{\sum }_{\substack{ N\in L_{n}^{\ast },R\in Z^{(n,r)} \\ 4N-R{\mathcal{M}}^{-1}\hspace{0.0pt}\text{}^{t\!}R\geqslant 0}}c(N,R)e(N\unicode[STIX]{x1D70F})e(\text{}^{t\!}Rz). & & \displaystyle \nonumber\end{eqnarray}$$
We remark that the second condition follows from the Koecher principle (cf. [Reference Ziegler18, Lemma 1.6]) if
$n>1$
. In condition (2), if
$\unicode[STIX]{x1D713}$
satisfies the condition
$c(N,R)=0$
unless
$4N-R{\mathcal{M}}^{-1t}\!R>0$
, then
$\unicode[STIX]{x1D713}$
is called a Jacobi cusp form.
We denote by
$J_{k,{\mathcal{M}}}^{(n)}$
(respectively,
$J_{k,{\mathcal{M}}}^{(n)\;\text{cusp}}$
) the
$\mathbb{C}$
-vector space of Jacobi forms (respectively, Jacobi cusp forms) of weight
$k$
of index
${\mathcal{M}}$
of degree
$n$
.
For
$\unicode[STIX]{x1D713}_{1}$
,
$\unicode[STIX]{x1D713}_{2}\in J_{k,{\mathcal{M}}}^{(n)\;\text{cusp}}$
, the Petersson inner product is defined by
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}\rangle :=\int _{{\mathcal{F}}_{n,r}}\unicode[STIX]{x1D713}_{1}(\unicode[STIX]{x1D70F},z)\overline{\unicode[STIX]{x1D713}_{2}(\unicode[STIX]{x1D70F},z)}\text{e}^{-4\unicode[STIX]{x1D70B}Tr({\mathcal{M}}v^{-1}[y])}\det (v)^{k-n-r-1}\,du\,dv\,dx\,dy, & & \displaystyle \nonumber\end{eqnarray}$$
where
${\mathcal{F}}_{n,r}:=\unicode[STIX]{x1D6E4}_{n,r}^{J}\backslash (\mathfrak{H}_{n}\times \mathbb{C}^{(n,r)})$
,
$\unicode[STIX]{x1D70F}=u+\text{i}v$
,
$z=x+\text{i}y$
,
$du=\prod _{i\leqslant j}u_{i,j}$
,
$dv=\prod _{i\leqslant j}v_{i,j}$
,
$dx=\prod _{i,j}x_{i,j}$
and
$dy=\prod _{i,j}y_{i,j}$
.
2.6 Jacobi forms of half-integral weight
We define a subgroup
$\unicode[STIX]{x1D6E4}_{n,1}^{J\ast }$
of
$\widetilde{G_{n,1}^{J}}$
by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{n,1}^{J\ast } & := & \displaystyle \{(M^{\ast },[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}])\in \widetilde{G_{n,1}^{J}}\mid M^{\ast }\in \unicode[STIX]{x1D6E4}_{0}^{(n)}(4)^{\ast },\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in \mathbb{Z}^{(n,1)},\unicode[STIX]{x1D705}\in \mathbb{Z}\}\nonumber\\ \displaystyle & \cong & \displaystyle \unicode[STIX]{x1D6E4}_{0}^{(n)}(4)^{\ast }\ltimes H_{n,1}(\mathbb{Z}),\nonumber\end{eqnarray}$$
where we put
$H_{n,1}(\mathbb{Z}):=H_{n,1}(\mathbb{R})\cap \mathbb{Z}^{(2n+2,2n+2)}$
, and where the group
$\unicode[STIX]{x1D6E4}_{0}^{(n)}(4)^{\ast }$
was defined in §2.2.
Definition 2.2. For integers
$k$
and
$m$
, a holomorphic function
$\unicode[STIX]{x1D719}$
on
$\mathfrak{H}_{n}\times \mathbb{C}^{(n,1)}$
is called a Jacobi form of weight
$k-\frac{1}{2}$
of index
$m$
of degree
$n$
, if
$\unicode[STIX]{x1D719}$
satisfies the following two conditions:
(1)
$\unicode[STIX]{x1D719}|_{k-1/2,m}\unicode[STIX]{x1D6FE}^{\ast }=\unicode[STIX]{x1D719}$
for any
$\unicode[STIX]{x1D6FE}^{\ast }\in \unicode[STIX]{x1D6E4}_{n,1}^{J\ast }$
;(2)
$\unicode[STIX]{x1D719}^{2}|_{2k-1,2m}\unicode[STIX]{x1D6FE}$
has the Fourier expansion, for any
$\unicode[STIX]{x1D6FE}\in \unicode[STIX]{x1D6E4}_{n,1}^{J}$
, with an integer$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D719}^{2}|_{2k-1,2m}\unicode[STIX]{x1D6FE})(\unicode[STIX]{x1D70F},z)=\mathop{\sum }_{\substack{ N\in L_{n}^{\ast },R\in \mathbb{Z}^{(n,1)} \\ 4Nm-hR\text{}^{t\!}R\geqslant 0}}C(N,R)e\biggl(\frac{1}{h}N\unicode[STIX]{x1D70F}\biggr)e(\text{}^{t\!}Rz), & & \displaystyle \nonumber\end{eqnarray}$$
$h>0$
, and where the slash operator
$|_{2k-1,2m}$
was defined in §2.4.
In condition (2), for any
$\unicode[STIX]{x1D6FE}$
, if
$\unicode[STIX]{x1D719}$
satisfies the condition
$C(N,R)=0$
unless
$4Nm-hR\text{}^{t\!}R>0$
, then
$\unicode[STIX]{x1D719}$
is called a Jacobi cusp form.
We denote by
$J_{k-1/2,m}^{(n)}$
the
$\mathbb{C}$
-vector space of Jacobi forms of weight
$k-\frac{1}{2}$
of index
$m$
of degree
$n$
.
For
$\unicode[STIX]{x1D719}_{1}$
,
$\unicode[STIX]{x1D719}_{2}\in J_{k-1/2,m}^{(n)\;\text{cusp}}$
, the Petersson inner product is defined by
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}\rangle & := & \displaystyle [\unicode[STIX]{x1D6E4}_{n}:\unicode[STIX]{x1D6E4}_{0}^{(n)}(4)]^{-1}\nonumber\\ \displaystyle & & \displaystyle \times \,\int _{{\mathcal{F}}_{n,1,4}}\unicode[STIX]{x1D719}_{1}(\unicode[STIX]{x1D70F},z)\overline{\unicode[STIX]{x1D719}_{2}(\unicode[STIX]{x1D70F},z)}\text{e}^{-4\unicode[STIX]{x1D70B}mv^{-1}[y]}\det (v)^{k-n-5/2}\,du\,dv\,dx\,dy,\nonumber\end{eqnarray}$$
where
${\mathcal{F}}_{n,1,4}:=\unicode[STIX]{x1D6E4}_{n,1}^{J}(4)\backslash (\mathfrak{H}_{n}\times \mathbb{C}^{(n,1)})$
,
$\unicode[STIX]{x1D70F}=u+\text{i}v$
,
$z=x+\text{i}y$
,
$du=\prod _{i\leqslant j}u_{i,j}$
,
$dv=\prod _{i\leqslant j}v_{i,j}$
,
$dx=\prod _{i}x_{i,1}$
,
$dy=\prod _{i}y_{i,1}$
and where
$[\unicode[STIX]{x1D6E4}_{n}:\unicode[STIX]{x1D6E4}_{0}^{(n)}(4)]$
denotes the index of
$\unicode[STIX]{x1D6E4}_{0}^{(n)}(4)$
in
$\unicode[STIX]{x1D6E4}_{n}$
, and we put
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6E4}_{n,1}^{J}(4) & := & \displaystyle \{(M,[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),\unicode[STIX]{x1D705}])\in \unicode[STIX]{x1D6E4}_{n,1}^{J}\mid M\in \unicode[STIX]{x1D6E4}_{0}^{(n)}(4),\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}\in \mathbb{Z}^{(n,1)},\unicode[STIX]{x1D705}\in \mathbb{Z}\}\nonumber\\ \displaystyle & \cong & \displaystyle \unicode[STIX]{x1D6E4}_{0}^{(n)}(4)\ltimes H_{n,1}(\mathbb{Z}).\nonumber\end{eqnarray}$$
Lemma 2.3. Let
$\unicode[STIX]{x1D719}\in J_{k-1/2,m}^{(n)}$
. Then
$\unicode[STIX]{x1D719}$
is a Jacobi cusp form if and only if the function
$$\begin{eqnarray}\displaystyle \det (v)^{(2k-1)/4}\text{e}^{-2\unicode[STIX]{x1D70B}mv^{-1}[y]}|\unicode[STIX]{x1D719}(\unicode[STIX]{x1D70F},z)| & & \displaystyle \nonumber\end{eqnarray}$$
is bounded on
$\mathfrak{H}_{n}\times \mathbb{C}^{n}$
. Here we put
$v=\text{Im}(\unicode[STIX]{x1D70F})$
and
$y=\text{Im}(z)$
.
Proof. This is analogous to [Reference Klingen15, p. 410 Lemma] and [Reference Dulinski4, Corollary to Proposition 1]. We omit the details. ◻
2.7 Index-shift maps of Jacobi forms
In this subsection we introduce index-shift maps for two kinds of Jacobi forms (of matrix index and of half-integral weight). These are generalizations of the
$V_{l}$
-map in the sense of Eichler and Zagier [Reference Eichler and Zagier5].
We define
$\text{GSp}_{n}^{+}(\mathbb{Z}):=\text{GSp}_{n}^{+}(\mathbb{R})\cap \mathbb{Z}^{(2n,2n)}$
and
$$\begin{eqnarray}\displaystyle \widetilde{\text{GSp}_{n}^{+}(\mathbb{Z})}:=\{(M,\unicode[STIX]{x1D711})\in \widetilde{\text{GSp}_{n}^{+}(\mathbb{R})}\mid M\in \text{GSp}_{n}^{+}(\mathbb{Z})\}. & & \displaystyle \nonumber\end{eqnarray}$$
First we define index-shift maps for Jacobi forms of integral weight with some matrix indices. These index-shift maps were introduced in [Reference Hayashida7, §2.7].
Let
${\mathcal{M}}=\big(\!\begin{smallmatrix}\ast & \ast \\ \ast & 1\end{smallmatrix}\!\big)\in L_{2}^{+}$
. We take a matrix
$X\in \text{GSp}_{n}^{+}(\mathbb{Z})$
such that the similitude of
$X$
is
$n(X)=N^{2}$
with a natural number
$N$
. For any
$\unicode[STIX]{x1D713}\in J_{k,{\mathcal{M}}}^{(n)}$
we define the function
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}|V(X) & := & \displaystyle \mathop{\sum }_{u,v\in (\mathbb{Z}/N\mathbb{Z})^{(n,1)}}\mathop{\sum }_{M\in \unicode[STIX]{x1D6E4}_{n}\backslash \unicode[STIX]{x1D6E4}_{n}X\unicode[STIX]{x1D6E4}_{n}}\unicode[STIX]{x1D713}|_{k,{\mathcal{M}}}\nonumber\\ \displaystyle & & \displaystyle \times \,\left(M\times \left(\begin{array}{@{}cccc@{}}N^{2} & 0 & 0 & 0\\ 0 & N & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & N\end{array}\right),[((0,u),(0,v)),0_{2}]\right),\nonumber\end{eqnarray}$$
where
$(0,u),(0,v)\in (\mathbb{Z}/N\mathbb{Z})^{(n,2)}$
. See §2.1 for the notation for the matrix
$$\begin{eqnarray}\left(M\times \left(\begin{array}{@{}cccc@{}}N^{2} & 0 & 0 & 0\\ 0 & N & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & N\end{array}\right),[((0,u),(0,v)),0_{2}]\right).\end{eqnarray}$$
The above summations are finite sums and do not depend on the choice of the representatives
$u$
,
$v$
and
$M$
. One can check that
$\unicode[STIX]{x1D713}|V(X)$
belongs to
$J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n)}$
. This means that
$V(X)$
is a map,
$$\begin{eqnarray}\displaystyle V(X):J_{k,{\mathcal{M}}}^{(n)}\rightarrow J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n)}. & & \displaystyle \nonumber\end{eqnarray}$$
Moreover, if
$\unicode[STIX]{x1D713}\in J_{k,{\mathcal{M}}}^{(n)\;\text{cusp}}$
, then
$\unicode[STIX]{x1D713}|V(X)\in J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n)\;\text{cusp}}$
. This fact is shown by the expression for the Fourier coefficients of
$\unicode[STIX]{x1D713}|V(X)$
.
For the sake of simplicity we set
$$\begin{eqnarray}\displaystyle V_{\unicode[STIX]{x1D6FC},n-\unicode[STIX]{x1D6FC}}(p^{2}):=V(\text{diag}(1_{\unicode[STIX]{x1D6FC}},p1_{n-\unicode[STIX]{x1D6FC}},p^{2}1_{\unicode[STIX]{x1D6FC}},p1_{n-\unicode[STIX]{x1D6FC}})) & & \displaystyle \nonumber\end{eqnarray}$$
for any prime
$p$
and for any
$\unicode[STIX]{x1D6FC}\;(0\leqslant \unicode[STIX]{x1D6FC}\leqslant n)$
.
Next we quote the definition of the index-shift maps for Jacobi forms of half-integral weight of integer index from [Reference Hayashida7, §2.7]. We assume that
$p$
is an odd prime. Let
$m$
be a positive integer. Let
$Y=(X,\unicode[STIX]{x1D711})\in \widetilde{\text{GSp}_{n}^{+}(\mathbb{Z})}$
with
$n(X)=p^{2\unicode[STIX]{x1D708}}$
, where
$\unicode[STIX]{x1D708}$
is a positive integer. For
$\unicode[STIX]{x1D719}\in J_{k-1/2,m}^{(n)}$
we define
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}|\widetilde{V}(Y) & := & \displaystyle n(X)^{n(2k-1)/4-n(n+1)/2}\nonumber\\ \displaystyle & & \displaystyle \times \,\mathop{\sum }_{\tilde{M}\in \unicode[STIX]{x1D6E4}_{0}^{(n)}(4)^{\ast }\backslash \unicode[STIX]{x1D6E4}_{0}^{(n)}(4)^{\ast }Y\unicode[STIX]{x1D6E4}_{0}^{(n)}(4)^{\ast }}\unicode[STIX]{x1D719}|_{k-1/2,m}(\tilde{M},[(0,0),0]),\nonumber\end{eqnarray}$$
where the above summation is a finite sum and does not depend on the choice of the representatives
$\tilde{M}$
. One can show by a direct computation that
$\unicode[STIX]{x1D719}|\widetilde{V}(Y)$
belongs to
$J_{k-1/2,mp^{2\unicode[STIX]{x1D708}}}^{(n)}$
.
For the sake of simplicity we set
$$\begin{eqnarray}\displaystyle \tilde{V}_{\unicode[STIX]{x1D6FC},n-\unicode[STIX]{x1D6FC}}(p^{2}):=\tilde{V}((\text{diag}(1_{\unicode[STIX]{x1D6FC}},p1_{n-\unicode[STIX]{x1D6FC}},p^{2}1_{\unicode[STIX]{x1D6FC}},p1_{n-\unicode[STIX]{x1D6FC}}),p^{\unicode[STIX]{x1D6FC}/2})) & & \displaystyle \nonumber\end{eqnarray}$$
for any odd prime
$p$
and for any
$\unicode[STIX]{x1D6FC}\;(0\leqslant \unicode[STIX]{x1D6FC}\leqslant n)$
.
For the prime
$p=2$
, the index-shift map
$\tilde{V}_{\unicode[STIX]{x1D6FC},n-\unicode[STIX]{x1D6FC}}(4)$
is defined for a certain subspace
$J_{k-1/2,m}^{+(n)}$
of
$J_{k-1/2,m}^{(n)}$
. This is a map from
$J_{k-1/2,m}^{+(n)}$
to
$J_{k-1/2,4m}^{+(n)}$
. The map
$\tilde{V}_{\unicode[STIX]{x1D6FC},n-\unicode[STIX]{x1D6FC}}(4)$
is defined through the linear isomorphism between
$J_{k,{\mathcal{M}}}^{(n)}$
and
$J_{k-1/2,m}^{+(n)}$
with
${\mathcal{M}}=\big(\!\begin{smallmatrix}\ast & \ast \\ \ast & 1\end{smallmatrix}\!\big)\in L_{2}^{+}$
such that
$\det (2{\mathcal{M}})=m$
. The definition of
$\tilde{V}_{\unicode[STIX]{x1D6FC},n-\unicode[STIX]{x1D6FC}}(4)$
is
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D704}_{{\mathcal{M}}}(\unicode[STIX]{x1D713})|\tilde{V}_{\unicode[STIX]{x1D6FC},n-\unicode[STIX]{x1D6FC}}(4):=2^{k(2n+1)-n(n+7/2)+(1/2)\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D704}_{{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}2 & \\ & 1\end{array}\!\big)\big]}(\unicode[STIX]{x1D713}|V_{\unicode[STIX]{x1D6FC},n-\unicode[STIX]{x1D6FC}}(4)) & & \displaystyle \nonumber\end{eqnarray}$$
for
$\unicode[STIX]{x1D713}\in J_{k,{\mathcal{M}}}^{(n)}$
. Here
$\unicode[STIX]{x1D704}_{{\mathcal{M}}}$
is the linear isomorphism map from
$J_{k,{\mathcal{M}}}^{(n)}$
to
$J_{k-1/2,m}^{+(n)}$
. See §5 for details on the map
$\unicode[STIX]{x1D704}_{{\mathcal{M}}}$
.
3 Ikeda lift and Siegel modular forms of half-integral weight
In this section we recall the Ikeda lift and construct Siegel modular forms of half-integral weight through the Eichler–Zagier–Ibukiyama correspondence.
Let
$2n$
and
$k$
be positive even integers such that
$k>2n+1$
. Let
$$\begin{eqnarray}\displaystyle g(z)=\mathop{\sum }_{\substack{ N\in \mathbb{Z} \\ N\equiv 0,(-1)^{n}\;\text{mod}\;4}}c(N)e(Nz)\in S_{k-n+1/2}^{+(1)} & & \displaystyle \nonumber\end{eqnarray}$$
be a Hecke eigenform. Let
$f$
be a normalized Hecke eigenform of elliptic cusp form of weight
$2k-2n$
which corresponds to
$g$
by the Shimura correspondence. Let
$F$
be a Ikeda lift of
$g$
which is a Siegel cusp form of weight
$k$
of degree
$2n$
given by
$$\begin{eqnarray}\displaystyle F(\unicode[STIX]{x1D70F})=\mathop{\sum }_{T\in L_{2n}^{+}}A(T)e(T\unicode[STIX]{x1D70F}), & & \displaystyle \nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D70F}\in \mathfrak{H}_{2n}$
and the Fourier coefficient
$A(T)$
is given by
$$\begin{eqnarray}\displaystyle A(T)=c(|D_{T}|)f_{T}^{k-n-1/2}\mathop{\prod }_{p|f_{T}}\tilde{F}_{p}(T,\unicode[STIX]{x1D6FC}_{p}), & & \displaystyle \nonumber\end{eqnarray}$$
where the fundamental discriminant
$D_{T}$
and the natural number
$f_{T}$
are determined by
$(-1)^{n}\det (2T)=D_{T}f_{T}^{2}$
, and where
$\tilde{F}_{p}(T,X)\in \mathbb{C}[X+X^{-1}]$
is a certain Laurent polynomial (see [Reference Ikeda12, p. 642] for the definition). Here
$\unicode[STIX]{x1D6FC}_{p}$
is the complex number determined by
$a_{f}(p)=(\unicode[STIX]{x1D6FC}_{p}+\unicode[STIX]{x1D6FC}_{p}^{-1})p^{k-n-1/2}$
, and where
$a_{f}(p)$
is the
$p$
th Fourier coefficient of
$f$
.
We take the Fourier–Jacobi expansion of
$F$
:
$$\begin{eqnarray}\displaystyle F\left(\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D70F} & z\\ \text{}^{t\!}z & \unicode[STIX]{x1D714}\end{array}\right)\right)=\mathop{\sum }_{N\in \mathbb{Z}}\unicode[STIX]{x1D713}_{N}(\unicode[STIX]{x1D70F},z)e(N\unicode[STIX]{x1D714}), & & \displaystyle \nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D70F}\in \mathfrak{H}_{2n-1}$
,
$\unicode[STIX]{x1D714}\in \mathfrak{H}_{1}$
and
$z\in \mathbb{C}^{(2n-1,1)}$
. The function
$\unicode[STIX]{x1D713}_{N}$
is a Jacobi form of weight
$k$
of index
$N$
of degree
$2n-1$
.
There exists
$G\in S_{k-1/2}^{+(2n-1)}$
which corresponds to
$\unicode[STIX]{x1D713}_{1}$
by the linear isomorphism between
$S_{k-1/2}^{+(2n-1)}$
and the space of Jacobi cusp forms of weight
$k$
of index
$1$
of degree
$2n-1$
(see [Reference Ibukiyama10] and Proposition 5.2 in §5; the form
$G$
is given by
$G=\unicode[STIX]{x1D704}_{1}(\unicode[STIX]{x1D713}_{1})$
).
We remark that the map
$$\begin{eqnarray}\displaystyle S_{k-n+1/2}^{+(1)}\rightarrow S_{k-1/2}^{+(2n-1)} & & \displaystyle \nonumber\end{eqnarray}$$
given as above is a linear map. If
$g$
is a Hecke eigenform, then
$G$
is also a Hecke eigenform (cf. [Reference Hayashida6, Theorem 1.1]).
4 Generalization of the Maass relation for Siegel modular forms of half-integral weight
Let
$g\in S_{k-n+1/2}^{+(1)}$
be a Hecke eigenform. Let
$G\in S_{k-1/2}^{+(2n-1)}$
be the lift of
$g$
as in §3. We take the Fourier–Jacobi expansion
$$\begin{eqnarray}\displaystyle G\left(\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D70F} & z\\ \text{}^{t\!}z & \unicode[STIX]{x1D714}\end{array}\right)\right)=\mathop{\sum }_{m\in \mathbb{Z}_{{>}0}}\unicode[STIX]{x1D719}_{m}(\unicode[STIX]{x1D70F},z)e(m\unicode[STIX]{x1D714}). & & \displaystyle \nonumber\end{eqnarray}$$
Since
$G$
belongs to the plus-space,
$\unicode[STIX]{x1D719}_{m}=0$
unless
$m\equiv 0$
,
$3\;\text{mod}\;4$
.
To explain a generalization of the Maass relation, we prepare some notation. For integers
$l\;(2\leqslant l)$
,
$\unicode[STIX]{x1D6FD}\;(0\leqslant \unicode[STIX]{x1D6FD}\leqslant l-1)$
and
$\unicode[STIX]{x1D6FC}\;(0\leqslant \unicode[STIX]{x1D6FC}\leqslant l)$
, we put
$$\begin{eqnarray}\displaystyle b_{\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FC}}:=b_{\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FC},l,p}(X)=\left\{\begin{array}{@{}ll@{}}(p^{l+1-\unicode[STIX]{x1D6FC}}-p^{-l-1+\unicode[STIX]{x1D6FC}})p^{1/2}\quad & \text{if }\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FC}-2,\\ (X+X^{-1})p\quad & \text{if }\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FC}-1,\\ p^{-l+\unicode[STIX]{x1D6FC}+3/2}\quad & \text{if }\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FC},\\ 0\quad & \text{otherwise},\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$
and we set
$B_{l,l+1}(X)$
as the
$l\times (l+1)$
matrix defined by
$$\begin{eqnarray}\displaystyle B_{l,l+1}(X):=(b_{\unicode[STIX]{x1D6FD},\unicode[STIX]{x1D6FC}})_{\substack{ \unicode[STIX]{x1D6FD}=0,\ldots ,l-1 \\ \unicode[STIX]{x1D6FC}=0,\ldots ,l}}=\left(\begin{array}{@{}cccc@{}}p^{-l+3/2} & b_{0,1} & \cdots \, & b_{0,l}\\ & \ddots & \ddots & \vdots \\ 0 & & p^{1/2} & b_{l-1,l}\end{array}\right) & & \displaystyle \nonumber\end{eqnarray}$$
with entries in
$\mathbb{R}[X+X^{-1}]$
. The
$2\times (n+1)$
matrix
$A_{2,2n-1}^{p}(X)$
is defined by
$$\begin{eqnarray}\displaystyle A_{2,n+1}^{p}(X) & := & \displaystyle \mathop{\prod }_{l=2}^{n}B_{l,l+1}(p^{(n+2)/2-l}X)\nonumber\\ \displaystyle & = & \displaystyle B_{2,3}(p^{(n+2)/2-2}X)B_{3,4}(p^{(n+2)/2-3}X)\cdots B_{n,n+1}(p^{(n+2)/2-n}X).\nonumber\end{eqnarray}$$
For
$a\in \mathbb{Z}_{{>}0}$
and for
$\unicode[STIX]{x1D719}\in J_{k-1/2,m}^{(n^{\prime })}$
we define the function
$\unicode[STIX]{x1D719}|U_{a}$
by
$$\begin{eqnarray}\displaystyle (\unicode[STIX]{x1D719}|U_{a})(\unicode[STIX]{x1D70F},z):=\unicode[STIX]{x1D719}(\unicode[STIX]{x1D70F},az). & & \displaystyle \nonumber\end{eqnarray}$$
In [Reference Hayashida7] we obtained the following generalization of the Maass relation.
Proposition 4.1. For any natural number
$m$
and for any prime
$p$
, we have the identity between the vectors
$$\begin{eqnarray}\displaystyle & & \displaystyle (\unicode[STIX]{x1D719}_{m}|\tilde{V}_{0,2n-2}(p^{2}),\unicode[STIX]{x1D719}_{m}|\tilde{V}_{1,2n-3}(p^{2}),\ldots ,\unicode[STIX]{x1D719}_{m}|\tilde{V}_{2n-2,0}(p^{2}))\nonumber\\ \displaystyle & & \displaystyle \quad =p^{k(2n-3)-2n^{2}-n+11/2}(\unicode[STIX]{x1D719}_{m/p^{2}}|U_{p^{2}},\unicode[STIX]{x1D719}_{m}|U_{p},\unicode[STIX]{x1D719}_{mp^{2}})\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\left(\begin{array}{@{}cc@{}}0 & p^{2k-3}\\ \displaystyle p^{k-2} & \displaystyle p^{k-2}\biggl(\frac{-m}{p}\biggr)\\ 0 & 1\end{array}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,A_{2,2n-1}^{p}(\unicode[STIX]{x1D6FC}_{p})\,\text{diag}(1,p^{1/2},p,\ldots ,p^{n-1}),\nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D6FC}_{p}$
is explained in §3.
Therefore we obtain
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D719}_{m}|\tilde{V}_{i,2n-2-i}(p^{2})\nonumber\\ \displaystyle & & \displaystyle \quad =a_{1}\unicode[STIX]{x1D719}_{m}|U_{p}+a_{2}\biggl(p^{2k-3}\unicode[STIX]{x1D719}_{m/p^{2}}\bigg|U_{p^{2}}+p^{k-2}\biggl(\frac{-m}{p}\biggr)\unicode[STIX]{x1D719}_{m}\bigg|U_{p}+\unicode[STIX]{x1D719}_{mp^{2}}\biggr)\qquad\end{eqnarray}$$
for any
$i\;(0\leqslant i\leqslant 2n-2)$
with certain constants
$a_{j}=a_{j,i,p}\;(j=1,2)$
which do not depend on the choice of
$m$
. In particular, if
$i=1$
, then by a straightforward calculation we obtain
$a_{2}=a_{2,1,p}=p^{k(2n-3)-4n^{2}+7n-3/2}\neq 0$
.
Definition 4.2. For any prime
$p$
we set
$$\begin{eqnarray}\displaystyle \hat{D}_{2n-2}(p^{2}):=a_{2}^{-1}(\tilde{V}_{1,2n-3}(p^{2})-a_{1}U_{p}), & & \displaystyle \nonumber\end{eqnarray}$$
where
$a_{1}$
and
$a_{2}$
are constants determined by the identity (4.1).
We remark that the map
$\hat{D}_{2n-2}(p^{2})$
depends on the choice of
$g$
, since
$a_{1}$
is determined by the value
$\{\unicode[STIX]{x1D6FC}_{p}^{\pm }\}$
. Remark also that
$\hat{D}_{2n-2}(p^{2})$
and
$\hat{D}_{2n-2}(q^{2})$
are compatible if two primes
$p$
and
$q$
are not the same. Moreover,
$\hat{D}_{2n-2}(p^{2})$
and
$U_{N}$
are compatible for any natural number
$N$
.
Let
$n^{\prime }=2n-2$
and let
$m$
be a natural number. We define the sequence of maps
$\{\hat{D}_{n^{\prime }}(N)\}_{N}$
and
$\{D_{n^{\prime }}(N)\}_{N}:=\{D_{n^{\prime },m}(N)\}_{N}$
through the two formal Dirichlet series
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{N=1}^{\infty }\frac{\hat{D}_{n^{\prime }}(N^{2})}{N^{s}}=\mathop{\prod }_{p}(I-\hat{D}_{n^{\prime }}(p^{2})p^{-s}+U_{p^{2}}p^{2k-3-2s})^{-1} & & \displaystyle \nonumber\end{eqnarray}$$
and
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{N=1}^{\infty }\frac{D_{n^{\prime }}(N^{2})}{N^{s}}=\bigg\{\mathop{\prod }_{p}\biggl(I-\biggl(\frac{-m}{p}\biggr)U_{p}p^{k-2-s}\biggr)\bigg\}\mathop{\sum }_{N=1}^{\infty }\frac{\hat{D}_{n^{\prime }}(N^{2})}{N^{s}}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$I$
denotes the identity map. We remark that the definitions of
$\hat{D}_{n^{\prime }}(N^{2})$
and
$D_{n^{\prime }}(N^{2})$
depend on the choice of
$g\in S_{k-n+1/2}^{+(1)}$
. The definition of
$\hat{D}_{n^{\prime }}(N^{2})$
is independent of the choice of
$m$
, but the definition of
$D_{n^{\prime }}(N^{2})$
depends on the choice of
$m$
. Because of the compatibility of
$\hat{D}_{n^{\prime }}(p^{2})$
and
$U_{N},$
the above definitions are well defined. Moreover,
$D_{n^{\prime }}(N^{2})$
and
$U_{M}$
are compatible for any natural numbers
$N$
and
$M$
.
Proposition 4.3. Let
$G\in S_{k-1/2}^{+(2n-1)}$
be as above. For any natural numbers
$N$
and for any negative fundamental discriminant
$-m$
, we define
$D_{n^{\prime }}(N^{2})$
as above. Then we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{mN^{2}}=\unicode[STIX]{x1D719}_{m}|D_{n^{\prime }}(N^{2}). & & \displaystyle \nonumber\end{eqnarray}$$
Proof. Due to the definition of
$D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})$
, we have
$$\begin{eqnarray}\displaystyle 0=p^{2k-3}D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}-2})U_{p^{2}}-D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})\hat{D}_{n^{\prime }}(p^{2})+D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}+2}), & & \displaystyle \nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D6FF}\geqslant 1$
, and
$$\begin{eqnarray}\displaystyle -\biggl(\frac{-m}{p}\biggr)p^{k-2}U_{p}=-\hat{D}_{n^{\prime }}(p^{2})+D_{n^{\prime }}(p^{2}). & & \displaystyle \nonumber\end{eqnarray}$$
Thus
$$\begin{eqnarray}\displaystyle 0=p^{2k-3}\unicode[STIX]{x1D719}_{m}|D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}-2})U_{p^{2}}-\unicode[STIX]{x1D719}_{m}|D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})\hat{D}_{n^{\prime }}(p^{2})+\unicode[STIX]{x1D719}_{m}|D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}+2}), & & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$
for any
$\unicode[STIX]{x1D6FF}\geqslant 1$
, and
$$\begin{eqnarray}\displaystyle -\biggl(\frac{-m}{p}\biggr)p^{k-2}\unicode[STIX]{x1D719}_{m}|U_{p}=-\unicode[STIX]{x1D719}_{m}|\hat{D}_{n^{\prime }}(p^{2})+\unicode[STIX]{x1D719}_{m}|D_{n^{\prime }}(p^{2}). & & \displaystyle\end{eqnarray}$$
On the other hand, since
$\hat{D}_{n^{\prime }}(p^{2})=-(a_{1}/a_{2})U_{p}+(1/a_{2})\tilde{V}_{1,2n-3}(p^{2})$
and due to (4.1) with
$i=1$
, we have
$$\begin{eqnarray}\displaystyle -\biggl(\frac{-m^{\prime }}{p}\biggr)p^{k-2}\unicode[STIX]{x1D719}_{m^{\prime }}|U_{p}=p^{2k-3}\unicode[STIX]{x1D719}_{m^{\prime }/p^{2}}|U_{p^{2}}-\unicode[STIX]{x1D719}_{m^{\prime }}|\hat{D}_{n^{\prime }}(p^{2})+\unicode[STIX]{x1D719}_{m^{\prime }p^{2}}\qquad & & \displaystyle\end{eqnarray}$$
for any natural number
$m^{\prime }$
.
We remark that
$\unicode[STIX]{x1D719}_{m/p^{2}}=0$
.
If
$m^{\prime }=m$
, then due to the identities (4.3) and (4.4), we have
$\unicode[STIX]{x1D719}_{mp^{2}}=\unicode[STIX]{x1D719}_{m}|D_{n^{\prime }}(p^{2})$
.
If
$m^{\prime }=mp^{2\unicode[STIX]{x1D6FF}}\;(\unicode[STIX]{x1D6FF}\geqslant 1)$
and if
$\unicode[STIX]{x1D719}_{mp^{2\unicode[STIX]{x1D6FF}-2j}}=\unicode[STIX]{x1D719}_{m}|D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}-2j})\;(j=0,1)$
is true, then due to the identities (4.2) and (4.4), we have
$\unicode[STIX]{x1D719}_{mp^{2\unicode[STIX]{x1D6FF}+2}}=\unicode[STIX]{x1D719}_{m}|D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}+2})$
. Thus, by induction we obtain
$\unicode[STIX]{x1D719}_{mp^{2\unicode[STIX]{x1D6FF}}}=\unicode[STIX]{x1D719}_{m}|D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})$
for any
$\unicode[STIX]{x1D6FF}\geqslant 0$
.
Let
$q$
be a prime which is different from
$p$
. Then
$(\frac{-mp^{2\unicode[STIX]{x1D6FF}}}{q})=(\frac{-m}{q})$
. Thus the definition of
$D_{n^{\prime }}(q^{2\unicode[STIX]{x1D6FE}})$
does not change, even if we replace
$m$
by
$mp^{2\unicode[STIX]{x1D6FF}}$
. The identities (4.2) and (4.3) are true, even if we replace
$m$
and
$p$
by
$mp^{2\unicode[STIX]{x1D6FF}}$
and by
$q$
, respectively. Therefore, we conclude this proposition by induction with respect to natural numbers
$p^{2\unicode[STIX]{x1D6FF}}$
and
$m$
.◻
Lemma 4.4. Let
$G\in S_{k-1/2}^{+(2n-1)}$
be as above. We fix a natural number
$m$
. We define
$\hat{D}_{n^{\prime }}(N)$
and
$D_{n^{\prime }}(N)$
as above. For any natural numbers
$N$
,
$M$
,
$\unicode[STIX]{x1D6FF}$
and
$\unicode[STIX]{x1D6FE}$
, and for any prime
$p$
, we have the following identities:
(1)
$$\begin{eqnarray}\displaystyle \hat{D}_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})\hat{D}_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FE}})=\mathop{\sum }_{i=0}^{\text{min}(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FE})}p^{(2k-3)i}U_{p^{2i}}\hat{D}_{n^{\prime }}(p^{2(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D6FE}-2i)}); & & \displaystyle \nonumber\end{eqnarray}$$
(2)
$$\begin{eqnarray}\displaystyle \hat{D}_{n^{\prime }}(N^{2})\hat{D}_{n^{\prime }}(M^{2})=\mathop{\sum }_{d|(N,M)}d^{(2k-3)}U_{d^{2}}\hat{D}_{n^{\prime }}\biggl(\frac{N^{2}M^{2}}{d^{4}}\biggr); & & \displaystyle \nonumber\end{eqnarray}$$
(3)
$$\begin{eqnarray}\displaystyle & & \displaystyle D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FE}})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{i=0}^{\text{min}(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FE})}p^{(2k-3)i}U_{p^{2i}}D_{n^{\prime }}(p^{2(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D6FE}-2i)})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\biggl(\frac{-m}{p}\biggr)p^{k-2}\mathop{\sum }_{i=0}^{\text{min}(\unicode[STIX]{x1D6FF},\unicode[STIX]{x1D6FE})-1}p^{(2k-3)i}U_{p^{2i+1}}D_{n^{\prime }}(p^{2(\unicode[STIX]{x1D6FF}+\unicode[STIX]{x1D6FE}-1-2i)});\nonumber\end{eqnarray}$$
(4)
where$$\begin{eqnarray}\displaystyle & & \displaystyle D_{n^{\prime }}(N^{2})D_{n^{\prime }}(M^{2})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{d|(N,M)}d^{2k-3}\mathop{\sum }_{d_{1}|(N,M)/d}\unicode[STIX]{x1D707}(d_{1})d_{1}^{k-2}\biggl(\frac{-m}{d_{1}}\biggr)U_{d^{2}d_{1}}D_{n^{\prime }}\biggl(\frac{N^{2}M^{2}}{d^{4}d_{1}^{2}}\biggr),\nonumber\end{eqnarray}$$
$\unicode[STIX]{x1D707}$
is the Möbius function.
Proof. The identity (1) follows from a straightforward calculation. The identity (3) follows from (1) and from the relation
$$\begin{eqnarray}\displaystyle D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})=\hat{D}_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})-\biggl(\frac{-m}{p}\biggr)p^{k-2}U_{p}\hat{D}_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}-2}). & & \displaystyle \nonumber\end{eqnarray}$$
The identities (2) and (4) follow from (1) and (3). ◻
5 Isomorphism between the spaces of Jacobi forms
In this section we review a generalization of the Eichler–Zagier–Ibukiyama correspondence shown in [Reference Hayashida8]. It is a linear isomorphism between certain spaces of Jacobi forms of integral weight and of half-integral weight.
Let
$n^{\prime }$
,
$k$
and
$r$
be natural numbers. We assume that
$k$
is an even integer and
$r\geqslant 1$
. We take a matrix
${\mathcal{M}}=\Big(\!\begin{smallmatrix}{\mathcal{M}}_{1} & \frac{1}{2}L\\ \frac{1}{2}\text{}^{t\!}L & 1\end{smallmatrix}\!\Big)\in L_{r}^{+}$
with
${\mathcal{M}}_{1}\in L_{r-1}^{+}$
and
$L\in M_{r-1,1}(\mathbb{Z})$
. We set
$$\begin{eqnarray}\displaystyle \mathfrak{M}:=\left\{\begin{array}{@{}ll@{}}4{\mathcal{M}}_{1}-L\text{}^{t\!}L\quad & \text{if }r\geqslant 2,\\ \emptyset \quad & \text{if }r=1.\end{array}\right. & & \displaystyle \nonumber\end{eqnarray}$$
If
$r=1$
, then
${\mathcal{M}}=1$
and, in an abuse of notation, we put
$\det (\mathfrak{M}):=\det (\emptyset )=1$
.
A plus-space
$J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })}$
for Jacobi forms is introduced in [Reference Hayashida8]. This is a subspace of
$J_{k-1/2,\mathfrak{M}}^{(n^{\prime })}$
and is a generalization of the generalized plus-space of Siegel modular forms to Jacobi forms, and where
$J_{k-1/2,\mathfrak{M}}^{(n^{\prime })}$
denotes the space of Jacobi forms of weight
$k-\frac{1}{2}$
of index
$\mathfrak{M}$
of degree
$n^{\prime }$
. (See the definition of
$J_{k-1/2,\mathfrak{M}}^{(n^{\prime })}$
in [Reference Hayashida8, §2.3] for the case
$r-1\geqslant 2$
. In this article we use the cases
$r-1=0$
and
$r-1=1$
.) The space
$J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })}$
is defined as follows. Let
$\unicode[STIX]{x1D719}\in J_{k-1/2,\mathfrak{M}}^{(n^{\prime })}$
be a Jacobi form of weight
$k-\frac{1}{2}$
of index
$\mathfrak{M}$
on
$\unicode[STIX]{x1D6E4}_{0}^{(n^{\prime })}(4)$
. We take the Fourier expansion of
$\unicode[STIX]{x1D719}$
,
for
$(\unicode[STIX]{x1D70F},z)\in \mathfrak{H}_{n^{\prime }}\times \mathbb{C}^{(n^{\prime },r-1)}$
, where
$N^{\prime }$
and
$R^{\prime }$
run over
$L_{n^{\prime }}^{\ast }$
and
$\mathbb{Z}^{(n^{\prime },r-1)}$
, respectively, such that
$4N^{\prime }-R^{\prime }\mathfrak{M}^{-1}$
$\text{}^{t\!}R^{\prime }\geqslant 0$
. Then
$\unicode[STIX]{x1D719}$
belongs to
$J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })}$
if and only if
$C_{\unicode[STIX]{x1D719}}(N^{\prime },R^{\prime })=0$
, unless
$$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}cc@{}}N^{\prime } & \frac{1}{2}R^{\prime }\\ \frac{1}{2}\text{}^{t\!}R^{\prime } & \mathfrak{M}\end{array}\right)+\unicode[STIX]{x1D706}\text{}^{t\!}\unicode[STIX]{x1D706}\in 4L_{n^{\prime }}^{\ast } & & \displaystyle \nonumber\end{eqnarray}$$
with some
$\unicode[STIX]{x1D706}\in \mathbb{Z}^{(n^{\prime }+r-1,1)}$
.
This condition requires a condition on
$\mathfrak{M}$
. For example, if
$r-1=1$
and if
$J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })}\neq \emptyset$
, then
$\mathfrak{M}\equiv 0,3\;\text{mod}\;4$
.
We define
$J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })\;\text{cusp}}:=J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })}\cap J_{k-1/2,\mathfrak{M}}^{(n^{\prime })\;\text{cusp}}$
.
If
$r=1$
, then
$J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })}=J_{k-1/2,\emptyset }^{+(n^{\prime })}=M_{k-1/2}^{+(n^{\prime })}$
and
$J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })\;\text{cusp}}=J_{k-1/2,\emptyset }^{+(n^{\prime })\;\text{cusp}}=S_{k-1/2}^{+(n^{\prime })}$
.
Lemma 5.1. Let
$F\in M_{k-1/2}^{+(n^{\prime }+r)}$
. We take the Fourier–Jacobi expansion
$$\begin{eqnarray}\displaystyle F\left(\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D70F} & z\\ \text{}^{t\!}z & \unicode[STIX]{x1D714}\end{array}\right)\right)=\mathop{\sum }_{\mathfrak{M}\in L_{r}^{\ast }}f_{\mathfrak{M}}(\unicode[STIX]{x1D70F},z)e(\mathfrak{M}\unicode[STIX]{x1D714}), & & \displaystyle \nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D70F}\in \mathfrak{H}_{n^{\prime }}$
,
$z\in \mathbb{C}^{(n^{\prime },r)}$
and
$\unicode[STIX]{x1D714}\in \mathfrak{H}_{r}$
. Then
$f_{\mathfrak{M}}\in J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })}$
. Moreover, if
$F\in S_{k-1/2}^{+(n^{\prime }+r)}$
, then
$f_{\mathfrak{M}}\in J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })\;\text{cusp}}$
.
Proof. This is obvious from the definition of the plus-space
$J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })}$
.◻
In particular, the function
$\unicode[STIX]{x1D719}_{m}$
in §4 belongs to the plus-space
$J_{k-1/2,m}^{+(2n-2)}$
.
Let
${\mathcal{M}}$
be as above. There exists a linear isomorphism map
$\unicode[STIX]{x1D704}_{{\mathcal{M}}}$
from
$J_{k,{\mathcal{M}}}^{(n)}$
to
$J_{k-1/2,\mathfrak{M}}^{(n)+}$
(cf. [Reference Eichler and Zagier5] (for
$r=n=1$
), [Reference Ibukiyama10] (for
$r=1$
,
$n>1$
), [Reference Hayashida8] (for
$r>1$
,
$n\geqslant 1$
)). This map
$\unicode[STIX]{x1D704}_{{\mathcal{M}}}$
is given as follows.
We assume
$\unicode[STIX]{x1D713}\in J_{k,{\mathcal{M}}}^{(n^{\prime })}$
and denote by
$C_{\unicode[STIX]{x1D713}}(\ast ,\ast )$
the Fourier coefficients of
$\unicode[STIX]{x1D713}$
. For
$\unicode[STIX]{x1D70F}\in \mathfrak{H}_{n^{\prime }}$
and for
$z=(z_{1},z_{2})\in \mathbb{C}^{(n^{\prime },r)}$
(
$z_{1}\in \mathbb{C}^{(n^{\prime },r-1)}$
,
$z_{2}\in \mathbb{C}^{(n^{\prime },1)}$
), we take the theta decomposition
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}(\unicode[STIX]{x1D70F},z)=\mathop{\sum }_{\substack{ R\in \mathbb{Z}^{(n^{\prime },1)} \\ R\;\text{mod}\;2\mathbb{Z}^{(n^{\prime },1)}}}f_{R}(\unicode[STIX]{x1D70F},z_{1})\unicode[STIX]{x1D717}_{1,R,L}(\unicode[STIX]{x1D70F},z_{1},z_{2}), & & \displaystyle \nonumber\end{eqnarray}$$
where
$$\begin{eqnarray}\displaystyle f_{R}(\unicode[STIX]{x1D70F},z_{1}) & = & \displaystyle \mathop{\sum }_{N_{1}\in L_{n^{\prime }}^{\ast },N_{3}\in \mathbb{Z}^{(n^{\prime },r-1)}}C_{\unicode[STIX]{x1D713}}(N_{1},(\begin{array}{@{}cc@{}}N_{3} & R\end{array}))\nonumber\\ \displaystyle & & \displaystyle \times \,e\biggl(\biggl(N_{1}-\frac{1}{4}R~\text{}^{t\!}R\biggr)\unicode[STIX]{x1D70F}+\biggl(N_{3}-\frac{1}{2}R~\text{}^{t\!}L\biggr)\text{}^{t\!}z_{1}\biggr)\nonumber\end{eqnarray}$$
and where the function
$\unicode[STIX]{x1D717}_{1,R,L}$
is defined by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D717}_{1,R,L}(\unicode[STIX]{x1D70F},z_{1},z_{2}) & := & \displaystyle \unicode[STIX]{x1D717}_{1,R}\biggl(\unicode[STIX]{x1D70F},\frac{1}{2}z_{1}L+z_{2}\biggr)\nonumber\\ \displaystyle & = & \displaystyle \mathop{\sum }_{\substack{ x\in \mathbb{Z}^{(n^{\prime },1)} \\ x\equiv R\;\text{mod}\;2}}e\biggl(\frac{1}{4}x~\text{}^{t\!}x\unicode[STIX]{x1D70F}+x~\text{}^{t\!}\biggl(\frac{1}{2}z_{1}L+z_{2}\biggr)\biggr)\nonumber\end{eqnarray}$$
(cf. [Reference Hayashida8, Lemma 4.1]). We put
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D704}_{{\mathcal{M}}}(\unicode[STIX]{x1D713})(\unicode[STIX]{x1D70F},z_{1}):=\mathop{\sum }_{R\in \mathbb{Z}^{(n^{\prime },1)}/(2\mathbb{Z}^{(n^{\prime },1)})}f_{R}(4\unicode[STIX]{x1D70F},4z_{1}). & & \displaystyle \nonumber\end{eqnarray}$$
Then
$\unicode[STIX]{x1D704}_{{\mathcal{M}}}(\unicode[STIX]{x1D713})$
belongs to
$J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })}$
(cf. [Reference Hayashida8, Proposition 4.4]). If
$\unicode[STIX]{x1D713}$
is a Jacobi cusp form, then
$\unicode[STIX]{x1D704}_{{\mathcal{M}}}(\unicode[STIX]{x1D713})$
is also a Jacobi cusp form. If
$r=1$
, then
${\mathcal{M}}=1$
and
$\unicode[STIX]{x1D704}_{1}(\unicode[STIX]{x1D713})$
is a Siegel modular form (cf. [Reference Eichler and Zagier5, Reference Ibukiyama10]).
Proposition 5.2 [Reference Hayashida8].
We take a matrix
${\mathcal{M}}=\big(\!\begin{smallmatrix}{\mathcal{M}}_{1} & \frac{1}{2}L\\ \frac{1}{2}\text{}^{t\!}L & 1\end{smallmatrix}\!\big)\in L_{r}^{+}$
. Let
$k$
be an even integer. The map
$\unicode[STIX]{x1D704}_{{\mathcal{M}}}$
gives the linear isomorphisms
$$\begin{eqnarray}\displaystyle & \displaystyle J_{k,{\mathcal{M}}}^{(n^{\prime })}\cong J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })}, & \displaystyle \nonumber\\ \displaystyle & \displaystyle J_{k,{\mathcal{M}}}^{(n^{\prime })\;\text{cusp}}\cong J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })\;\text{cusp}}. & \displaystyle \nonumber\end{eqnarray}$$
In the case of
$r=1$
(which means
${\mathcal{M}}=1$
,
$\mathfrak{M}=\emptyset$
and
$J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })}=M_{k-1/2}^{+(n^{\prime })}$
), these isomorphisms were shown in [Reference Eichler and Zagier5] (for
$n^{\prime }=1$
) and in [Reference Ibukiyama10] (for
$n^{\prime }>1$
).
We quote a lemma about the Petersson inner products.
Lemma 5.3. For
$\unicode[STIX]{x1D713}_{i}\in J_{k,{\mathcal{M}}}^{(n^{\prime })\;\text{cusp}}\;(i=1,2)$
we set
$\unicode[STIX]{x1D719}_{i}=\unicode[STIX]{x1D704}_{{\mathcal{M}}}(\unicode[STIX]{x1D713}_{i})\in J_{k-1/2,\mathfrak{M}}^{+(n^{\prime })\;\text{cusp}}$
. Then we have
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}\rangle =(1+\unicode[STIX]{x1D6FF}_{1,r})^{-1}2^{2n^{\prime }(k-1)}\langle \unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}\rangle . & & \displaystyle \nonumber\end{eqnarray}$$
Proof. This lemma was shown in [Reference Kohnen and Zagier17, pp. 189–191] and in [Reference Eichler and Zagier5, Theorem 5.4] (for
$r=1$
and
$n^{\prime }=1$
), in [Reference Katsurada and Kawamura14, p. 2051] (for
$r=1$
and
$n^{\prime }>1$
), and in [Reference Hayashida9, Lemma 3.1] (for
$r>1$
and
$n^{\prime }\geqslant 1$
). Observe that the constant
$(1+\unicode[STIX]{x1D6FF}_{1,r})^{-1}$
in the identity is needed. (See the remark in [Reference Hayashida9, Lemma 3.1].)◻
In what follows we consider the case
$r=2$
.
Let
$a\in \mathbb{Q}_{{>}0}$
,
$b\in \mathbb{Q}$
and
${\mathcal{M}}\in L_{2}^{+}$
. For
$\unicode[STIX]{x1D713}\in J_{k,{\mathcal{M}}}^{(n^{\prime })}$
we define the function
$$\begin{eqnarray}\displaystyle \Big(\unicode[STIX]{x1D713}|U_{\big(\!\begin{array}{@{}cc@{}}a & 0\\ b & 1\end{array}\!\big)}\Big)(\unicode[STIX]{x1D70F},z):=\unicode[STIX]{x1D713}\left(\unicode[STIX]{x1D70F},z\left(\begin{array}{@{}cc@{}}a & 0\\ b & 1\end{array}\right)\right). & & \displaystyle \nonumber\end{eqnarray}$$
Lemma 5.4. Let
$r=2$
and
${\mathcal{M}}=\big(\!\begin{smallmatrix}\ast & \ast \\ \ast & 1\end{smallmatrix}\!\big)\in L_{2}^{+}$
. Let
$\unicode[STIX]{x1D713}\in J_{k,{\mathcal{M}}}^{(n^{\prime })}$
. Then, for any
$a\in \mathbb{Z}_{{>}0}$
and for any
$b\in \mathbb{Z}$
, we have
$\unicode[STIX]{x1D713}|U_{\big(\!\begin{array}{@{}cc@{}}a & 0\\ b & 1\end{array}\!\big)}\in J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}a & 0\\ b & 1\end{array}\!\big)\big]}^{(n^{\prime })}$
and
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D704}_{{\mathcal{M}}}(\unicode[STIX]{x1D713})|U_{a}=\unicode[STIX]{x1D704}_{{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}a & 0\\ b & 1\end{array}\!\big)\big]}\Big(\unicode[STIX]{x1D713}|U_{\big(\!\begin{array}{@{}cc@{}}a & 0\\ b & 1\end{array}\!\big)}\Big). & & \displaystyle \nonumber\end{eqnarray}$$
Proof. The first statement follows directly from the definition of Jacobi forms. The second statement can be shown by comparing the Fourier coefficients of both sides. The reader is referred to [Reference Hayashida7, Proposition 4.3] for the details of the calculation for the second statement. ◻
Lemma 5.5. Let
$r=2$
and
${\mathcal{M}}=\big(\!\begin{smallmatrix}\ast & \ast \\ \ast & 1\end{smallmatrix}\!\big)\in L_{2}^{+}$
. For any odd prime
$p$
and for
$0\leqslant \unicode[STIX]{x1D6FC}\leqslant n^{\prime }$
, let
$\tilde{V}_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}(p^{2})$
and
$V_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}(p^{2})$
be index-shift maps defined in §2.7. Then, for any
$\unicode[STIX]{x1D713}\in J_{k,{\mathcal{M}}}^{(n^{\prime })}$
, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D704}_{{\mathcal{M}}}(\unicode[STIX]{x1D713})|\tilde{V}_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}(p^{2})\nonumber\\ \displaystyle & & \displaystyle \quad =p^{k(2n^{\prime }+1)-n^{\prime }(n^{\prime }+7/2)+(1/2)\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D704}_{{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}p & \\ & 1\end{array}\!\big)\big]}(\unicode[STIX]{x1D713}|V_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}(p^{2})).\end{eqnarray}$$
Proof. The identity can be shown by comparing the Fourier coefficients of both sides. The proof is the same as in [Reference Hayashida7, Proposition 4.4]. ◻
We remark that
$\tilde{V}_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}(4)$
has been defined via the identity (5.1) (see §2.7).
6 Adjoint maps
In this section we introduce some adjoint maps for Jacobi forms with respect to the Petersson inner product.
We assume
${\mathcal{M}}=\big(\!\begin{smallmatrix}\ast & \ast \\ \ast & 1\end{smallmatrix}\!\big)\in L_{2}^{+}$
such that
$\det (2{\mathcal{M}})=m$
. We remark that
$m\equiv 0,3\;\text{mod}\;4$
. There exists
${\mathcal{M}}$
for any such natural number
$m$
.
Let
$n^{\prime }$
and
$N$
be natural numbers.
6.1 Adjoint map
$U^{\ast }$
for Jacobi forms of integral weight
For any
$\unicode[STIX]{x1D713}\in J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}$
we define the function
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }:=N^{-2n^{\prime }}\mathop{\sum }_{\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D707}_{1}\in (\mathbb{Z}/N\mathbb{Z})^{(n^{\prime },1)}}\unicode[STIX]{x1D713}\Big|U_{\big(\!\begin{array}{@{}cc@{}}N^{-1} & 0\\ 0 & 1\end{array}\!\big)}\Big|_{k,{\mathcal{M}}}[((\unicode[STIX]{x1D706}_{1},0),(\unicode[STIX]{x1D707}_{1},0)),0_{2}]. & & \displaystyle \nonumber\end{eqnarray}$$
Lemma 6.1. We obtain a map
$$\begin{eqnarray}\displaystyle U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }:J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}\rightarrow J_{k,{\mathcal{M}}}^{(n^{\prime })}. & & \displaystyle \nonumber\end{eqnarray}$$
Moreover, if
$\unicode[STIX]{x1D713}\in J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })\;\text{cusp}}$
, then
$\unicode[STIX]{x1D713}|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }\in J_{k,{\mathcal{M}}}^{(n^{\prime })\;\text{cusp}}$
.
Proof. One can check the first statement by a straightforward calculation. For the second statement we need to show that
$\unicode[STIX]{x1D713}|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }$
is a Jacobi cusp form. This is shown by the expression for the Fourier coefficients of
$\unicode[STIX]{x1D713}|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }$
by using the Fourier coefficients of
$\unicode[STIX]{x1D713}$
.◻
Lemma 6.2. The map
$U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }$
is the adjoint map of
$U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}$
with respect to the Petersson inner product. This means that we have
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D713}_{1}|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)},\unicode[STIX]{x1D713}_{2}\rangle =\langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }\rangle & & \displaystyle \nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D713}_{1}\in J_{k,{\mathcal{M}}}^{(n^{\prime })\;\text{cusp}}$
and any
$\unicode[STIX]{x1D713}_{2}\in J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })\,\text{cusp}}$
.
Proof. We fix a fundamental domain
${\mathcal{F}}_{n^{\prime },2}$
of
$\unicode[STIX]{x1D6E4}_{n^{\prime },2}^{J}\backslash (\mathfrak{H}_{n^{\prime }}\times \mathbb{C}^{(n^{\prime },2)})$
. We put
$$\begin{eqnarray}{\mathcal{F}}_{n^{\prime },2}(N)=\left\{\left.\left(\unicode[STIX]{x1D70F},z\left(\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\right)\right)\in \mathfrak{H}_{n^{\prime }}\times \mathbb{C}^{(n^{\prime },2)}\right|(\unicode[STIX]{x1D70F},z)\in {\mathcal{F}}_{n^{\prime },2}\right\}.\end{eqnarray}$$
We have
$\text{vol}({\mathcal{F}}_{n^{\prime },2}(N))=N^{2n^{\prime }}\text{vol}({\mathcal{F}}_{n^{\prime },2})$
. Here
$\text{vol}(\ast )$
is the volume with the measure
$\det (v)^{-n^{\prime }-3}\,du\,dv\,dx\,dy$
. Let
$X=[((\unicode[STIX]{x1D706}_{1},0),(\unicode[STIX]{x1D707}_{1},0)),0_{2}]\in \mathbb{Z}^{(n^{\prime },2)}\times \mathbb{Z}^{(n^{\prime },2)}\times \mathbb{Z}^{(2,2)}$
. We write
$z=x+\sqrt{-1}y=(z_{1},z_{2})\in \mathbb{C}^{(n^{\prime },2)}$
,
$z_{i}=x_{i}+\sqrt{-1}y_{i}\;(i=1,2)$
. Then, by the substitution
$z_{1}\rightarrow z_{1}+(1/N)\unicode[STIX]{x1D70F}\unicode[STIX]{x1D706}_{1}+(1/N)\unicode[STIX]{x1D707}_{1}$
, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \langle \unicode[STIX]{x1D713}_{1}|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)},\unicode[STIX]{x1D713}_{2}\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =\int _{{\mathcal{F}}_{n^{\prime },2}}\unicode[STIX]{x1D713}_{1}(\unicode[STIX]{x1D70F},(Nz_{1},z_{2}))\overline{\unicode[STIX]{x1D713}_{2}(\unicode[STIX]{x1D70F},z)}\det (v)^{k-n^{\prime }-3}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,e\left(\frac{-4\unicode[STIX]{x1D70B}}{2\unicode[STIX]{x1D70B}\text{i}}{\mathcal{M}}\left[\left(\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\right)\right]v^{-1}[y]\right)\,du\,dv\,dx\,dy\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{{\mathcal{F}}_{n^{\prime },2}}\unicode[STIX]{x1D713}_{1}(\unicode[STIX]{x1D70F},(Nz_{1}+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D706}_{1}+\unicode[STIX]{x1D707}_{1},z_{2}))\overline{\unicode[STIX]{x1D713}_{2}\biggl(\unicode[STIX]{x1D70F},\biggl(z_{1}+\frac{1}{N}\unicode[STIX]{x1D70F}\unicode[STIX]{x1D706}_{1}+\frac{1}{N}\unicode[STIX]{x1D707}_{1},z_{2}\biggr)\biggr)}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\det (v)^{k-n^{\prime }-3}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,e\left(\frac{-4\unicode[STIX]{x1D70B}}{2\unicode[STIX]{x1D70B}\text{i}}{\mathcal{M}}\left[\left(\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\right)\right]v^{-1}\left[y+\left(\frac{1}{N}v\unicode[STIX]{x1D706}_{1},0\right)\right]\right)\,du\,dv\,dx\,dy,\nonumber\end{eqnarray}$$
and by the substitution
$z_{1}\rightarrow (1/N)z_{1}$
,
$$\begin{eqnarray}\displaystyle & = & \displaystyle \int _{{\mathcal{F}}_{n^{\prime },2}(N)}N^{-2n^{\prime }}\unicode[STIX]{x1D713}_{1}(\unicode[STIX]{x1D70F},(z_{1}+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D706}_{1}+\unicode[STIX]{x1D707}_{1},z_{2}))\nonumber\\ \displaystyle & & \displaystyle \times \,\overline{\unicode[STIX]{x1D713}_{2}(\unicode[STIX]{x1D70F},(N^{-1}(z_{1}+\unicode[STIX]{x1D70F}\unicode[STIX]{x1D706}_{1}+\unicode[STIX]{x1D707}_{1}),z_{2}))}\det (v)^{k-n^{\prime }-3}\nonumber\\ \displaystyle & & \displaystyle \times \,e\biggl(\frac{-4\unicode[STIX]{x1D70B}}{2\unicode[STIX]{x1D70B}\text{i}}{\mathcal{M}}v^{-1}[y+(v\unicode[STIX]{x1D706}_{1},0)]\biggr)\,du\,dv\,dx\,dy\nonumber\\ \displaystyle & = & \displaystyle \int _{{\mathcal{F}}_{n^{\prime },2}(N)}N^{-2n^{\prime }}\unicode[STIX]{x1D713}_{1}(\unicode[STIX]{x1D70F},z)\overline{\left(\unicode[STIX]{x1D713}_{2}\left(\ast ,\ast \left.\left(\begin{array}{@{}cc@{}}N^{-1} & 0\\ 0 & 1\end{array}\right)\right)\right|X\right)(\unicode[STIX]{x1D70F},z)}\det (v)^{k-n^{\prime }-3}\nonumber\\ \displaystyle & & \displaystyle \times \,e\biggl(\frac{-4\unicode[STIX]{x1D70B}}{2\unicode[STIX]{x1D70B}\text{i}}{\mathcal{M}}v^{-1}[y]\biggr)\,du\,dv\,dx\,dy.\nonumber\end{eqnarray}$$
Thus we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \langle \unicode[STIX]{x1D713}_{1}|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)},\unicode[STIX]{x1D713}_{2}\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =N^{-2n^{\prime }}\mathop{\sum }_{\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D707}_{1}\in (\mathbb{Z}/N\mathbb{Z})^{(n^{\prime },1)}}\int _{{\mathcal{F}}_{n^{\prime },2}(N)}N^{-2n^{\prime }}\unicode[STIX]{x1D713}_{1}(\unicode[STIX]{x1D70F},z)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\overline{\left(\unicode[STIX]{x1D713}_{2}\left(\ast ,\ast \left.\left(\begin{array}{@{}cc@{}}N^{-1} & 0\\ 0 & 1\end{array}\right)\right)\right|X\right)(\unicode[STIX]{x1D70F},z)}\det (v)^{k-n^{\prime }-3}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,e\biggl(\frac{-4\unicode[STIX]{x1D70B}}{2\unicode[STIX]{x1D70B}\text{i}}{\mathcal{M}}v^{-1}[y]\biggr)\,du\,dv\,dx\,dy\nonumber\\ \displaystyle & & \displaystyle \quad =N^{-2n^{\prime }}\int _{{\mathcal{F}}_{n^{\prime },2}(N)}\unicode[STIX]{x1D713}_{1}(\unicode[STIX]{x1D70F},z)\overline{\Big(\unicode[STIX]{x1D713}_{2}|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }\Big)(\unicode[STIX]{x1D70F},z)}\det (v)^{k-n^{\prime }-3}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,e\biggl(\frac{-4\unicode[STIX]{x1D70B}}{2\unicode[STIX]{x1D70B}\text{i}}{\mathcal{M}}v^{-1}[y]\biggr)\,du\,dv\,dx\,dy\nonumber\\ \displaystyle & & \displaystyle \quad =\int _{{\mathcal{F}}_{n^{\prime },2}}\unicode[STIX]{x1D713}_{1}(\unicode[STIX]{x1D70F},z)\overline{\Big(\unicode[STIX]{x1D713}_{2}|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }\Big)(\unicode[STIX]{x1D70F},z)}\det (v)^{k-n^{\prime }-3}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,e\biggl(\frac{-4\unicode[STIX]{x1D70B}}{2\unicode[STIX]{x1D70B}\text{i}}{\mathcal{M}}v^{-1}[y]\biggr)\,du\,dv\,dx\,dy\nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }\rangle .\nonumber\end{eqnarray}$$
Thus we conclude the lemma. ◻
Lemma 6.3. For any natural numbers
$N$
and
$M$
, we have
(1)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{1}\Big|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}\Big|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }=\unicode[STIX]{x1D713}_{1}, & & \displaystyle \nonumber\end{eqnarray}$$
(2)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{2}\Big|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }=\unicode[STIX]{x1D713}_{2}\Big|U_{\big(\!\begin{array}{@{}cc@{}}M & 0\\ 0 & 1\end{array}\!\big)}\Big|U_{\big(\!\begin{array}{@{}cc@{}}NM & 0\\ 0 & 1\end{array}\!\big)}^{\ast } & & \displaystyle \nonumber\end{eqnarray}$$
for any prime
$p$
and for any
$\unicode[STIX]{x1D713}_{1}\in J_{k,{\mathcal{M}}}^{(n^{\prime })}$
and any
$\unicode[STIX]{x1D713}_{2}\in J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}$
. Moreover, if a natural number
$d$
is coprime to
$N$
, then
(3)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{2}\Big|U_{\big(\!\begin{array}{@{}cc@{}}d & 0\\ 0 & 1\end{array}\!\big)}\Big|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }=\unicode[STIX]{x1D713}_{2}\Big|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }\Big|U_{\big(\!\begin{array}{@{}cc@{}}d & 0\\ 0 & 1\end{array}\!\big)}, & & \displaystyle \nonumber\end{eqnarray}$$
(4)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}_{3}\Big|U_{\big(\!\begin{array}{@{}cc@{}}d & 0\\ 0 & 1\end{array}\!\big)}^{\ast }\Big|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }=\unicode[STIX]{x1D713}_{3}\Big|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }\Big|U_{\big(\!\begin{array}{@{}cc@{}}d & 0\\ 0 & 1\end{array}\!\big)}^{\ast } & & \displaystyle \nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D713}_{2}\in J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}$
and any
$\unicode[STIX]{x1D713}_{3}\in J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}Nd & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}$
.
Proof. The identity (1) is obvious because of the definitions. The identity (2) follows from the definitions of
$U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}$
and
$U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }$
, since
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D713}\Big|U_{\big(\!\begin{array}{@{}cc@{}}M & 0\\ 0 & 1\end{array}\!\big)}\Big|U_{\big(\!\begin{array}{@{}cc@{}}(NM)^{-1} & 0\\ 0 & 1\end{array}\!\big)}|_{k,{\mathcal{M}}}[(\unicode[STIX]{x1D706}_{1}+N\unicode[STIX]{x1D706}_{1}^{\prime },0),(\unicode[STIX]{x1D707}_{1}+N\unicode[STIX]{x1D707}_{1}^{\prime },0),0_{2}]\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D713}\Big|U_{\big(\!\begin{array}{@{}cc@{}}N^{-1} & 0\\ 0 & 1\end{array}\!\big)}\Big|_{k,{\mathcal{M}}}[(\unicode[STIX]{x1D706}_{1},0),(\unicode[STIX]{x1D707}_{1},0),0_{2}]\nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D706}_{1}$
,
$\unicode[STIX]{x1D706}_{1}^{\prime }$
,
$\unicode[STIX]{x1D707}_{1}$
,
$\unicode[STIX]{x1D707}_{1}^{\prime }\in \mathbb{Z}^{(n^{\prime },1)}$
. The identities (3) and (4) also follow from straightforward calculations.◻
6.2 Adjoint map
$U^{\ast }$
for Jacobi forms of half-integral weight
Definition 6.4. By combining Proposition 5.2 and Lemma 6.2 we define the adjoint map
$U_{N}^{\ast }$
of
$U_{N}$
with respect to the Petersson inner product. For
$\unicode[STIX]{x1D719}\in J_{k-1/2,mN^{2}}^{+(n^{\prime })}$
we define
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}|U_{N}^{\ast }:=\unicode[STIX]{x1D704}_{{\mathcal{M}}}\Big(\Big(\unicode[STIX]{x1D704}_{{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{-1}(\unicode[STIX]{x1D719})\Big)\Big|U_{\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)}^{\ast }\Big). & & \displaystyle \nonumber\end{eqnarray}$$
Then, due to Proposition 5.2 and Lemma 6.2, we have
$\unicode[STIX]{x1D719}|U_{N}^{\ast }\in J_{k-1/2,m}^{+(n^{\prime })}$
. And if
$\unicode[STIX]{x1D719}\in J_{k-1/2,mN^{2}}^{+(n^{\prime })\;\text{cusp}}$
, then
$\unicode[STIX]{x1D719}|U_{N}^{\ast }\in J_{k-1/2,m}^{+(n^{\prime })\;\text{cusp}}$
.
Lemma 6.5. The map
$U_{N}^{\ast }$
is the adjoint map of
$U_{N}$
with respect to the Petersson inner product. This means that we have
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D719}_{1}|U_{N},\unicode[STIX]{x1D719}_{2}\rangle =\langle \unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}|U_{N}^{\ast }\rangle & & \displaystyle \nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D719}_{1}\in J_{k-1/2,m}^{(n^{\prime })\;\text{cusp}}$
and any
$\unicode[STIX]{x1D719}_{2}\in J_{k-1/2,mN^{2}}^{(n^{\prime })\;\text{cusp}}$
.
Proof. Due to Lemmas 5.3, 5.4 and 6.2, we have
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D719}_{1}|U_{N},\unicode[STIX]{x1D719}_{2}\rangle & = & \displaystyle 2^{-2n^{\prime }(k-1)}\langle \unicode[STIX]{x1D704}_{{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{-1}(\unicode[STIX]{x1D719}_{1}|U_{N}),\unicode[STIX]{x1D704}_{{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{-1}(\unicode[STIX]{x1D719}_{2})\rangle \nonumber\\ \displaystyle & = & \displaystyle 2^{-2n^{\prime }(k-1)}\langle \unicode[STIX]{x1D704}_{{\mathcal{M}}}^{-1}(\unicode[STIX]{x1D719}_{1})|U_{\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]},\unicode[STIX]{x1D704}_{{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{-1}(\unicode[STIX]{x1D719}_{2})\rangle \nonumber\\ \displaystyle & = & \displaystyle 2^{-2n^{\prime }(k-1)}\langle \unicode[STIX]{x1D704}_{{\mathcal{M}}}^{-1}(\unicode[STIX]{x1D719}_{1}),\unicode[STIX]{x1D704}_{{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{-1}(\unicode[STIX]{x1D719}_{2})|U_{\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{\ast }\rangle \nonumber\\ \displaystyle & = & \displaystyle \langle \unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}|U_{N}^{\ast }\rangle .\Box \nonumber\end{eqnarray}$$
Lemma 6.6. For any natural numbers
$N$
and
$M$
, we have
(1)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{1}|U_{N}|U_{N}^{\ast }=\unicode[STIX]{x1D719}_{1}, & & \displaystyle \nonumber\end{eqnarray}$$
(2)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{2}|U_{N}^{\ast } & = & \displaystyle \unicode[STIX]{x1D719}_{2}|U_{M}|U_{NM}^{\ast }\nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D719}_{1}\in J_{k-1/2,m}^{+(n^{\prime })}$
and any
$\unicode[STIX]{x1D719}_{2}\in J_{k-1/2,mN^{2}}^{+(n^{\prime })}$
. Moreover, if a natural number
$d$
is coprime to
$N$
, then
(3)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{2}|U_{d}|U_{N}^{\ast }=\unicode[STIX]{x1D719}_{2}|U_{N}^{\ast }|U_{d}, & & \displaystyle \nonumber\end{eqnarray}$$
(4)
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{3}|U_{d}^{\ast }|U_{N}^{\ast }=\unicode[STIX]{x1D719}_{3}|U_{N}^{\ast }|U_{d}^{\ast } & & \displaystyle \nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D719}_{2}\in J_{k-1/2,mN^{2}}^{+(n^{\prime })}$
and any
$\unicode[STIX]{x1D719}_{3}\in J_{k-1/2,mN^{2}d^{2}}^{+(n^{\prime })}$
.
Lemma 6.7. If
$(N,2)=1$
, then the map
$U_{N}^{\ast }$
is given by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}|U_{N}^{\ast }=(2N)^{-2n^{\prime }}\mathop{\sum }_{\unicode[STIX]{x1D706}\in (\mathbb{Z}/N\mathbb{Z})^{n^{\prime }}}\mathop{\sum }_{\unicode[STIX]{x1D707}\in (\mathbb{Z}/4N\mathbb{Z})^{n^{\prime }}}(\unicode[STIX]{x1D719}|U_{1/N})\bigg|_{m}\biggl[\biggl(\unicode[STIX]{x1D706},\frac{1}{4}\unicode[STIX]{x1D707}\biggr),0\biggr] & & \displaystyle \nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D719}\in J_{k-1/2,mN^{2}}^{+(n^{\prime })}$
.
Proof. By comparing the Fourier coefficients on the both sides, we obtain the identity. ◻
6.3 Adjoint map
$V^{\ast }$
for Jacobi forms of integral weight
Let
$X\in \text{GSp}_{n}^{+}(\mathbb{Z})$
be a matrix such that the similitude of
$X$
is
$n(X)=N^{2}$
with a natural number
$N$
. For any
$\unicode[STIX]{x1D713}\in J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}$
we define the function
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D713}|V^{\ast }(X)\nonumber\\ \displaystyle & & \displaystyle \quad :=N^{2k-4n^{\prime }}\mathop{\sum }_{\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D707}_{1}\in (\mathbb{Z}/N^{2}\mathbb{Z})^{(n^{\prime },1)}}\mathop{\sum }_{\unicode[STIX]{x1D706}_{2},\unicode[STIX]{x1D707}_{2}\in (\mathbb{Z}/N\mathbb{Z})^{(n^{\prime },1)}}\mathop{\sum }_{M\in \unicode[STIX]{x1D6E4}_{n^{\prime }}\backslash \unicode[STIX]{x1D6E4}_{n^{\prime }}X\unicode[STIX]{x1D6E4}_{n^{\prime }}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\unicode[STIX]{x1D713}|_{k,{\mathcal{M}}}\left(M\times \left(\begin{array}{@{}cccc@{}}1 & 0 & 0 & 0\\ 0 & N & 0 & 0\\ 0 & 0 & N^{2} & 0\\ 0 & 0 & 0 & N\end{array}\right),[((\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2}),(\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2})),0_{2}]\right),\nonumber\end{eqnarray}$$
where
$(\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2}),(\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2})\in (\mathbb{Z}/N^{2}\mathbb{Z})^{(n^{\prime },1)}\times (\mathbb{Z}/N\mathbb{Z})^{(n^{\prime },1)}$
. See §2.1 for the notation for the matrix
$$\begin{eqnarray}\left(M\times \left(\begin{array}{@{}cccc@{}}1 & 0 & 0 & 0\\ 0 & N & 0 & 0\\ 0 & 0 & N^{2} & 0\\ 0 & 0 & 0 & N\end{array}\right),[(\unicode[STIX]{x1D706},\unicode[STIX]{x1D707}),0_{2}]\right).\end{eqnarray}$$
The above summations are finite sums and do not depend on the choice of the representatives
$(\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2})$
,
$(\unicode[STIX]{x1D707}_{1},\unicode[STIX]{x1D707}_{2})$
and
$M$
.
Let
$V(X)$
be the index-shift map for Jacobi forms of matrix index defined in §2.7.
Lemma 6.8. We have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D713}|V^{\ast }(X)=\unicode[STIX]{x1D713}|V(X)|U_{\big(\!\begin{array}{@{}cc@{}}N^{2} & 0\\ 0 & 1\end{array}\!\big)}^{\ast }. & & \displaystyle \nonumber\end{eqnarray}$$
In particular, the function
$\unicode[STIX]{x1D713}|V^{\ast }(X)$
belongs to
$J_{k,{\mathcal{M}}}^{(n^{\prime })}$
. This means that
$V^{\ast }(X)$
is a map
$$\begin{eqnarray}\displaystyle V^{\ast }(X):J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}\rightarrow J_{k,{\mathcal{M}}}^{(n^{\prime })}. & & \displaystyle \nonumber\end{eqnarray}$$
Moreover, if
$\unicode[STIX]{x1D713}\in J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })\;\text{cusp}}$
, then
$\unicode[STIX]{x1D713}|V^{\ast }(X)\in J_{k,{\mathcal{M}}}^{(n^{\prime })\;\text{cusp}}$
.
Proof. The first identity follows from the definitions of
$V^{\ast }(X)$
,
$V(X)$
and
$U_{\big(\!\begin{array}{@{}cc@{}}N^{2} & 0\\ 0 & 1\end{array}\!\big)}^{\ast }$
. The other statements follow from this identity.◻
Lemma 6.9. The map
$V^{\ast }(X)$
is the adjoint map of
$V(X)$
with respect to the Petersson inner product. This means that we have
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D713}_{1}|V(X),\unicode[STIX]{x1D713}_{2}\rangle =\langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}|V^{\ast }(X)\rangle & & \displaystyle \nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D713}_{1}\in J_{k,{\mathcal{M}}}^{(n^{\prime })\;\text{cusp}}$
and any
$\unicode[STIX]{x1D713}_{2}\in J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })\;\text{cusp}}$
.
Proof. The proof is similar to that of Lemma 6.2. ◻
For the sake of simplicity we set
$$\begin{eqnarray}\displaystyle V_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}^{\ast }(p^{2}):=V^{\ast }(\text{diag}(1_{\unicode[STIX]{x1D6FC}},p1_{n^{\prime }-\unicode[STIX]{x1D6FC}},p^{2}1_{\unicode[STIX]{x1D6FC}},p1_{n^{\prime }-\unicode[STIX]{x1D6FC}})) & & \displaystyle \nonumber\end{eqnarray}$$
for any prime
$p$
and for any
$\unicode[STIX]{x1D6FC}\;(0\leqslant \unicode[STIX]{x1D6FC}\leqslant n^{\prime })$
.
6.4 Adjoint map
$\tilde{V}^{\ast }$
for Jacobi forms of half-integral weight
We define the index-shift map
$\tilde{V}_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}^{\ast }(p^{2})$
for Jacobi forms of half-integral weight.
Definition 6.10. For any prime
$p$
and for any
$\unicode[STIX]{x1D719}\in J_{k-1/2,mp^{2}}^{+(n^{\prime })}$
, we define
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D719}|\tilde{V}_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}^{\ast }(p^{2})\nonumber\\ \displaystyle & & \displaystyle \quad :=p^{k(2n^{\prime }+1)-n^{\prime }(n^{\prime }+7/2)+(1/2)\unicode[STIX]{x1D6FC}}\unicode[STIX]{x1D704}_{{\mathcal{M}}}\Big(\Big(\unicode[STIX]{x1D704}_{{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}p & 0\\ 0 & 1\end{array}\!\big)\big]}^{-1}(\unicode[STIX]{x1D719})\Big)\Big|V_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}^{\ast }(p^{2})\Big).\nonumber\end{eqnarray}$$
Lemma 6.11. For any prime
$p$
, for any
$\unicode[STIX]{x1D719}_{1}\in J_{k-1/2,m}^{+(n^{\prime })\;\text{cusp}}$
and for any
$\unicode[STIX]{x1D719}_{2}\in J_{k-1/2,mp^{2}}^{+(n^{\prime })\;\text{cusp}}$
, we have
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D719}_{1}|\tilde{V}_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}(p^{2}),\unicode[STIX]{x1D719}_{2}\rangle =\langle \unicode[STIX]{x1D719}_{1},\unicode[STIX]{x1D719}_{2}|\tilde{V}_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}^{\ast }(p^{2})\rangle . & & \displaystyle \nonumber\end{eqnarray}$$
Lemma 6.12. We have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}|\tilde{V}_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}^{\ast }(p^{2})=\unicode[STIX]{x1D719}|\tilde{V}_{\unicode[STIX]{x1D6FC},n^{\prime }-\unicode[STIX]{x1D6FC}}(p^{2})|U_{p^{2}}^{\ast } & & \displaystyle \nonumber\end{eqnarray}$$
for any prime
$p$
and for any
$\unicode[STIX]{x1D719}\in J_{k-1/2,mp^{2}}^{+(n^{\prime })\;\text{cusp}}$
.
6.5 Adjoint map
$D^{\ast }$
for Jacobi forms of half-integral weight
Let
$n^{\prime }=2n-2$
and let the symbol
$D_{n^{\prime }}(N^{2})$
be as in §4.
In this section we will give the adjoint map of
$D_{n^{\prime }}(N^{2})$
with respect to the Petersson inner product.
Proposition 6.13. The adjoint map
$D_{n^{\prime }}^{\ast }(N^{2})$
of
$D_{n^{\prime }}(N^{2})$
with respect to the Petersson inner product is given by
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}|D_{n^{\prime }}^{\ast }(N^{2})=\unicode[STIX]{x1D719}|D_{n^{\prime }}(N^{2})|U_{N^{2}}^{\ast } & & \displaystyle \nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D719}\in J_{k-1/2,mN^{2}}^{+(n^{\prime })}$
. This means we have
$$\begin{eqnarray}\displaystyle \langle \unicode[STIX]{x1D713}_{1}|D_{n^{\prime }}(N^{2}),\unicode[STIX]{x1D713}_{2}\rangle =\langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}|D_{n^{\prime }}(N^{2})|U_{N^{2}}^{\ast }\rangle & & \displaystyle \nonumber\end{eqnarray}$$
for any
$\unicode[STIX]{x1D713}_{1}\in J_{k-1/2,m}^{+(n^{\prime })\;\text{cusp}}$
and any
$\unicode[STIX]{x1D713}_{2}\in J_{k-1/2,mN^{2}}^{+(n^{\prime })\;\text{cusp}}$
.
Proof. For a prime
$p$
we write
$V_{n^{\prime }}(p^{2})=\tilde{V}_{1,2n-3}(p^{2})$
and
$V_{n^{\prime }}^{\ast }(p^{2})=\tilde{V}_{1,2n-3}^{\ast }(p^{2})$
for the sake of simplicity. Let
$\unicode[STIX]{x1D6FF}$
and
$l$
be natural numbers. Due to Lemmas 6.5, 6.6, 6.11 and 6.12, for Jacobi cusp forms
$\unicode[STIX]{x1D713}_{1}$
and
$\unicode[STIX]{x1D713}_{2}$
with suitable indices, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \langle \unicode[STIX]{x1D713}_{1}|U_{p^{\unicode[STIX]{x1D6FF}-l}}|V_{n^{\prime }}(p^{2})^{l},\unicode[STIX]{x1D713}_{2}\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D713}_{1}|U_{p^{\unicode[STIX]{x1D6FF}-l}}|V_{n^{\prime }}(p^{2})^{l-1},\unicode[STIX]{x1D713}_{2}|V_{n^{\prime }}^{\ast }(p^{2})\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D713}_{1}|U_{p^{\unicode[STIX]{x1D6FF}-l}}|V_{n^{\prime }}(p^{2})^{l-1},\unicode[STIX]{x1D713}_{2}|V_{n^{\prime }}(p^{2})|U_{p^{2}}^{\ast }\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D713}_{1}|U_{p^{\unicode[STIX]{x1D6FF}-l}}|V_{n^{\prime }}(p^{2})^{l-1}|U_{p^{2}},\unicode[STIX]{x1D713}_{2}|V_{n^{\prime }}(p^{2})\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D713}_{1}|U_{p^{\unicode[STIX]{x1D6FF}-l+2}}|V_{n^{\prime }}(p^{2})^{l-1},\unicode[STIX]{x1D713}_{2}|V_{n^{\prime }}(p^{2})\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =\cdots =\langle \unicode[STIX]{x1D713}_{1}|U_{p^{\unicode[STIX]{x1D6FF}+l}},\unicode[STIX]{x1D713}_{2}|V_{n^{\prime }}(p^{2})^{l}\rangle =\langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}|V_{n^{\prime }}(p^{2})^{l}|U_{p^{\unicode[STIX]{x1D6FF}+l}}^{\ast }\rangle \nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}|V_{n^{\prime }}(p^{2})^{l}|U_{p^{\unicode[STIX]{x1D6FF}-l}}|U_{p^{2\unicode[STIX]{x1D6FF}}}^{\ast }\rangle =\langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}|U_{p^{\unicode[STIX]{x1D6FF}-l}}|V_{n^{\prime }}(p^{2})^{l}|U_{p^{2\unicode[STIX]{x1D6FF}}}^{\ast }\rangle .\nonumber\end{eqnarray}$$
Since
$D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})$
is a linear combination of compositions of some
$V_{n^{\prime }}(p^{2})$
and
$U_{p}$
, and since
$V_{n^{\prime }}(p^{2})$
and
$\{{U_{N}\}}_{N}$
are compatible, we can write
$D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})=\sum _{l=0}^{\unicode[STIX]{x1D6FF}}c_{l}U_{p^{\unicode[STIX]{x1D6FF}-l}}V_{n^{\prime }}(p^{2})^{l}$
with certain constants
$c_{l}$
. Remark that
$c_{l}$
are real numbers. By virtue of the above calculation, we obtain
$\langle \unicode[STIX]{x1D713}_{1}|D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}}),\unicode[STIX]{x1D713}_{2}\rangle =\langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}|D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})|U_{p^{2\unicode[STIX]{x1D6FF}}}^{\ast }\rangle$
. Since
$D_{n^{\prime }}(p^{2\unicode[STIX]{x1D6FF}})$
and
$D_{n^{\prime }}(q^{2\unicode[STIX]{x1D6FE}})$
for primes
$p$
,
$q$
are compatible, we conclude
$\langle \unicode[STIX]{x1D713}_{1}|D_{n^{\prime }}(N^{2}),\unicode[STIX]{x1D713}_{2}\rangle =\langle \unicode[STIX]{x1D713}_{1},\unicode[STIX]{x1D713}_{2}|D_{n^{\prime }}(N^{2})|U_{N^{2}}^{\ast }\rangle$
.◻
7 Jacobi–Eisenstein series and index-shift maps
Let
$n^{\prime }$
be a natural number. In this section we introduce Fourier–Jacobi coefficients
$\{e_{k-1/2,m}^{(n^{\prime })}\}_{m}$
of a generalized Cohen–Eisenstein series. Here
$e_{k-1/2,m}^{(n^{\prime })}\in J_{k-1/2,m}^{+(n^{\prime })}$
. Moreover, we shall give a formula for
$e_{k-1/2,mN^{2}}^{(n^{\prime })}|U_{N}^{\ast }$
, where
$U_{N}^{\ast }$
is the adjoint map introduced in §6.2.
7.1 Jacobi–Eisenstein series
In this subsection we review some Jacobi–Eisenstein series and their Fourier–Jacobi coefficients.
Let
$r>0$
be an integer and let
${\mathcal{M}}\in L_{r}^{+}$
. For an even integer
$k>n^{\prime }+r+1$
, the Jacobi–Eisenstein series of weight
$k$
of index
${\mathcal{M}}$
of degree
$n^{\prime }$
is defined by
$$\begin{eqnarray}\displaystyle E_{k,{\mathcal{M}}}^{(n^{\prime })}:=\mathop{\sum }_{M\in \unicode[STIX]{x1D6E4}_{\infty }^{(n^{\prime })}\backslash \unicode[STIX]{x1D6E4}_{n^{\prime }}}\mathop{\sum }_{\unicode[STIX]{x1D706}\in \mathbb{Z}^{(n^{\prime },r)}}1|_{k,{\mathcal{M}}}([(\unicode[STIX]{x1D706},0),0_{r}],M). & & \displaystyle \nonumber\end{eqnarray}$$
We take
${\mathcal{M}}=1\in \mathbb{Z}_{{>}0}$
. We define
$$\begin{eqnarray}\displaystyle {\mathcal{H}}_{k-1/2}^{(n^{\prime })}:=\unicode[STIX]{x1D704}_{1}(E_{k,1}^{(n^{\prime })}), & & \displaystyle \nonumber\end{eqnarray}$$
where the linear map
$\unicode[STIX]{x1D704}_{1}:J_{k,1}^{(n^{\prime })}\rightarrow M_{k-1/2}^{+(n^{\prime })}$
is defined in §5. The form
${\mathcal{H}}_{k-1/2}^{(n^{\prime })}$
belongs to the plus-space
$M_{k-1/2}^{+(n^{\prime })}$
. In this article we call
${\mathcal{H}}_{k-1/2}^{(n^{\prime })}$
a generalized Cohen–Eisenstein series,
${\mathcal{H}}_{k-1/2}^{(1)}$
having been introduced by Cohen [Reference Cohen3] (see also [Reference Arakawa1]).
For
$m\in \mathbb{Z}$
we denote by
$e_{k-1/2,m}^{(n^{\prime })}$
the
$m$
th Fourier–Jacobi coefficient of
${\mathcal{H}}_{k-1/2}^{(n^{\prime }+1)}$
, which means that
$$\begin{eqnarray}\displaystyle {\mathcal{H}}_{k-1/2}^{(n^{\prime }+1)}\left(\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D70F} & z\\ \text{}^{t\!}z & \unicode[STIX]{x1D714}\end{array}\right)\right)=\mathop{\sum }_{m\in \mathbb{Z}}e_{k-1/2,m}^{(n^{\prime })}(\unicode[STIX]{x1D70F},z)e(m\unicode[STIX]{x1D714}), & & \displaystyle \nonumber\end{eqnarray}$$
for
$\unicode[STIX]{x1D70F}\in \mathfrak{H}_{n^{\prime }}$
,
$z\in \mathbb{C}^{n^{\prime }}$
and
$\unicode[STIX]{x1D714}\in \mathfrak{H}_{1}$
. We remark that
$e_{k-1/2,m}^{(n^{\prime })}=0$
unless
$m\equiv 0$
,
$3\;\text{mod}\;4$
, since
${\mathcal{H}}_{k-1/2}^{(n^{\prime }+1)}$
belongs to the plus-space
$M_{k-1/2}^{+(n^{\prime }+1)}$
. Remark also that
$e_{k-1/2,m}^{(n^{\prime })}$
is a Jacobi form which belongs to
$J_{k-1/2,m}^{+(n^{\prime })}$
.
On the other hand, we take a Fourier–Jacobi expansion of
$E_{k,1}^{(n^{\prime }+1)}$
,
$$\begin{eqnarray}\displaystyle & & \displaystyle E_{k,1}^{(n^{\prime }+1)}\left(\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D70F} & z_{1}\\ \text{}^{t\!}z_{1} & \unicode[STIX]{x1D714}_{1}\end{array}\right),\left(\begin{array}{@{}c@{}}z_{2}\\ \unicode[STIX]{x1D714}_{2}\end{array}\right)\right)e(\unicode[STIX]{x1D714}_{3})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{\substack{ {\mathcal{M}}\in L_{2}^{\ast } \\ {\mathcal{M}}=\big(\!\begin{smallmatrix}\ast & \ast \\ \ast & 1\end{smallmatrix}\!\big)}}e_{k,{\mathcal{M}}}^{(n^{\prime })}(\unicode[STIX]{x1D70F},(z_{1},z_{2}))e\left({\mathcal{M}}\left(\begin{array}{@{}cc@{}}\unicode[STIX]{x1D714}_{1} & \unicode[STIX]{x1D714}_{2}\\ \text{}^{t\!}\unicode[STIX]{x1D714}_{2} & \unicode[STIX]{x1D714}_{3}\end{array}\right)\right),\nonumber\end{eqnarray}$$
where
$\bigg(\!\begin{smallmatrix}\unicode[STIX]{x1D70F} & z_{1} & z_{2}\\ \text{}^{t\!}z_{1} & \unicode[STIX]{x1D714}_{1} & \unicode[STIX]{x1D714}_{2}\\ \text{}^{t\!}z_{2} & \text{}^{t\!}\unicode[STIX]{x1D714}_{2} & \unicode[STIX]{x1D714}_{3}\end{smallmatrix}\!\bigg)\in \mathfrak{H}_{n^{\prime }+2}$
,
$\unicode[STIX]{x1D70F}\in \mathfrak{H}_{n^{\prime }}$
,
$\unicode[STIX]{x1D714}_{1}\in \mathfrak{H}_{1}$
and
$\unicode[STIX]{x1D714}_{3}\in \mathfrak{H}_{1}$
. The form
$e_{k,{\mathcal{M}}}^{(n^{\prime })}$
is a Jacobi form which belongs to
$J_{k,{\mathcal{M}}}^{(n^{\prime })}$
.
Lemma 7.1. For
${\mathcal{M}}=\big(\!\begin{smallmatrix}\ast & \ast \\ \ast & 1\end{smallmatrix}\!\big)\in L_{2}^{+}$
we put
$m=\det (2{\mathcal{M}})$
. Then we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D704}_{{\mathcal{M}}}(e_{k,{\mathcal{M}}}^{(n^{\prime })})=e_{k-1/2,m}^{(n^{\prime })}. & & \displaystyle \nonumber\end{eqnarray}$$
Proof. From the definition of two linear maps
$\unicode[STIX]{x1D704}_{1}$
and
$\unicode[STIX]{x1D704}_{{\mathcal{M}}}$
and from the definition of the Fourier–Jacobi expansions, the diagram
is commutative, where the two down arrows are given by the Fourier–Jacobi expansions. Thus this lemma follows from the definitions of
$e_{k-1/2,m}^{(n^{\prime })}$
and
$e_{k,{\mathcal{M}}}^{(n^{\prime })}$
.◻
We now describe
$e_{k,{\mathcal{M}}}^{(n^{\prime })}$
as a linear combination of the Jacobi–Eisenstein series
$\{E_{k,{\mathcal{M}}^{\prime }}^{(n^{\prime })}\}_{{\mathcal{M}}^{\prime }}$
.
We denote by
$h_{k-1/2}(m)$
the
$m$
th Fourier coefficient of the Cohen–Eisenstein series
${\mathcal{H}}_{k-1/2}^{(1)}$
. This means that
${\mathcal{H}}_{k-1/2}^{(1)}(\unicode[STIX]{x1D70F})=\sum _{m}h_{k-1/2}(m)e(m\unicode[STIX]{x1D70F})$
.
Let
$m$
be a natural number such that
$-m=D_{0}f^{2}$
with a fundamental discriminant
$D_{0}$
and with a natural number
$f$
. It is obvious that
$m\equiv 0,3\;\text{mod}\;4$
. We define
$$\begin{eqnarray}\displaystyle g_{k}(m):=\mathop{\sum }_{d|f}\unicode[STIX]{x1D707}(d)h_{k-1/2}\biggl(\frac{m}{d^{2}}\biggr), & & \displaystyle \nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D707}$
is the Möbius function.
We will use the following lemma for the proof of Proposition 7.8.
Lemma 7.2. Let
$m^{\prime }$
be a natural number such that
$-m^{\prime }\equiv 0$
,
$1\;\text{mod}\;4$
. Then for any prime
$p$
we have
$$\begin{eqnarray}\displaystyle g_{k}(p^{2}m^{\prime })=\biggl(p^{2k-3}-\biggl(\frac{-m^{\prime }}{p}\biggr)p^{k-2}\biggr)g_{k}(m^{\prime }). & & \displaystyle \nonumber\end{eqnarray}$$
Proof. See [Reference Hayashida7, Lemma 3.2]. ◻
Lemma 7.3. For
${\mathcal{M}}=\big(\!\begin{smallmatrix}\ast & \ast \\ \ast & 1\end{smallmatrix}\!\big)\in L_{2}^{+}$
we set
$m=\det (2{\mathcal{M}})$
. Let
$D_{0}$
and
$f$
be as above. If
$k>n^{\prime }+3$
, then
$$\begin{eqnarray}\displaystyle e_{k,{\mathcal{M}}}^{(n^{\prime })}(\unicode[STIX]{x1D70F},z)=\mathop{\sum }_{d|f}g_{k}\biggl(\frac{m}{d^{2}}\biggr)E_{k,{\mathcal{M}}[{W_{d}}^{-1}]}^{(n^{\prime })}(\unicode[STIX]{x1D70F},z\text{}^{t\!}W_{d}), & & \displaystyle \nonumber\end{eqnarray}$$
where we chose a matrix
$W_{d}\in \mathbb{Z}^{(2,2)}$
for each
$d$
which satisfies the conditions
$\det (W_{d})=d$
,
${\mathcal{M}}[{W_{d}}^{-1}]\in L_{2}^{+}$
and
${\mathcal{M}}[{W_{d}}^{-1}]=\big(\!\begin{smallmatrix}\ast & \ast \\ \ast & 1\end{smallmatrix}\!\big)$
. If
$W_{d}$
satisfies these conditions, the right-hand side of the identity does not depend on the choice of
$W_{d}$
.
Proof. The reader is referred to [Reference Hayashida7, Proposition 3.3]. This formula was originally given in [Reference Böcherer2, Satz 7]. ◻
Definition 7.4. For
${\mathcal{M}}=\big(\!\begin{smallmatrix}\ast & \ast \\ \ast & 1\end{smallmatrix}\!\big)\in L_{2}^{+}$
we set
$m=\det (2{\mathcal{M}})$
. We define
$$\begin{eqnarray}\displaystyle E_{k-1/2,m}^{(n^{\prime })}:=\unicode[STIX]{x1D704}_{{\mathcal{M}}}(E_{k,{\mathcal{M}}}^{(n^{\prime })}). & & \displaystyle \nonumber\end{eqnarray}$$
Lemma 7.5. Let the symbols
${\mathcal{M}}$
,
$m$
and
$f$
be as above. Then we have
$E_{k-1/2,m}^{(n^{\prime })}\in J_{k-1/2,m}^{+(n^{\prime })}$
and
$$\begin{eqnarray}\displaystyle e_{k-1/2,m}^{(n^{\prime })}(\unicode[STIX]{x1D70F},z)=\mathop{\sum }_{d|f}g_{k}\biggl(\frac{m}{d^{2}}\biggr)E_{k-1/2,m/d^{2}}^{(n^{\prime })}(\unicode[STIX]{x1D70F},dz). & & \displaystyle \nonumber\end{eqnarray}$$
Lemma 7.6. In the above notation, we have
$$\begin{eqnarray}\displaystyle E_{k-1/2,mN^{2}}^{(n^{\prime })}|U_{N}^{\ast }=N^{-n^{\prime }}E_{k-1/2,m}^{(n^{\prime })} & & \displaystyle \nonumber\end{eqnarray}$$
for any natural number
$N$
, where
$U_{N}^{\ast }$
is the adjoint map of
$U_{N}$
introduced in §6.2.
Proof. Since
$$\begin{eqnarray}\displaystyle E_{k-1/2,mN^{2}}^{(n^{\prime })}|U_{N}^{\ast } & = & \displaystyle \unicode[STIX]{x1D704}_{{\mathcal{M}}}\Big(\unicode[STIX]{x1D704}_{{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{-1}(E_{k-1/2,mN^{2}}^{(n^{\prime })})|U_{\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{\ast }\Big)\nonumber\\ \displaystyle & = & \displaystyle \unicode[STIX]{x1D704}_{{\mathcal{M}}}\Big(E_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}\Big|U_{\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{\ast }\Big),\nonumber\end{eqnarray}$$
it is enough to show
$$\begin{eqnarray}\displaystyle E_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}\Big|U_{\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{\ast }=N^{-n^{\prime }}E_{k,{\mathcal{M}}}^{(n^{\prime })}. & & \displaystyle \nonumber\end{eqnarray}$$
From the definition we have
$$\begin{eqnarray}\displaystyle & & \displaystyle E_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}(\unicode[STIX]{x1D70F},z)\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{M\in \unicode[STIX]{x1D6E4}_{\infty }^{(n^{\prime })}\backslash \unicode[STIX]{x1D6E4}_{n^{\prime }}}\mathop{\sum }_{\unicode[STIX]{x1D706}\in \mathbb{Z}^{(n^{\prime },2)}}J_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}(([(\unicode[STIX]{x1D706},0),0_{2}],M),(\unicode[STIX]{x1D70F},z))^{-1}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{M\in \unicode[STIX]{x1D6E4}_{\infty }^{(n^{\prime })}\backslash \unicode[STIX]{x1D6E4}_{n^{\prime }}}\mathop{\sum }_{\unicode[STIX]{x1D706}\in \mathbb{Z}^{(n^{\prime },2)}}J_{k,{\mathcal{M}}}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\left(\left(\left[\left(\unicode[STIX]{x1D706}\left(\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\right),0\right),0_{2}\right],M\right),\left(\unicode[STIX]{x1D70F},z\left(\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\right)\right)\right)^{-1}.\nonumber\end{eqnarray}$$
We write
$X_{\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D707}^{\prime }}=[((\unicode[STIX]{x1D706}^{\prime },0),(\unicode[STIX]{x1D707}^{\prime },0)),0_{2}]$
with
$\unicode[STIX]{x1D706}^{\prime }$
,
$\unicode[STIX]{x1D707}^{\prime }\in \mathbb{Z}^{(n^{\prime },1)}$
. Then
$$\begin{eqnarray}\displaystyle & & \displaystyle \Big(E_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}\Big|U_{\big(\!\begin{array}{@{}cc@{}}N^{-1} & 0\\ 0 & 1\end{array}\!\big)}\Big|_{k,{\mathcal{M}}}X_{\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D707}^{\prime }}\Big)(\unicode[STIX]{x1D70F},z)\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{M\in \unicode[STIX]{x1D6E4}_{\infty }^{(n^{\prime })}\backslash \unicode[STIX]{x1D6E4}_{n^{\prime }}}\mathop{\sum }_{\unicode[STIX]{x1D706}\in \mathbb{Z}^{(n^{\prime },2)}}J_{k,{\mathcal{M}}}(\unicode[STIX]{x1D6FE},(\unicode[STIX]{x1D70F},z))^{-1},\nonumber\end{eqnarray}$$
where we write
$\unicode[STIX]{x1D6FE}=([(\unicode[STIX]{x1D706}\big(\!\begin{smallmatrix}N & 0\\ 0 & 1\end{smallmatrix}\!\big)+(\unicode[STIX]{x1D706}^{\prime \prime },0),(\unicode[STIX]{x1D707}^{\prime \prime },0)),0_{2}],M)$
and where
$$\begin{eqnarray}\displaystyle \left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D706}^{\prime \prime }\\ \unicode[STIX]{x1D707}^{\prime \prime }\end{array}\right)=\text{}^{t\!}M^{-1}\left(\begin{array}{@{}c@{}}\unicode[STIX]{x1D706}^{\prime }\\ \unicode[STIX]{x1D707}^{\prime }\end{array}\right). & & \displaystyle \nonumber\end{eqnarray}$$
Thus we obtain
$$\begin{eqnarray}\displaystyle & & \displaystyle E_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}\Big|U_{\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{\ast }\nonumber\\ \displaystyle & & \displaystyle \quad =N^{-2n^{\prime }}\mathop{\sum }_{\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D707}^{\prime }\in (\mathbb{Z}/N\mathbb{Z})^{(n^{\prime },1)}}\Big(E_{k,{\mathcal{M}}\big[\big(\!\begin{array}{@{}cc@{}}N & 0\\ 0 & 1\end{array}\!\big)\big]}^{(n^{\prime })}\Big|U_{\big(\!\begin{array}{@{}cc@{}}N^{-1} & 0\\ 0 & 1\end{array}\!\big)}\Big|_{k,{\mathcal{M}}}X_{\unicode[STIX]{x1D706}^{\prime },\unicode[STIX]{x1D707}^{\prime }}\Big)\nonumber\\ \displaystyle & & \displaystyle \quad =N^{-n^{\prime }}E_{k,{\mathcal{M}}}^{(n^{\prime })}.\Box \nonumber\end{eqnarray}$$
Definition 7.7. Let
$N$
be a natural number and let
$\unicode[STIX]{x1D708}=\text{ord}_{p}N$
be the largest integer such that
$p^{\unicode[STIX]{x1D708}}|N$
. We define
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6F9}_{p}^{(n^{\prime })}(N,X) & := & \displaystyle \unicode[STIX]{x1D6F9}_{p,D_{0}}^{(n^{\prime })}(N,X)\nonumber\\ \displaystyle & = & \displaystyle \frac{X^{\unicode[STIX]{x1D708}+1}-X^{-(\unicode[STIX]{x1D708}+1)}}{X-X^{-1}}-\biggl(\frac{D_{0}}{p}\biggr)p^{-n^{\prime }/2-1/2}\frac{X^{\unicode[STIX]{x1D708}}-X^{-\unicode[STIX]{x1D708}}}{X-X^{-1}}.\nonumber\end{eqnarray}$$
Proposition 7.8. Let the symbols
$m$
,
$D_{0}$
and
$f$
be as in Lemma 7.3. Recall that
$D_{0}$
is the fundamental discriminant such that
$f=\sqrt{m/|D_{0}|}$
is a natural number. If natural number
$N$
is coprime to
$f$
, then we obtain
$$\begin{eqnarray}\displaystyle e_{k-1/2,mN^{2}}^{(n^{\prime })}|U_{N}^{\ast }=N^{k-n^{\prime }/2-3/2}\mathop{\prod }_{p|N}\unicode[STIX]{x1D6F9}_{p}^{(n^{\prime })}(p^{\unicode[STIX]{x1D708}},p^{k-n^{\prime }/2-3/2})e_{k-1/2,m}^{(n^{\prime })}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$\unicode[STIX]{x1D708}=\text{ord}_{p}N$
.
Proof. We assume that
$f$
and
$p$
are coprime. Due to Lemmas 6.6, 7.2, 7.5 and 7.6, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle e_{k-1/2,mp^{2\unicode[STIX]{x1D708}}}^{(n^{\prime })}|U_{p^{\unicode[STIX]{x1D708}}}^{\ast }=\mathop{\sum }_{d|f}\mathop{\sum }_{i=0}^{\unicode[STIX]{x1D708}}g_{k}\biggl(\frac{m}{d^{2}}p^{2(\unicode[STIX]{x1D708}-i)}\biggr)E_{k-1/2,(m/d^{2})p^{2(\unicode[STIX]{x1D708}-i)}}^{(n^{\prime })}|U_{dp^{i}}|U_{p^{\unicode[STIX]{x1D708}}}^{\ast }\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{d|f}g_{k}\biggl(\frac{m}{d^{2}}\biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\bigg\{\mathop{\sum }_{i=0}^{\unicode[STIX]{x1D708}-1}\biggl(p^{(2k-3)(\unicode[STIX]{x1D708}-i)}-\biggl(\frac{D_{0}}{p}\biggr)p^{k-2+(2k-3)(\unicode[STIX]{x1D708}-i-1)}\biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,E_{k-1/2,(m/d^{2})p^{2(\unicode[STIX]{x1D708}-i)}}^{(n^{\prime })}|U_{d}|U_{p^{\unicode[STIX]{x1D708}-i}}^{\ast }+E_{k-1/2,m/d^{2}}^{(n^{\prime })}|U_{d}\bigg\}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{d|f}g_{k}\biggl(\frac{m}{d^{2}}\biggr)E_{k-1/2,m/d^{2}}^{(n^{\prime })}|U_{d}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\bigg\{1+\mathop{\sum }_{i=0}^{\unicode[STIX]{x1D708}-1}\biggl(p^{(2k-3)(\unicode[STIX]{x1D708}-i)}-\biggl(\frac{D_{0}}{p}\biggr)p^{k-2+(2k-3)(\unicode[STIX]{x1D708}-i-1)}\biggr)p^{-n^{\prime }(\unicode[STIX]{x1D708}-i)}\bigg\}\nonumber\\ \displaystyle & & \displaystyle \quad =e_{k-1/2,m}^{(n^{\prime })}\bigg\{p^{(k-n^{\prime }/2-3/2)\unicode[STIX]{x1D708}}\frac{p^{(k-n^{\prime }/2-3/2)(\unicode[STIX]{x1D708}+1)}-p^{-(k-n^{\prime }/2-3/2)(\unicode[STIX]{x1D708}+1)}}{p^{(k-n^{\prime }/2-3/2)}-p^{-(k-n^{\prime }/2-3/2)}}\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\biggl(\frac{D_{0}}{p}\biggr)p^{k-2-n^{\prime }+(k-n^{\prime }/2-3/2)(\unicode[STIX]{x1D708}-1)}\frac{p^{(k-n^{\prime }/2-3/2)\unicode[STIX]{x1D708}}-p^{-(k-n^{\prime }/2-3/2)\unicode[STIX]{x1D708}}}{p^{(k-n^{\prime }/2-3/2)}-p^{-(k-n^{\prime }/2-3/2)}}\bigg\}.\nonumber\end{eqnarray}$$
Thus we have
$$\begin{eqnarray}\displaystyle e_{k-1/2,mp^{2\unicode[STIX]{x1D708}}}^{(n^{\prime })}|U_{p^{\unicode[STIX]{x1D708}}}^{\ast }=p^{(k-n^{\prime }/2-3/2)\unicode[STIX]{x1D708}}\unicode[STIX]{x1D6F9}_{p}^{(n^{\prime })}(p^{\unicode[STIX]{x1D708}},p^{k-n^{\prime }/2-3/2})e_{k-1/2,m}^{(n^{\prime })}. & & \displaystyle \nonumber\end{eqnarray}$$
By induction with respect to
$m$
and
$p$
, we obtain the proposition.◻
8 Proof of main theorem
In this section we prove Theorem 1.1. Let the symbols
$G$
and
$\unicode[STIX]{x1D719}_{m}$
be as in §§1 and 3. The adjoint map
$U_{N}^{\ast }$
was introduced in Definition 6.4. The index-shift map
$D_{2n-2}(N^{2})$
appears in §4. The adjoint map
$\tilde{D}_{2n-2}(N^{2})^{\ast }$
of
$D_{2n-2}(N^{2})$
was obtained in Proposition 6.13.
We write
$-m=D_{0}f^{2}$
with a fundamental discriminant
$D_{0}$
and with a natural number
$f$
.
Lemma 8.1. We assume that
$f$
and
$N$
are coprime. Then we have
$$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D719}_{mN^{2}}|U_{N}^{\ast }=N^{k-n-1/2}\mathop{\prod }_{p|N}\unicode[STIX]{x1D6F9}_{p}^{(2n-2)}(p^{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D6FC}_{p})\unicode[STIX]{x1D719}_{m}. & & \displaystyle \nonumber\end{eqnarray}$$
Proof. Due to Proposition 7.8 we have
$$\begin{eqnarray}\displaystyle e_{k^{\prime }-1/2,mN^{2}}^{(2n-2)}|U_{N}^{\ast }=N^{k^{\prime }-n-1/2}\mathop{\prod }_{p|N}\unicode[STIX]{x1D6F9}_{p}^{(2n-2)}(p^{\unicode[STIX]{x1D708}},p^{k^{\prime }-n-1/2})e_{k^{\prime }-1/2,m}^{(2n-2)} & & \displaystyle \nonumber\end{eqnarray}$$
for infinitely many natural numbers
$k^{\prime }$
. By a standard argument for Ikeda lifts (cf. [Reference Ikeda12, pp. 664–665], [Reference Hayashida7, proof of Theorem 8.2]), we obtain the lemma.◻
Lemma 8.2. We assume that
$f$
and
$N$
are coprime. Then we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D719}_{m}|D_{2n-2}(N^{2})|D_{2n-2}^{\ast }(N^{2})\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{d|N}(Nd^{-1})^{2k-2n-1}d^{2k-3}\mathop{\sum }_{d_{1}|(N/d)}\unicode[STIX]{x1D707}(d_{1})\biggl(\frac{D_{0}}{d_{1}}\biggr)d_{1}^{n-3/2}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\prod }_{p|(N^{2}/d^{2}d_{1})}\unicode[STIX]{x1D6F9}_{p}^{(2n-2)}(p^{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D6FC}_{p})\unicode[STIX]{x1D719}_{m}.\nonumber\end{eqnarray}$$
Proof. By virtue of Proposition 6.13, Lemma 4.4(4), Proposition 4.3, Lemma 6.6(2) and Lemma 8.1, we obtain
$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D719}_{m}|D_{2n-2}(N^{2})|D_{2n-2}^{\ast }(N^{2})\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D719}_{m}|D_{2n-2}(N^{2})|D_{2n-2}(N^{2})|U_{N^{2}}^{\ast }\nonumber\\ \displaystyle & & \displaystyle \quad =\unicode[STIX]{x1D719}_{m}\bigg|\mathop{\sum }_{d|N}d^{2k-3}\mathop{\sum }_{d_{1}|(N/d)}\unicode[STIX]{x1D707}(d_{1})\biggl(\frac{D_{0}}{d_{1}}\biggr)d_{1}^{k-2}U_{d^{2}d_{1}}\bigg|D_{2n-2}\biggl(\frac{N^{4}}{d^{4}d_{1}^{2}}\biggr)\bigg|U_{N^{2}}^{\ast }\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{d|N}d^{2k-3}\mathop{\sum }_{d_{1}|(N/d)}\unicode[STIX]{x1D707}(d_{1})\biggl(\frac{D_{0}}{d_{1}}\biggr)d_{1}^{k-2}\unicode[STIX]{x1D719}_{mN^{4}/d^{4}d_{1}^{2}}|U_{d^{2}d_{1}}|U_{N^{2}}^{\ast }\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{d|N}d^{2k-3}\mathop{\sum }_{d_{1}|(N/d)}\unicode[STIX]{x1D707}(d_{1})\biggl(\frac{D_{0}}{d_{1}}\biggr)d_{1}^{k-2}\unicode[STIX]{x1D719}_{mN^{4}/d^{4}d_{1}^{2}}|U_{N^{2}/d^{2}d_{1}}^{\ast }\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{d|N}d^{2k-3}\mathop{\sum }_{d_{1}|(N/d)}\unicode[STIX]{x1D707}(d_{1})\biggl(\frac{D_{0}}{d_{1}}\biggr)d_{1}^{k-2}(N^{2}d^{-2}d_{1}^{-1})^{k-n-1/2}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\prod }_{p|(N^{2}/d^{2}d_{1})}\unicode[STIX]{x1D6F9}_{p}^{(2n-2)}(p^{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D6FC}_{p})\unicode[STIX]{x1D719}_{m}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{d|N}(Nd^{-1})^{2k-2n-1}d^{2k-3}\mathop{\sum }_{d_{1}|(N/d)}\unicode[STIX]{x1D707}(d_{1})\biggl(\frac{D_{0}}{d_{1}}\biggr)d_{1}^{n-3/2}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\prod }_{p|(N^{2}/d^{2}d_{1})}\unicode[STIX]{x1D6F9}_{p}^{(2n-2)}(p^{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D6FC}_{p})\unicode[STIX]{x1D719}_{m}.\Box \nonumber\end{eqnarray}$$
We now prove Theorem 1.1. We have
$$\begin{eqnarray}\displaystyle \mathop{\sum }_{\substack{ m\in \mathbb{Z}_{{>}0}-m\equiv 0,1\;\text{mod}\;4}}\frac{\langle \unicode[STIX]{x1D719}_{m},\unicode[STIX]{x1D719}_{m}\rangle }{m^{s+k-n-1/2}}=\mathop{\sum }_{D_{0}}\frac{1}{|D_{0}|^{s+k-n-1/2}}\mathop{\sum }_{N=1}^{\infty }\frac{\langle \unicode[STIX]{x1D719}_{|D_{0}|N^{2}},\unicode[STIX]{x1D719}_{|D_{0}|N^{2}}\rangle }{N^{2(s+k-n-1/2)}}, & & \displaystyle \nonumber\end{eqnarray}$$
where
$D_{0}$
runs over all negative fundamental discriminants. Due to Proposition 4.3, we have
$\unicode[STIX]{x1D719}_{|D_{0}|N^{2}}=\unicode[STIX]{x1D719}_{|D_{0}|}|D_{2n-2}(N^{2})$
for any natural number
$N$
. Here
$D_{2n-2}(N^{2})$
is defined for a fixed
$|D_{0}|$
. Thus, by virtue of Lemma 8.2, we have
$$\begin{eqnarray}\displaystyle & & \displaystyle \mathop{\sum }_{N=1}^{\infty }\frac{\langle \unicode[STIX]{x1D719}_{|D_{0}|N^{2}},\unicode[STIX]{x1D719}_{|D_{0}|N^{2}}\rangle }{N^{2s}}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{N=1}^{\infty }\frac{\langle \unicode[STIX]{x1D719}_{|D_{0}|}|D_{2n-2}(N^{2}),\unicode[STIX]{x1D719}_{|D_{0}|}|D_{2n-2}(N^{2})\rangle }{N^{2s}}\nonumber\\ \displaystyle & & \displaystyle \quad =\mathop{\sum }_{N=1}^{\infty }\frac{\langle \unicode[STIX]{x1D719}_{|D_{0}|}|D_{2n-2}(N^{2})D_{2n-2}^{\ast }(N^{2}),\unicode[STIX]{x1D719}_{|D_{0}|}\rangle }{N^{2s}}\nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D719}_{|D_{0}|},\unicode[STIX]{x1D719}_{|D_{0}|}\rangle \mathop{\sum }_{N=1}^{\infty }\mathop{\sum }_{d|N}\biggl(\frac{N}{d}\biggr)^{-2s+2k-2n-1}d^{-2s+2k-3}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{d_{1}|(N/d)}\unicode[STIX]{x1D707}(d_{1})\biggl(\frac{D_{0}}{d_{1}}\biggr)d_{1}^{n-3/2}\mathop{\prod }_{p|(N^{2}/d^{2}d_{1})}\unicode[STIX]{x1D6F9}_{p}^{(2n-2)}(p^{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D6FC}_{p})\nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D719}_{|D_{0}|},\unicode[STIX]{x1D719}_{|D_{0}|}\rangle \unicode[STIX]{x1D701}(2s-2k+3)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\sum }_{N=1}^{\infty }N^{-2s+2k-2n-1}\mathop{\sum }_{d_{1}|N}\unicode[STIX]{x1D707}(d_{1})\biggl(\frac{D_{0}}{d_{1}}\biggr)d_{1}^{n-3/2}\mathop{\prod }_{p|(N^{2}/d_{1})}\unicode[STIX]{x1D6F9}_{p}^{(2n-2)}(p^{\unicode[STIX]{x1D708}},\unicode[STIX]{x1D6FC}_{p})\nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D719}_{|D_{0}|},\unicode[STIX]{x1D719}_{|D_{0}|}\rangle \unicode[STIX]{x1D701}(2s-2k+3)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\prod }_{p}\bigg\{1+\mathop{\sum }_{\unicode[STIX]{x1D6FF}=1}^{\infty }p^{\unicode[STIX]{x1D6FF}(-2s+2k-2n-1)}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\biggl(\unicode[STIX]{x1D6F9}_{p}^{(2n-2)}(p^{2\unicode[STIX]{x1D6FF}},\unicode[STIX]{x1D6FC}_{p})-\biggl(\frac{D_{0}}{p}\biggr)p^{n-3/2}\unicode[STIX]{x1D6F9}_{p}^{(2n-2)}(p^{2\unicode[STIX]{x1D6FF}-1},\unicode[STIX]{x1D6FC}_{p})\biggr)\bigg\}\nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D719}_{|D_{0}|},\unicode[STIX]{x1D719}_{|D_{0}|}\rangle \unicode[STIX]{x1D701}(2s-2k+3)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\prod }_{p}\{(1-\unicode[STIX]{x1D6FC}_{p}^{2}p^{-2s+2k-2n-1})(1-\unicode[STIX]{x1D6FC}_{p}^{-2}p^{-2s+2k-2n-1})\}^{-1}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\bigg\{1+p^{-2s+2k-2n-1}-\biggl(\frac{D_{0}}{p}\biggr)p^{-2s+2k-3n-1/2}(\unicode[STIX]{x1D6FC}_{p}+\unicode[STIX]{x1D6FC}_{p}^{-1})\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\biggl(\frac{D_{0}}{p}\biggr)^{2}p^{-2s+2k-2n-2}(1+p^{-2s+2k-2n-1})\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\biggl(\frac{D_{0}}{p}\biggr)p^{-2s+2k-n-5/2}(\unicode[STIX]{x1D6FC}_{p}+\unicode[STIX]{x1D6FC}_{p}^{-1})\bigg\}\nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D719}_{|D_{0}|},\unicode[STIX]{x1D719}_{|D_{0}|}\rangle \unicode[STIX]{x1D701}(2s-2k+3)L(2s-2k+2n+1,f,Ad)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\prod }_{p}\bigg\{(1-p^{-4s+4k-4n-2})\biggl(1+\biggl(\frac{D_{0}}{p}\biggr)^{2}p^{-2s+2k-2n-2}\biggr)\nonumber\\ \displaystyle & & \displaystyle \qquad -\,\biggl(\frac{D_{0}}{p}\biggr)(1-p^{-2s+2k-2n-1})(1+p^{2n-2})p^{-2s+2k-3n-1/2}(\unicode[STIX]{x1D6FC}_{p}+\unicode[STIX]{x1D6FC}_{p}^{-1})\bigg\}\nonumber\\ \displaystyle & & \displaystyle \quad =\langle \unicode[STIX]{x1D719}_{|D_{0}|},\unicode[STIX]{x1D719}_{|D_{0}|}\rangle \unicode[STIX]{x1D701}(2s-2k+3)\unicode[STIX]{x1D701}(4s-4k+4n+2)^{-1}\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,L(2s-2k+2n+1,f,Ad)\nonumber\\ \displaystyle & & \displaystyle \qquad \times \,\mathop{\prod }_{p\nmid D_{0}}\bigg\{1+p^{-2s+2k-2n-2}-\frac{\big(\frac{D_{0}}{p}\big)(1+p^{2n-2})p^{-k+1}a_{f}(p)}{p^{2s-2k+2n+1}+1}\bigg\}.\nonumber\end{eqnarray}$$
Therefore we conclude Theorem 1.1.
Acknowledgements
The author would like to express his sincere thanks to Professor Hidenori Katsurada and Yoshinori Mizuno for their variable comments. Also the author would like to express his sincere gratitude to the anonymous referee for many suggestions, which helped the author to improve the paper. This work was supported by JSPS KAKENHI grant 80597766.
