Proofs of Ibukiyama’s conjectures on Siegel modular forms of half-integral weight and of degree 2

Abstract

We prove Ibukiyama’s conjectures on Siegel modular forms of half-integral weight and of degree 2 by using Arthur’s multiplicity formula on the split odd special orthogonal group ${\text {SO}}_5$ and Gan–Ichino’s multiplicity formula on the metaplectic group ${\text {Mp}}_4$. In the proof, the representation theory of the Jacobi groups also plays an important role.

The adelic lift of \(F \in S_{k-\frac{1}{2}, j}^+(\varGamma _0(4), (\frac{-1}{\cdot })^l)\)

In this appendix, we consider the adelic lifts of Siegel cusp forms of half-integral weight. Note that any argument here is not needed to prove Theorem 2.1. Let $j\ge 0$ and $k\ge 3$ be integers, $l \in {\mathbb {Z}}/2{\mathbb {Z}}$, and $\psi$ the nontrivial additive character of ${\mathbb {Q}}\backslash {\mathbb {A}}$ such that $\psi _\infty =\mathbf{e }$. If j is odd, then the plus space $S_{k-\frac{1}{2},j}^+(\varGamma _0(4), \left( \frac{-1}{\cdot }\right) ^l)$ is zero, and the arguments in this section are trivially true. Hence we assume that j is even.

For any prime number p, the restriction of the Weil representation $\omega _{\psi _p}$ of ${\text {Mp}}_4({\mathbb {Q}}_p)$ to the metaplectic covering $\widetilde{\varGamma }_0(4)_p$ over

defines a genuine character $\varepsilon _p$ of $\widetilde{\varGamma }_0(4)_p$ by

$$\begin{aligned} \omega _{\psi _p}(\gamma ) 1_{{\mathbb {Z}}_p^2}&=\varepsilon _p(\gamma )^{-1}1_{{\mathbb {Z}}_p^2},&\gamma&\in \widetilde{\varGamma }_0(4)_p. \end{aligned}$$

Note that if $p\ne 2$, $\varepsilon _p$ is quadratic and defines the splitting (5) over $\varGamma _0(4)_p={\text {Sp}}_4({\mathbb {Z}}_p)$. Then, these characters define a genuine character $\varepsilon _{ fin }=\prod _p \varepsilon _p$ of a subgroup

$$\begin{aligned} \widetilde{\varGamma }_0(4)_{ fin }= \widetilde{\varGamma }_0(4)_2 \times \prod _{p\ne 2} \varGamma _0(4)_p \end{aligned}$$

of ${\text {Mp}}_4({\mathbb {A}})$. At the real place, the Weil representation defines the factor of automorphy $j_\infty (g,Z)$ by

$$\begin{aligned} \omega _{\psi _\infty }(g)\varphi _Z =j_\infty (g,Z)^{-1}\varphi _{gZ}, \end{aligned}$$

for $g \in {\text {Mp}}_4({\mathbb {R}})$ and $Z \in {\mathfrak {H}}_2$, where

$$\begin{aligned} \varphi _Z(x) =\mathbf{e }({}^{\mathrm t}{}{x}Zx) \in {\mathcal {S}}({\mathbb {R}}^2). \end{aligned}$$

Note that the weight of the character $j_\infty (-, i) : \widetilde{K_\infty } \rightarrow {\mathbb {C}}^\times$ relative to the additive character $\psi _\infty =\mathbf{e }$ is $(\frac{1}{2}, \ldots , \frac{1}{2})$. Then it is known that for any $\gamma \in \varGamma _0(4)$, we have

$$\begin{aligned} \frac{\theta (\gamma Z)}{\theta (Z)} =j_\infty ((\gamma ,1),Z) \varepsilon _{ fin }((\gamma ,1)). \end{aligned}$$

The strong approximation theorem for ${\text {Sp}}_4$ induces that for ${\text {Mp}}_4$, hence we have

$$\begin{aligned} {\text {Mp}}_4({\mathbb {A}}) ={\text {Sp}}_4({\mathbb {Q}}) {\text {Mp}}_4({\mathbb {R}}) \widetilde{\varGamma _0}(4)_{ fin }. \end{aligned}$$

Let $F\in S_{k-\frac{1}{2},j}^+(\varGamma _0(4), \left( \frac{-1}{\cdot }\right) ^l)$ be a Siegel modular form. Now the adelic lift $\varPhi _F$ of F is defined by

$$\begin{aligned} \varPhi _F(g) =\left( \frac{-1}{\kappa } \right) ^l \left\{ j_\infty (g_\infty , i) \varepsilon _{ fin }(\kappa ) \right\} ^{-(2k-1)} {\text {Sym}}_j(J(g_\infty ,i))^{-1}F(g_\infty i), \end{aligned}$$

where $g=\gamma g_\infty \kappa \in {\text {Mp}}_4({\mathbb {A}})={\text {Sp}}_4({\mathbb {Q}}) {\text {Mp}}_4({\mathbb {R}}) \widetilde{\varGamma _0}(4)_{ fin }$. Here, note that $\left( \frac{-1}{\kappa } \right) =\left( \frac{-1}{\kappa _2} \right)$, ($\kappa =(\kappa _p)_p$). The function $\varPhi _F$ is well-defined.

Proposition A.1

The function $\varPhi _F : {\text {Mp}}_4({\mathbb {A}}) \rightarrow V_j$ satisfies the following:

• (1) $\varPhi _F$ is left ${\text {Sp}}_4({\mathbb {Q}})$-invariant;

• (2) $\varPhi _F$ is right ${\text {Sp}}_4({\mathbb {Z}}_p)$-invariant for $p\ne 2$;

• (3) $\varPhi _F(gx) = j_\infty (x,i)^{-2k+1} {\text {Sym}}_j(J(x,i))^{-1}\varPhi _F(g)$, for any $g\in {\text {Mp}}_4({\mathbb {R}})$, $x\in \widetilde{K_\infty }$;

• (4) ${\mathfrak {p}}_{\mathbb {C}}^- \cdot \varPhi _F = 0$;

• (5) $\varPhi _F$ is cuspidal.

Proof

The proof is similar to that of [6, Theorem 1]. $\square$

Since the complex conjugate $\overline{{\text {Sym}}_j}$ is isomorphic to the contragredient representation of ${\text {Sym}}_j$, by a similar argument to [8, §4.5 and §6.3.4], we can construct an automorphic cuspidal representation

$$\begin{aligned} \pi _F=\bigotimes _v \pi _{F,v} \subset L^2_\text {disc}({\text {Mp}}_4). \end{aligned}$$

and check that it is a direct sum $\oplus _i \pi _i$ of a finite number of irreducible automorphic cuspidal representations $\pi _i=\otimes _v \pi _{i,v}$ such that

• $\pi _{i, p}$ are unramified and isomorphic to each other for any odd prime p;

• $\pi _{i,\infty } =\pi _{(k+j-\frac{1}{2}, k-\frac{1}{2}), \mathbf{e }} =\pi _{(-k+\frac{1}{2}, -j-k+\frac{1}{2}), {\overline{\mathbf{e }}}}$ for any i.

In particular, $\pi _i$ are nearly equivalent. Moreover, we have the following lemma.

Lemma A.2

Let p be an odd prime.

• (1) Assume that $l\equiv k \pmod 2$, and let $(\alpha _1, \alpha _2)$ be the Satake parameter of $\pi _{i, p}$ with respect to ${\overline{\psi }}_p$. Then we have

  $$\begin{aligned} \eta (p)&=\left( \frac{-1}{p}\right) ^k p^2(\alpha _1+\alpha _2+\alpha _1^{-1}+\alpha _2^{-1}),\\ \omega (p)&=p^3(\alpha _1\alpha _2+\alpha _1\alpha _2^{-1}+\alpha _1^{-1}\alpha _2+\alpha _1^{-1}\alpha _2^{-1}+1-p^{-2}), \end{aligned}$$

  and hence

  $$\begin{aligned} L(s, F)_p =L(s-k-j+\tfrac{3}{2}, \pi _{i,p}, {\overline{\psi }}_p). \end{aligned}$$

• (2) Assume that $l\equiv k-1 \pmod 2$, and let $(\alpha _1, \alpha _2)$ be the Satake parameter of $\pi _{i, p}$ with respect to $\psi _p$. Then we have

  $$\begin{aligned} \eta (p)&=\left( \frac{-1}{p}\right) ^{k+1} p^2(\alpha _1+\alpha _2+\alpha _1^{-1}+\alpha _2^{-1}),\\ \omega (p)&=p^3(\alpha _1\alpha _2+\alpha _1\alpha _2^{-1}+\alpha _1^{-1}\alpha _2+\alpha _1^{-1}\alpha _2^{-1}+1-p^{-2}), \end{aligned}$$

  and hence

  $$\begin{aligned} L(s,F)_p =L(s-k-j+\tfrac{3}{2}, \pi _{i,p}, \psi _p). \end{aligned}$$

Proof

Complete systems $\{\widetilde{g}_{s,t}\}_t$ of representatives of the right coset decompositions

$$\begin{aligned} \widetilde{\varGamma }_0(4) (K_s(p^2), p^{1-\frac{s}{2}}) \widetilde{\varGamma }_0(4) =\bigsqcup _t \widetilde{\varGamma }_0(4) \widetilde{g}_{s,t} \end{aligned}$$

are explicitly given by Zhuravlev [31]. Thus the assertions are proved by straightforward calculation. $\square$

Let us recall from Theorem 4.1 that we have an isomorphism $\varPsi$. Put

$$\begin{aligned} F'=\varPsi ^{-1}(F) \in {\left\{ \begin{array}{ll} J_{(k,j),1}^\text {hol, cusp},&{}\text { when }l\equiv k \pmod 2,\\ J_{(k,j),1}^\text {skew, cusp},&{}\text { when }l\equiv k-1 \pmod 2, \end{array}\right. } \end{aligned}$$

and write

$$\begin{aligned} \pi _{F'} =\pi ' \otimes \pi _{ SW , \psi } =\bigotimes _v \left( \pi '_p \otimes \pi _{ SW , \psi _p} \right) . \end{aligned}$$

Then $\pi _F$ and $\pi '$ are related as follows.

Theorem A.3

The automorphic representation $\pi _F$ is irreducible, and if we write

$$\begin{aligned} \pi _F=\bigotimes _v \pi _{F, v}, \end{aligned}$$

then we have the following.

• (1) Assume that $l\equiv k \pmod 2$. Then we have $\pi _{F, v}\cong \pi '_v$ for every place v, i.e., $\pi _F \cong \pi '$.

• (2) Assume that $l\equiv k-1 \pmod 2$. Then the L-parameter and the character of its component group associated to $\pi _{F,v}$ relative to $\psi _v$ coincide with those associated to $\pi '_v$ relative to ${\overline{\psi }}_v$.

Proof

(1) At the real place, by (19) we have

$$\begin{aligned} \pi _{i,\infty } \cong \pi _{(k+j-\frac{1}{2}, k-\frac{1}{2}), \mathbf{e }} \cong \pi '_\infty . \end{aligned}$$

At every finite place p except 2, by Lemmas 6.7 and A.2 we have

$$\begin{aligned} L(s-k-j+\tfrac{3}{2}, \pi _{i,p}, {\overline{\psi }}_p) =L(s, F)_p =L(s-k-j+\tfrac{3}{2}, \pi '_p, {\overline{\psi }}_p). \end{aligned}$$

Since both of $\pi _{i,p}$ and $\pi '_p$ are unramified representation of ${\text {Mp}}_4({\mathbb {Q}}_p)$, they are isomorphic. In particular, $\pi _i$ and $\pi '$ are nearly equivalent, and hence the A-parameter of $\pi _i$ relative to ${\overline{\psi }}$ is that of $\pi '$ relative to ${\overline{\psi }}$. By the proof of Lemma 6.9, the A-parameter of $\pi '$ relative to ${\overline{\psi }}$ is tempered. (Note that the case (5) may occur.) Since the local L-packet of an unramified L-parameter for ${\text {Mp}}_4$ is a singleton, this implies that

$$\begin{aligned} \pi _{i,2} \cong \pi '_2. \end{aligned}$$

Hence $\pi _F$ is isomorphic to a direct sum of $\pi '$. Since Gan–Ichino’s multiplicity formula tells us that $\pi '$ appears in $L^2_\text {disc}({\text {Mp}}_4)$ with multiplicity one, the assertion follows.

(2) At the real place, since

$$\begin{aligned} \pi _{i,\infty }&\cong \pi _{(k+j-\frac{1}{2}, k-\frac{1}{2}), \mathbf{e }},&\pi '_\infty&\cong \pi _{(k+j-\frac{1}{2},k-\frac{1}{2}), {\overline{\mathbf{e }}}}, \end{aligned}$$

the L-parameter and the character of its component group associated to $\pi _{i,\infty }$ relative to $\psi _\infty$ coincide with those associated to $\pi '_\infty$ relative to ${\overline{\psi }}_\infty$. At every finite place p except 2, by Lemmas 6.7 and A.2 we have

$$\begin{aligned} L(s-k-j+\tfrac{3}{2}, \pi _{i,p}, \psi _p) =L(s,F)_p =L(s-k-j+\tfrac{3}{2}, \pi '_p, {\overline{\psi }}_p). \end{aligned}$$

Since the local L-packet of an unramified L-parameter for ${\text {Mp}}_4$ is a singleton, there is a similar relation between $\pi _{i,p}$ and $\pi '_p$. Hence the A-parameter of $\pi _i$ relative to $\psi$ is that of $\pi '$ relative to ${\overline{\psi }}$, which is tempered. Then the assertion follows from the same argument as the proof of (1). $\square$