The weight reduction of mod p Siegel modular forms for $GSp_4$ and theta operators

Abstract

In this paper we investigate the (classical) weights of mod p Siegel modular forms of degree 2 toward studying Serre’s conjecture for $GSp_4/\mathbb {Q}$. We first construct various theta operators on the space of such forms à la Katz and define the theta cycles for the specific theta operators. Secondly, we study the partial Hasse invariants on each Ekedahl–Oort stratum and their local behaviors. This enables us to obtain a kind of weight reduction theorem for mod p Siegel modular forms of degree 2 without increasing the level.

Appendix A

In this section we give an explicit form of Pieri’s decomposition. Let R be an $\overline{\mathbb {F}}_p$-algebra. Put $\textrm{St}_2(R)=Re_1\oplus Re_2$ and let $GL_2(R)$ acts on $St_2$ by

$$\begin{aligned} ge_1=ae_1+c e_2,\ ge_2=be_1+de_2,\ g=\begin{pmatrix} a &{}\quad b \\ c&{}\quad d \end{pmatrix}. \end{aligned}$$

For a positive integer n, let $V(n)=\textrm{Sym}^n \textrm{St}_2(R)$ be the n-th symmetric representation of $GL_2(R)$. Put $V(n,m)=V(n)\otimes _R \det ^n(St_2(R))$ for an integer $m\ge 0$.

We are concerned with an explicit decomposition of $V(n)\otimes _R V(2)$. Consider the basis $u_i=e^i_1e^{n-i}_2,\ i=0,\ldots ,n$ (resp. $v_2=e^2_1,\ v_1=e_1e_2,\ v_0=e^2_2$) of V(n) (resp. V(2)). We define the operators E, F on V(n) by $Eu_i=iu_{i-1},\ Fu_i=(n-i)u_{i+1}$. We also define the same operators on $V(n)\otimes _R V(n')$ by Leibniz rule $E(u_i\otimes u_{j})=Eu_i\otimes u_j+u_i\otimes Eu_j,\ F(u_i\otimes u_{j})=Fu_i\otimes u_j+u_i\otimes Fu_j$.

We first assume that $n+3\le p$. Put

$$\begin{aligned} w_0=u_n\otimes v_2,\ w_1=u_n\otimes v_1-u_{n-1}\otimes v_2,\ w_2=u_n\otimes v_0-2u_{n-1}\otimes v_1+u_{n-2}\otimes v_2 \end{aligned}$$

Since $Fw_i=0$ for $i=0,1,2$, we expect that these vectors would be highest weight vectors. Put

$$\begin{aligned} W_0(n)= & {} \left\langle f^{(0)}_i:=\frac{1}{(n+2)_i}E^iw_0,\ 0\le i\le n+2 \right\rangle _R,\\{} & {} W_1(n)=\left\langle f^{(1)}_i:=\frac{1}{(n)_i}E^iw_1,\ 0\le i\le n \right\rangle _R, \end{aligned}$$

and

$$\begin{aligned} W_2(n)=\left\langle f^{(2)}_i:=\frac{1}{(n-2)_i}E^iw_2,\ 0\le i\le n-2 \right\rangle _R \end{aligned}$$

where $(x)_i=x!/(x-i)!$ and we set $(*)_0:=1$.

Recall $V(n,m)=V(n)\otimes _R \det ^m$ and denote by $\{u_i\}_i$ its basis by abuse of notation. By direct computation, we have the following:

Proposition 7.1

For $j=0,1,2$, as $GL_2(R)$-modules,

$$\begin{aligned} V(n+2-2j,j){\mathop {\longrightarrow }\limits ^{\sim }} W_j(n),\ u_i\mapsto f^{(j)}_i. \end{aligned}$$

When $n=p-1$ or $p-2$, the denominators of the coefficients appearing in $f^{(2)}_i:=\frac{1}{(n-2)_i}E^iw_2$ are not divisible by p. Hence only $W_2(n)$ still makes sense and so does the isomorphism $V(n-2,2){\mathop {\longrightarrow }\limits ^{\sim }} W_2(n)$. Clearly $W_2(n)$ gives a splitting of the surjection $V(n)\otimes _R V(2)\longrightarrow V(n-2,2)$.

To end this section we give an explicit realization of an isomorphism $V(n)\otimes _R V(2)\simeq W_2(n)\oplus W_1(n)\oplus W_0(n)$ in terms of our basis. Note that $\theta ^{\underline{k}}_1$ is related to $W_2(n)$. We identify $v=\sum _{\begin{array}{c} 0\le i\le n \\ 0\le j\le 2 \end{array}}a_{ij}u_i\otimes v_j\in V(n)\otimes _R V(2)$ with the low vector

$$\begin{aligned} (a_{n2},a_{n1},a_{n0},a_{(n-1)2},a_{(n-1)1},a_{(n-1)0},\ldots ,a_{12},a_{11},a_{10},a_{02},a_{01},a_{00}). \end{aligned}$$

Let us first assume that $n<p-2$. If we write

$$\begin{aligned} v=\sum _{i=0}^{n+2}b^{(0)}_if^{(0)}_i+\sum _{i=0}^{n}b^{(1)}_if^{(1)}_i+\sum _{i=0}^{n-2}b^{(2)}_if^{(2)}_i, \end{aligned}$$

then we have

$$\begin{aligned} {}^t(b^{(0)}_i)_{0\le i\le n+2}= \left( \begin{array}{c} a_{n2} \\ a_{(n-1)2}+a_{n1} \\ a_{(n-2)2}+a_{(n-1)1}+a_{n0} \\ a_{(n-3)2}+a_{(n-2)1}+a_{(n-1)0} \\ \vdots \\ a_{12}+a_{21}+a_{30} \\ a_{02}+a_{11}+a_{20} \\ a_{01}+a_{10} \\ a_{00} \end{array} \right) \end{aligned}$$

(7.1)

where the superscript “t" stands for the transpose.

As for $b^{(1)}_i$ we have

$$\begin{aligned} b^{(1)}_i=\left\{ \begin{array}{lc} -\displaystyle \frac{2}{n+2}a_{(n-1)2}+\frac{n}{n+2}a_{n1}&{}\quad (i=0) \\ &{} \\ -\displaystyle \frac{2+2i}{n+2}a_{(n-1-i)2}+\frac{n-2i}{n+2}a_{(n-i)1}+\frac{2n+2-2i}{n+2}a_{(n+1-i)0},\ &{}\quad (i=1,\ldots ,n-1) \\ &{} \\ -\displaystyle \frac{n}{n+2}a_{01}+\frac{2}{n+2}a_{10} &{}\quad (i=n) \end{array} \right. \nonumber \\ \end{aligned}$$

(7.2)

For the remaining coefficients $b^{(2)}_i$, we have

$$\begin{aligned} b^{(2)}_i= & {} \frac{(i+1)(i+2)}{n(n+1)}a_{(n-2-i)2}- \frac{(i+1)(n-i-1)}{n(n+1)}a_{(n-1-i)1}\nonumber \\{} & {} + \frac{(n-i-1)(n-i)}{n(n+1)}a_{(n-i)0} \end{aligned}$$

(7.3)

for $i=0,\ldots , n-2$. When $n=p-1$ or $p-2$ we have a splitting projection $V(n)\otimes _R V(2)\longrightarrow W_2(n)$ which is given by replacing $b^{(2)}_i$ in (7.3) with

$$\begin{aligned} b^{(2)}_i=(i+1)(i+2)a_{(n-2-i)2}- (i+1)(n-i-1)a_{(n-1-i)1}+ (n-i-1)(n-i)a_{(n-i)0}\nonumber \\ \end{aligned}$$

(7.4)

obtained by multiplying $n(n+1)$. Notice that $n(n+1)$ is zero if $n=p-1$ but we can justify by working over $\mathbb {Z}[1/(n+2)!]$ at first and then by multiplying $n(n+1)$. This yields (7.4) from (7.3).