A reformulation of the Siegel series and intersection numbers

Abstract

In this paper, we will explain a conceptual reformulation and inductive formula of the Siegel series. Using this, we will explain that both sides of the local intersection multiplicities of Gross and Keating (Invent Math 112(225–245):2051, 1993) and the Siegel series have the same inherent structures, beyond matching values. As an application, we will prove a new identity between the intersection number of two modular correspondences over $\mathbb {F}_p$ and the sum of the Fourier coefficients of the Siegel-Eisenstein series for $\mathrm {Sp}_4/\mathbb {Q}$ of weight 2, which is independent of $p \left( > 2\right)$. In addition, we will explain a description of the local intersection multiplicities of the special cycles over $\mathbb {F}_p$ on the supersingular locus of the ‘special fiber’ of the Shimura varieties for $\mathrm {GSpin}(n,2), n\le 3$ in terms of the Siegel series directly.

Appendix A: The table of intersection numbers

Table 1 Intersection numbers

For each positive integer m we denote by $\psi _{m/n^2}$ the polynomial in $\mathbb {Z}[x,y]$ which appears as a factor of $\varphi _m=\displaystyle \prod \nolimits _{n^2|m}\psi _{m/n^2}$ (cf. p.2 of [43]). In this appendix we give a table for

$$\begin{aligned} \mathrm{dim}_\mathbb {C}\mathbb {C}[x,y]/(\psi _{m_1},\psi _{m_2}),\ \mathrm{dim}_{\mathbb {F}_p}\mathbb {F}_p[x,y]/(\psi _{m_1},\psi _{m_2}) \end{aligned}$$

for $2\le m_1\le m_2 \le 9$ such that $m_1m_2$ is not a square and $2\le p<50$. Yokoyama (cf. [49]) kindly computed both quantities and checked they coincide directly. From this computation with Corollary 7.2 and Theorem 2.1 of [43] it is easy to see that

$$\begin{aligned} (T_{m_1,p},T_{m_2,p})= & {} \sum _{n^2_1|m_1,n^2_2|m_2}\mathrm{dim}_{\mathbb {F}_p}\mathbb {F}_p[x,y]/\left( \psi _{\frac{m_1}{n^2_1}},\psi _{\frac{m_2}{n^2_2}}\right) \\= & {} \sum _{n^2_1|m_1,n^2_2|m_2}\mathrm{dim}_{\mathbb {C}}\mathbb {C}[x,y]/\left( \psi _{\frac{m_1}{n^2_1}},\psi _{\frac{m_2}{n^2_2}}\right) =(T_{m_1,\mathbb {C}},T_{m_2,\mathbb {C}}) \end{aligned}$$

for $m_1,m_2,p$ above including the case where $p=2$. Put $d(m_1,m_2):=\mathrm{dim}_\mathbb {C}\mathbb {C}[x,y]/(\psi _{m_1},\psi _{m_2})=\mathrm{dim}_{\mathbb {F}_p}\mathbb {F}_p[x,y]/(\psi _{m_1},\psi _{m_2})$. We list up all of them as below (Table 1):

Let us remark that $d(1,m)=d(m,1)=\displaystyle \sum \nolimits _{d|m}\max \{d,\frac{m}{d}\}$. By using this we see, for example, that $(T_{2,p},T_{4,p})=(T_{2,\mathbb {C}},T_{4,\mathbb {C}})=d(2,1)+d(2,4)=4+28=32$ for any $p<50$.