## Arithmetic Geometry and Number Theory RTG Seminar: Arithmetic Siegel-Weil formula for orthogonal Shimura varieties

Seminar | March 4 | 3-5 p.m. | 748 Evans Hall

Tonghai Yang, University of Wisconsin

Department of Mathematics

After reviewing Siegel-Weil formula and progress on arithmetic Siegel-Weil formula, I will talk about my new work with Jan Bruinier on this subject. Let $L$ be an integral lattice of signature $(n, 2)$ over $\mathbb Q$, and let $T$ be a non-singular symmetric integral matrix. Associated to it are two objects. One is the $T$-th Fourier coefficient $a(T)$ of the derivative of some `incoherent’ Siegel Eisenstein series. The other one is an arithmetic $0$ divisor $\widehat{\mathcal Z}(T)$, which can only be supported at one prime (including $\infty$). The arithmetic Siegel-Weil formula claims that $a(T)$ is equal to the degree of the arithmetic $0$-divisor $\widehat{\mathcal Z}(T)$. In this joint work, we proved that it is true when the support is at $\infty$ or at prime $p$ when $L$ is unimodular and $\mathcal Z(T)(\bar{\mathbb F}_P)$ is finite. I should mention that Garcia and Sankaran have a very different proof when the support is at the infinity.

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