Weak Siegel-Weil formula for M_2(Q) and arithmetic on quaternions

TL;DR

This paper establishes a weak Siegel-Weil formula for M_2(Q), linking Hecke correspondences, representation numbers, and Eisenstein series, and applies it to reprove the four-square theorem.

Contribution

It proves the weak Siegel-Weil formula for M_2(Q) and connects arithmetic on quaternions with automorphic forms in a novel way.

Findings

Explicit formulas for Hecke correspondence degrees

Average representation numbers over genus

Reproof of the four-square theorem

Abstract

In this paper, we prove the weak Siegel-Weil formula for the space M_2(Q) . We study the Hecke correspondence and representation numbers associated to Eichler orders, and give the explicit formula for degree of Hecke correspondence and average representation numbers over genus. This formula could recover the main results in [DuYang]. We could identify these numbers with the Fourier coefficients of Eisenstein series via Siegel-Weil formula(weak Siegel-Weil formula for M_2(Q). In the last part of this article, we reprove four square sum Theorem via Siegel-Weil formula and Kudla's matching.