From sum of two squares to arithmetic Siegel-Weil formulas

TL;DR

This paper surveys recent advances in the arithmetic Siegel-Weil formula, connecting classical problems like sums of two squares to modern arithmetic geometry and its applications to conjectures such as Beilinson-Bloch.

Contribution

It provides an accessible overview of recent progress on the arithmetic Siegel-Weil formula and its applications to Shimura varieties and related conjectures.

Findings

Recent proof of the arithmetic Siegel-Weil formula for arbitrary-dimensional Shimura varieties

Applications to the arithmetic inner product formula

Connections to the Beilinson-Bloch conjecture

Abstract

The main goal of this expository article is to survey recent progress on the arithmetic Siegel-Weil formula and its applications. We begin with the classical sum of two squares problem and put it in the context of the Siegel-Weil formula. We then motivate the geometric and arithmetic Siegel-Weil formula using the classical example of the product of modular curves. After explaining the recent result on the arithmetic Siegel-Weil formula for Shimura varieties of arbitrary dimension, we discuss some aspects of the proof and its application to the arithmetic inner product formula and the Beilinson-Bloch conjecture. Rather than intended to be a complete survey of this vast field, this article focuses more on examples and background to provide easier access to several recent works by the author with W. Zhang [LZ22a, LZ22b] and Y. Liu [LL21, LL22].