From sum of two squares to arithmetic Siegel-Weil formulas

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].