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In loving memory of my mother, Xiaoping Mao (1965-2022)
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 being intended as 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 and Y. Liu.
1.1. Sum of Two Squares
Which prime p can be written as the sum of two squares? For the first few primes we find that …