Skip to Main Content

Advancing Research. Creating Connections.

The American Mathematical Society is dedicated to advancing research and connecting the diverse global mathematical community through publications, meetings and conferences, MathSciNet, professional services, advocacy, and awareness programs.

Learn more and join »

From Bulletin of the AMS

From sum of two squares to arithmetic Siegel–Weil formulas

by Chao Li

Two divisors Zm shown in red and Z1 shown in blue, on the surface X with four intersection points
Geometric intersection number

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 …

Read more »


More from Bulletin»


Feature Stories

More Feature Stories »



28-3 May/Jun

May 28-Jun 3, 2023

Mathematics Research Community (MRC) Week 1

Ricci Curvatures of Graphs and Applications to Data Science

4-10 Jun

Jun 4-10, 2023

Mathematics Research Community (MRC) Week 2

2a: Explicit Computations with Stacks; 2b: Derived Categories, Arithmetic and Geometry


Jun 18-24, 2023

Mathematics Research Community (MRC) Week 3

Complex Social Systems


Sep 9-10, 2023

2023 Fall Eastern Sectional

University of Buffalo (SUNY), Buffalo, NY


Oct 7-8, 2023

2023 Fall Central Sectional

Creighton University, Omaha, NE


Oct 13-15, 2023

2023 Fall Southeastern Sectional

University of South Alabama, Mobile, AL



31 May


AMS Prize & Award Nominations

Birkhoff, Chevalley, Cole Algebra, Conant, Difference, Exemplary, Grenander, Impact, Morgan, Public Service, Russell, Whiteman

30 Jun


AMS Prize & Award Nominations

Babuška, Stein

18 Jul


2023 Fall Eastern Sectional Abstract Submissions

8 Aug


2023 Fall Central Sectional Abstract Submissions

15 Aug


2023 Fall Southeastern Sectional Abstract Submissions

15 Sep


AMS Prize & Award Nominations

JPBM Communications Award

30 Sep


2025 Mathematics Research Communities (MRC) Organizers

Submit expressions of interest in organizing a 2025 MRC by Sep 30, 2023. Complete proposals due November 30.

Member Journals

Notices of the American Mathematical Society Bulletin of the American Mathematical Society