Asian Journal of Mathematics

Volume 21 (2017)

Number 4

Genus periods, genus points and congruent number problem

Pages: 721 – 774



Ye Tian (Academy of Mathematics and Systems Science, Morningside Center of Mathematics, Chinese Academy of Sciences, Beijing, China)

Xinyi Yuan (Department of Mathematics, University of California at Berkeley)

Shou-Wu Zhang (Department of Mathematics, Princeton University, Princeton, New Jersey, U.S.A.)


In this paper, based on an idea of Tian we establish a new sufficient condition for a positive integer $n$ to be a congruent number in terms of the Legendre symbols for the prime factors of $n$. Our criterion generalizes previous results of Heegner, Birch–Stephens, Monsky, and Tian, and conjecturally provides a list of positive density of congruent numbers. Our method of proving the criterion is to give formulae for the analytic Tate–Shafarevich number $\mathcal{L}(n)$ in terms of the so-called genus periods and genus points. These formulae are derived from the Waldspurger formula and the generalized Gross–Zagier formula of Yuan–Zhang–Zhang.


congruent number, Birch and Swinnerton–Dyer conjecture, Tate–Shafarevich group, Heegner point, Selmer group, Gross–Zagier formula, Waldspurger formula, L-function

2010 Mathematics Subject Classification

11D25, 11G05, 11G40

Received 28 November 2015

Published 25 August 2017