# MathSciDoc: An Archive for Mathematician ∫

#### Algebraic GeometryNumber Theorymathscidoc:1808.07018

Asian Journal of Mathematics, 21, (4), 721-774, 2017
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.
Genus periods, genus points, congruent number problem
@inproceedings{ye2017genus,
title={Genus periods, genus points and congruent number problem},
author={Ye Tian, Xinyi Yuan, and Shou-Wu Zhang},
url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180816232849742109144},
booktitle={Asian Journal of Mathematics},
volume={21},
number={4},
pages={721-774},
year={2017},
}

Ye Tian, Xinyi Yuan, and Shou-Wu Zhang. Genus periods, genus points and congruent number problem. 2017. Vol. 21. In Asian Journal of Mathematics. pp.721-774. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20180816232849742109144.