Genus periods, genus points and congruent number problem

Ye Tiant, Xinyi Yuan, Shou Wu Zhang

Research output: Contribution to journalArticlepeer-review

14 Scopus citations


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 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.

Original languageEnglish (US)
Pages (from-to)721-774
Number of pages54
JournalAsian Journal of Mathematics
Issue number4
StatePublished - 2017

All Science Journal Classification (ASJC) codes

  • General Mathematics
  • Applied Mathematics


  • Birch and swinnerton-dyer conjecture
  • Congruent number
  • Gross-Zagier formula
  • Heegner point
  • L-function
  • Selmer group
  • Tate-shafarevich group
  • Waldspurger formula


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