Abstract
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 language | English (US) |
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Pages (from-to) | 721-774 |
Number of pages | 54 |
Journal | Asian Journal of Mathematics |
Volume | 21 |
Issue number | 4 |
DOIs | |
State | Published - 2017 |
All Science Journal Classification (ASJC) codes
- General Mathematics
- Applied Mathematics
Keywords
- Birch and swinnerton-dyer conjecture
- Congruent number
- Gross-Zagier formula
- Heegner point
- L-function
- Selmer group
- Tate-shafarevich group
- Waldspurger formula