GENUS PERIODS, GENUS POINTS AND CONGRUENT NUMBER PROBLEM

被引：11

作者：

Tian, Ye ^{[1
]}

Yuan, Xinyi ^{[2
]}

Zhang, Shou-Wu ^{[3
]}

机构：

[1] Chinese Acad Sci, Acad Math & Syst Sci, Morningside Ctr Math, Beijing 100190, Peoples R China

[2] Univ Calif Berkeley, Dept Math, Berkeley, CA 94720 USA

[3] Princeton Univ, Dept Math, Princeton, NJ 08544 USA

来源：

ASIAN JOURNAL OF MATHEMATICS | 2017年 / 21卷 / 04期

基金：

美国国家科学基金会;

关键词：

Congruent number; Birch and Swinnerton-Dyer conjecture; Tate-Shafarevich group; Heegner point; Selmer group; Gross-Zagier formula; Waldspurger formula; L-function; ELLIPTIC-CURVES; HEEGNER POINTS; SWINNERTON-DYER; DERIVATIVES; CONJECTURE; CHARACTERS; SYMMETRY; FORMULA; GL(2); BIRCH;

D O I：

10.4310/AJM.2017.v21.n4.a5

中图分类号：

O29 [应用数学];

学科分类号：

070104 ;

摘要：

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.

引用

页码：721 / 774

页数：54

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