Rank-one quadratic twists of an infinite family of elliptic curves

被引：11

作者：

Byeon, Dongho ^{[1
]}

Jeon, Daeyeol ^{[2
]}

Kim, Chang Heon ^{[3
]}

机构：

[1] Seoul Natl Univ, Dept Math, Seoul, South Korea

[2] Kongju Natl Univ, Dept Math Educ, Kong Ju 314701, South Korea

[3] Hanyang Univ, Dept Math, Seoul 133791, South Korea

来源：

JOURNAL FUR DIE REINE UND ANGEWANDTE MATHEMATIK | 2009年 / 633卷

关键词：

MODULAR L-FUNCTIONS; L-SERIES; POINTS;

D O I：

10.1515/CRELLE.2009.060

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

A conjecture of Goldfeld implies that a positive proportion of quadratic twists of an elliptic curve E/Q has (analytic) rank 1. This assertion has been confirmed by Vatsal [V1] and the first author [By] for only two elliptic curves. Here we confirm this assertion for infinitely many elliptic curves E/Q using the Heegner divisors, the 3-part of the class groups of quadratic fields, and a variant of the binary Goldbach problem for polynomials.

引用

页码：67 / 76

页数：10

共 22 条

[1]

Birch B, 1984, E HORWOOD SER MATH A, P13

[2] On the modularity of elliptic curves over Q: Wild 3-adic exercises [J].