Elliptic curves of rank 1 satisfying the 3-part of the Birch and Swinnerton-Dyer conjecture

JOURNAL OF NUMBER THEORY

卷:130

Article

Byeon, Dongho

关键词: Elliptic curves;    Quadratic fields;    Birch and Swinnerton-Dyer conjecture;   

DOI  :  10.1016/j.jnt.2010.07.001

来源: Elsevier

【 摘 要 】

Let E be an elliptic curve over Q of conductor N and K be an imaginary quadratic field, where all prime divisors of N split. If the analytic rank of E over K is equal to 1, then the Gross and Zagier formula for the value of the derivative of the L-function of E over K. when combined with the Birch and Swinnerton-Dyer conjecture, gives a conjectural formula for the order of the Shafarevich-Tate group of E over K. In this paper, we show that there are infinitely many elliptic curves E such that for a positive proportion of imaginary quadratic fields K, the 3-part of the conjectural formula is true. (C) 2010 Elsevier Inc. All rights reserved.

【 授权许可】

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