Elliptic curves with everywhere good reduction

JOURNAL OF NUMBER THEORY

卷:161

Article

Clemm, Amanda¹  Trebat-Leder, Sarah¹ 

[1] Emory Univ, Dept Math, Atlanta, GA 30322 USA

关键词: Elliptic curves;    Quadratic fields;    Diophantine equations;    Everywhere good reduction;   

DOI  :  10.1016/j.jnt.2015.07.001

来源: Elsevier

【 摘 要 】

We consider the question of which quadratic fields have elliptic curves with everywhere good reduction. By revisiting work of Setzer, we expand on congruence conditions that determine the real and imaginary quadratic fields with elliptic curves of everywhere good reduction and rational j-invariant. Using this, we determine the density of such real and imaginary quadratic fields. If R(X) denotes the number of real quadratic fields K = Q[root m] such that vertical bar Delta(K)vertical bar < X and for which there exists an elliptic curve E/K with rational j-invariant that has everywhere good reduction, then R(X) >> X/root log(X) We also obtain a similar result for imaginary quadratic fields. To obtain these estimates we explicitly construct quadratic fields over which we can construct elliptic curves with everywhere good reduction. The estimates then follow from elementary multiplicative number theory. In addition, we obtain infinite families of real and imaginary quadratic fields such that there are no elliptic curves with everywhere good reduction over these fields. (C) 2015 Elsevier Inc. All rights reserved.

【 授权许可】

Free