Asian Journal of Mathematics

Volume 20 (2016)

Number 3

Non-vanishing theorems for quadratic twists of elliptic curves

Pages: 475 – 502

DOI: http://dx.doi.org/10.4310/AJM.2016.v20.n3.a4

Author

Shuai Zhai (School of Mathematics, Shandong University, Jinan, Shandong, China; and Department of Pure Mathematics and Mathematical Statistics, University of Cambridge, United Kingdom)

Abstract

In this paper, we use rather classical results on modular symbols to prove that, for certain families of elliptic curves defined over $\mathbb{Q}$, there always exists a large class of explicit quadratic twists whose complex $L$-series does not vanish at $s = 1$. We also prove the $2$-part of the conjecture of Birch and Swinnerton–Dyer for many of these quadratic twists.

Keywords

Birch–Swinnerton–Dyer conjecture, elliptic curves, non-vanishing

2010 Mathematics Subject Classification

11G05, 11G40

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