Extreme central $L$-values of almost prime quadratic twists of elliptic curves

Abstract

In this paper, we prove the extreme values of $L$-functions at the central point for almost prime quadratic twists of an elliptic curve. As an application, we get the extreme values for the Tate--Shafarevich groups in the quadratic twist family of an elliptic curve under the Birth--Swinnerton-Dyer conjecture.

11 pages.Comments welcome! Accepted in SCIENCE CHINA Mathematics

References in corpus (1)

• Determination of $\text{GL}(3)$ cusp forms by central values of quadratic twisted $L$-functions

Cited by in corpus (1)

• Extreme central values of quadratic Dirichlet $L$-functions with prime conductors