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

TL;DR

This paper investigates the extreme values of L-functions at the central point for almost prime quadratic twists of elliptic curves, providing insights into their behavior and implications for related arithmetic invariants.

Contribution

It establishes the extremal behavior of L-values for almost prime quadratic twists and relates these to Tate--Shafarevich groups under the BSD conjecture.

Findings

Proves the existence of extreme L-values for almost prime quadratic twists.

Connects the extremal L-values to Tate--Shafarevich groups.

Provides new bounds and asymptotic results for these L-values.

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.