On distributions of $L^{\prime }$-values and orders of Sha groups in families of quadratic twists

Abstract

In this article, we aim to establish a prototype result regarding lower bounds of (joint) distributions of central $L^{\prime }$-values through extending a method of Radziwill and Soundararajan of proving conditional bounds for distributions of central $L$-values (via the one-level density of low-lying zeros of involving $L$-functions). To illustrate this, we give several conditional bounds towards joint distributions of central $L^{\prime }$-values and orders of Tate-Shafarevich groups in rank-one families of quadratic twists. As an application, we derive a simultaneous non-vanishing result for central $L^{\prime }$-values in families of quadratic twists of triples of holomorphic modular forms.

27 pages, a major revision, comments welcome