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arXiv · 2403.18382

On distributions of $L'$-values and orders of Sha groups in families of quadratic twists

Peng-Jie Wong
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Abstract

In this article, we aim to establish a prototype result regarding lower bounds of (joint) distributions of central $L'$-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 $L'$-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 $L'$-values in families of quadratic twists of triples of holomorphic modular forms.

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Peng-Jie Wong. 2024-03-27. On distributions of $L'$-values and orders of Sha groups in families of quadratic twists. https://arxiv.org/abs/2403.18382

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