On distributions of \(L'\)-values and orders of Sha groups in families of quadratic twists

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-ly...

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Bibliographic Details
Published in:arXiv.org 2024-11
Main Author: Peng-Jie, Wong
Format: Article
Language:English
Subjects:
Curves
Lower bounds
Online Access:Get full text
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Summary: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.
ISSN:2331-8422