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On the coefficients of the Taylor expansion of L-functions of elliptic curves

组织者

刁晗生 , 杜衡 , 胡悦科 , Bin Xu , Yihang Zhu

演讲者

翟帅

时间

2025年04月24日 10:00 至 11:00

地点

Shuangqing-C654

摘要

Let $E$ be an elliptic curve over $\mathbb{Q}$. By the modularity theorem, the complex $L$-series associated to $E$ admits a holomorphic continuation to the entire complex plane. Consequently, it admits a Taylor expansion at the central point. The Birch and Swinnerton-Dyer conjecture predicts a deep connection between the leading coefficient of this expansion and the arithmetic invariants of $E$. Thanks to the breakthrough of Yun--Zhang, an extension of the Gross--Zagier and Waldspurger formulae for higher derivatives of such $L$-functions over function fields are established. Over number fields, there is still no known arithmetic explanation for the other coefficients. In this talk, I will investigate these coefficients within the family of quadratic twists, from an analytic perspective.