Twist-minimal trace formulas for Hecke operators : Modular forms

Hecke 算子的扭曲最小迹公式：模形式

基本信息

批准号：
2117877
负责人：
金额：
--
依托单位：
University of Bristol
依托单位国家：
英国
项目类别：
Studentship
财政年份：
2018
资助国家：
英国
起止时间：
2018 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=studentship-2117877
关键词：
Twist minimal trace formulas Hecke

项目摘要

Holomorphic modular forms have been studied actively in both algebraic number theory and analytic number theory, related with the class field theory, Galois representations and elliptic curves. They are providing both new methods and a source of problems in analytic number theory. In particular modular forms are important examples of automorphic L-functions. The theory of modular forms have been well-established, but still, there are unsolved conjectures.The main object of this project is to provide a method to compute the space of holomorphic modular forms for arbitrary weight, level and Nebentypus characters. There are several ways, including the method of modular symbols, but our tool is based on the Eichler-Selberg trace formula. In the recently published book by H. Cohen and F. Strömberg, the newform trace formulas (based on the newform theory of modular forms) and their applications (e.g. Pari/GP package, which can evaluate L-functions for modular forms and other numerical values associated with modular forms) are explained. When the level is non-square free, the corresponding space of modular forms becomes more complicated. The newform spaces can be further sieved down to the twist-minimal forms, i.e., those newforms whose conductor cannot be reduced by twisting with Dirichlet characters. Our aim is to derive twist-minimal Eichler-Selberg trace formulas and to find applications including computing the space, evaluating associated L-functions, etc.

全纯模形式在代数数论和解析数论中得到了积极的研究，与类域论、伽罗瓦表示和椭圆曲线相关，它们为解析数论提供了新的方法和问题的根源。是自同构 L 函数的重要例子。模形式的理论已经很成熟，但仍然存在未解决的猜想。该项目的主要目标是提供一种计算空间的方法。任意重量、级别和 Nebentypus 字符的全纯模形式有多种方法，包括模符号方法，但我们的工具基于 H. Cohen 和 F. 最近出版的书中的 Eichler-Selberg 迹公式。 Strömberg，newform 迹公式（基于模形式的 newform 理论）及其应用（例如 Pari/GP 包，可以评估模形式的 L 函数以及与模形式相关的其他数值）解释说，当水平非自由平方时，模形式的相应空间变得更加复杂，新形式空间可以进一步筛选为最小扭曲形式，即那些不能通过狄利克雷特征扭曲来减少的新形式。我们的目标是推导最小扭曲 Eichler-Selberg 迹公式，并找到包括计算空间、评估相关 L 函数等在内的应用。