Topics in Automorphic forms and Spectral Theory

自守形式和谱论主题

• 批准号: 2417008
• 金额: --
• 依托单位: University College London
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2020
• 资助国家: 英国
• 起止时间: 2020 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2417008
• 关键词: Topics; Automorphic; forms; Spectral; Theory

Research Areas: Number Theory, Mathematical AnalysisThe student will work on the spectral theory of automorphic forms associated to locally symmetric spaces. In two dimensions these are realised as H/G, where G is a discrete cofinite subgroup of SL(2, R). To understand the structure of G, one studies the lattice-point problem, the prime geodesic theorem and Weyl's law. In the last three years a number of mathematicians have worked on the distribution of modular symbols attached to classical modular forms using various techniques: spectral theory, dynamical systems (Lee-Sun and Bettin-Drappeau), second moments of L-functions (Blomer et al), approximate functional equations (Diamantis et al), and extending the results to higher weight modular forms and period polynomials (Nordentoft).The motivation is to study the excess rank of elliptic curves as we vary the field over cyclotomic extensions of Q. The modular symbols are related to the twisted central value of the L-function a holomorphic cusp form of weight 2 for a Hecke group. The modular symbols themselves are central values of L-functions twisted by additive characters. Recently Mazur and Rubin explained the motivation and stated various conjectures (many still open) about the distribution of modular symbols, theta constants, and their arithmetic implications. Using spectral theory, Petridis and Risager in Arithmetic Statistics of modular symbols, Invent. math. 212 (2018) 997-1053 investigated two conjectures of Mazur and Rubin and one conjecture of Mazur, Rubin, and Stein. The student will concentrate on the case of totally real fields. The background is based on previous work of Bruggeman-Miatello, Gon, Gorodnik-Nevo, Nelson, Petridis-Risager, and Risager-Truelsen.

研究领域：数字理论，数学分析学生将在与本地对称空间相关的自动形式的光谱理论上工作。在二维中，这些被实现为H/G，其中G是SL（2，R）的离散辅助亚组。为了理解G的结构，一个人研究了晶格问题，主要的测量定理和Weyl定律。在过去的三年中为了研究椭圆形曲线的过度等级，因为我们在Q的环体延伸范围内改变了场。模块化符号与Hecke组的holomorthic cusp重量2的L函数的扭曲中心值有关。模块化符号本身是添加字符扭曲的L功能的中心值。最近，马祖尔（Mazur）和鲁宾（Rubin）解释了动机，并陈述了各种猜想（许多仍然开放），介绍了模块化符号，theta常数及其算术含义的分布。在模块化符号的算术统计中，使用光谱理论，petridis和risager，发明。数学。 212（2018）997-1053研究了马祖尔和鲁宾的两个猜想，以及一个猜想的马祖尔，鲁宾和斯坦因。学生将专注于完全真实领域的情况。背景基于Bruggeman-Miatello，Gon，Gorodnik-Nevo，Nelson，Petridis-Risager和Risager-Truelsen的先前工作。