Topics in Automorphic forms and Spectral Theory

研究领域：数论，数学分析学生将研究与局部对称空间相关的自守形式的谱理论。在二维空间中，它们被实现为H/G，其中G是SL（2，R）的离散余有限子群。为了理解G的结构，人们研究了格点问题、素测地线定理和外尔定律。在过去的三年里，许多数学家使用各种技术研究了附加在经典模形式上的模符号的分布：谱理论，动力系统（Lee-Sun和Bettin-Drappeau），L-函数的二阶矩（Blomer等人），近似函数方程（Diamantis等人），并将结果推广到更高权的模形式和周期多项式（Nordentoft）。动机是研究当我们在Q的分圆扩张上改变场时椭圆曲线的超额秩。模符号与Hecke群的权为2的全纯尖点形式的L-函数的扭曲中心值有关。模符号本身是加性特征标扭曲的L-函数的中心值。最近Mazur和Rubin解释了动机，并陈述了关于模符号分布、theta常数及其算术含义的各种解释（许多仍然开放）。使用谱理论，Petridis和Risager在算术统计的模块化符号，发明。212（2018）997-1053研究了Mazur和Rubin的两个猜想以及Mazur、Rubin和Stein的一个猜想。学生们将专注于完全真实的场的情况。背景是基于Bruggeman-Miatello、Gon、Gorodnik-Nevo、纳尔逊、Petridis-Risager和Risager-Truelsen以前的工作。

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.