Topics in Automorphic forms and Spectral Theory

Research Areas: Number Theory, Mathematical Analysis

The 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.