Around the conjectures of Mazur and Rubin on the distribution of modular symbols

Abstract: In 2016 Mazur and Rubin put forth a number of conjectures concerning the arithmetic distribution of modular symbols motivated by questions in Diophantine stability of elliptic curves. One of these conjectures predicts that modular symbols should follow a normal distribution and another one is concerned with the residual distribution of modular symbols. In this talk I will give an introduction to these conjectures and discuss different results related to them. In particular, I will present an automorphic methods for proving residual equidistribution of modular symbols and for computing the variance (with a surprising connection to perturbation theory). If time permits, I will explain how these results can be generalized to classes in the first cohomology of arithmetic subgroups of $\mathrm{SO}(n,1)$.

Part of the talk is joint work with Petru Constantinescu (UCL).

algebraic geometrynumber theory

Audience: researchers in the topic

Séminaire de géométrie arithmétique et motivique (Paris Nord)