Residual equidistribution of modular symbols and cohomology classes for quotients of hyperbolic $n$-space

TL;DR

This paper introduces an automorphic approach using Eisenstein series to study the equidistribution of modular symbols and cohomology classes, proving average distribution results and revealing biases with connections to perturbation theory.

Contribution

It develops a new automorphic method for equidistribution, proving an average conjecture of Mazur and Rubin, and extends results to hyperbolic n-space cohomology classes.

Findings

Modular symbols are asymptotically jointly equidistributed mod p.

Residual equidistribution results are obtained for Dedekind sums.

The variance of the distribution shows a surprising bias linked to perturbation theory.

Abstract

We provide a new and simple automorphic method using Eisenstein series to study the equidistribution of modular symbols modulo primes, which we apply to prove an average version of a conjecture of Mazur and Rubin. More precisely, we prove that modular symbols corresponding to a Hecke basis of weight 2 cusp forms are asymptotically jointly equidistributed mod $p$ while we allow restrictions on the location of the cusps. As an application, we obtain a residual equidistribution result for Dedekind sums. Furthermore, we calculate the variance of the distribution and show a surprising bias with connections to perturbation theory. Additionally, we prove the full conjecture in some particular cases using a connection to Eisenstein congruences. Finally, our methods generalise to equidistribution results for cohomology classes of finite volume quotients of $n$-dimensional hyperbolic space.