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

Asbjorn Christian Nordentoft, Petru Constantinescu

Published 2020-10-23Version 1

We provide a 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. Additionally, we prove the full conjecture in some particular cases using a connection to Eisenstein congruences. We also obtain residual equidistribution results for modular symbols where we order by the length of the corresponding geodesic. Finally, and most importantly, our methods generalise to equidistribution results for cohomology classes of finite volume quotients of $n$-dimensional hyperbolic space.