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

2020/10/23 by Nordentoft, Asbjorn Christian, Constantinescu, Petru
#FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2010.12403

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.

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