Bianchi period polynomials: Hecke action and congruences

Abstract

Let $\Gamma$ be a Bianchi group associated to one of the five Euclidean imaginary quadratic fields. We show that the space of weight $k$ period polynomials for $\Gamma$ is ``dual'' to the space of weight $k$ modular symbols for $\Gamma$, reflecting the duality between the first and second cohomology groups of Bianchi groups. Using this result, we describe the action of Hecke operators on the space of period polynomials for $\Gamma$ via the Heilbronn matrices. In the second part of the paper, we numerically investigate congruences between level 1 Bianchi eigenforms via computer programs which implement the Hecke action on spaces of Bianchi period polynomials. Computations with the Hecke action are used to indicate moduli of congruences between the underlying Bianchi forms; we then prove the congruences using the period polynomials. From this we find congruences between genuine Bianchi modular forms and both a base-change Bianchi form and an Eisenstein series. We believe these congruences are the first of their kind in the literature.