On Ihara's lemma for degree one and two cohomology over imaginary quadratic fields

TL;DR

This paper extends Ihara's Lemma to degree 1 and 2 cuspidal cohomology over imaginary quadratic fields with torsion coefficients, using new methods and relating to level-raising congruences.

Contribution

It proves a generalized version of Ihara's Lemma for higher cohomological degrees over imaginary quadratic fields with torsion coefficients, employing different techniques from previous work.

Findings

Established Ihara's Lemma for degrees 1 and 2 cohomology

Connected the theorem to level-raising congruences

Extended previous results to arbitrary weights (k≥2)

Abstract

We prove a version of Ihara's Lemma for degree q=1,2 cuspidal cohomology of the symmetric space attached to automorphic forms of arbitrary weight (k\geq 2) over an imaginary quadratic field with torsion (prime power) coefficients. This extends an earlier result of the author which concerned the case k=2, q=1. Our method here is different and uses results of Diamond and Blasius-Franke-Grunewald. We discuss the relationship of our main theorem to the problem of the existence of level-raising congruences.