ON IHARA'S LEMMA FOR DEGREE ONE AND TWO COHOMOLOGY OVER IMAGINARY QUADRATIC FIELDS

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 ≥ 2) over an imaginary quadratic field with torsion (prime power) coefficients. This extends an earlier result of the author [Ihara's lemma for imaginary quadratic fields, J. Number Theory128(8) (2008) 2251–2262] which concerned the case k = 2, q = 1. Our method is different from [Ihara's lemma for imaginary quadratic fields, J. Number Theory128(8) (2008) 2251–2262] and uses results of Diamond [Congruence primes for cusp forms of weight k ≥ 2, Astérisque196–197 (1991) 205–213] and Blasius–Franke–Grunewald [Cohomology of S-arithmetic subgroups in the number field case, Invent. Math.116(1–3) (1994) 75–93]. We discuss the relationship of our main theorem to the problem of the existence of level-raising congruences.