Non-Eisenstein cohomology of locally symmetric spaces for $GL_2$ over a CM field

TL;DR

This paper proves that, under mild conditions, the modulo p cohomology of certain locally symmetric spaces for GL_2 over a CM field is concentrated in specific degrees, revealing structural insights into their cohomology groups.

Contribution

It establishes the concentration of modulo p cohomology in the Borel-Wallach range for these spaces, under mild assumptions, and deduces consequences for Hecke algebra module structures.

Findings

Cohomology is concentrated in degrees [q_0, q_0 + l_0] after localization.

Results hold under mild conditions and for level prime to p.

Implications for the structure of first and last cohomology groups.

Abstract

Let $F$ be a CM field, let $p$ be a prime number. The goal of this paper is to show, under mild conditions, that the modulo $p$ cohomology of the locally symmetric spaces $X$ for $GL_2(F)$ with level prime to $p$ is concentrated in degrees belonging to the Borel-Wallach range $[q_0,q_0+\ell_0]$ after localizing at a "strongly non-Eisenstein" maximal ideal of the Hecke algebra. From this result, we deduce expected consequences on the structure of the first and last cohomology groups as modules over the Hecke algebra.