  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.
