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

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