On the ℓ-adic cohomology of varieties over number fields and its Galois cohomology

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Abstract

If X is a smooth, projective variety over a number field k, then the absolute Galois group Gk = Gal(/k) acts on the étale cohomology groups Hi(, ℚ/ℤ(n)), where = X Xk for an algebraic closure of k. In this paper I study some properties of these Gk-modules; in particular, I am interested in the corank of the Galois cohomology groups $${H^v}\,({G_k},{H^i}(\bar X,\,{Q_\ell }/{Z_\ell }(n))).$$