### Q_\ell versus F_\ell coefficients in the Grothendieck-Serre/Tate conjectures

Abstract
This is a joint work with Anna Cadoret and Akio Tamagawa. We investigate the relation between the Grothendieck-Serre/Tate conjectures with Q_\ell and F_\ell-coefficients for all sufficiently large \ell. In particular, when X is a smooth projective variety defined over a finitely generated field K of characteristic p>0, we prove that the Tate conjecture with Q_\ell-coefficients for divisors of X for all \ell not equal to p is equivalent to the finiteness of the Galois-fixed part of the prime-to-p torsion subgroup of the geometric Brauer group Br(X_{\overline K}). The equivalence when K is finite is a result of Tate.