Let X be a smooth projective variety over a local field of characteristic $(0,p)$. The Brauer group of X is a torsion group whose $l$-part is of cofinite type of some corank.
Is it know that the $l$-corank is independent of $l \not= p$, and that the $p$-corank is always larger than the $l$-corank?
If X is a curve, then both questions are affirmative by Lichtenbaum duality to the Picard group. By a result of Colliot-Thelene and Saito and the proper base change theorem, the first question is affirmative if one assumes the Tate conjecture over finite fields.
Are there any other known results in this direction?
More generally, I am interested in etale motivic cohomology groups, but the Brauer group is the first interesting example.
