Uniform boundedness for Brauer groups of forms in positive characteristic

Abstract: Let

k

be a finitely generated field of characteristic

p > 0

and

X

a smooth and proper scheme over

k

. Recent works of Cadoret, Hui and Tamagawa show that, if

X

satisfies the

e l l

-adic Tate conjecture for divisors for every prime

e l l e q p

, the Galois invariant subgroup

B r (X_{o v e r l i n e k}) [p^{'}]^{p i_{1} (k)}

of the prime-to-

p

torsion of the geometric Brauer group of

X

is finite. The main result of this note is that, for every integer

d g e q 1

, there exists a constant

C : = C (X, d)

such that for every finite field extension

k s u b s e t e q k^{'}

with

[k^{'} : k] l e q d

and every

(o v e r l i n e k / k^{'})

-form

Y

of

X

one has

| (B r (Y i m e s_{k^{'}} o v e r l i n e k) [p^{'}]^{p i_{1} (k^{'})} | l e q C

. The theorem is a consequence of general results on forms of compatible systems of

p i_{1} (k)

-representations and it extends to positive characteristic a recent result of Orr and Skorobogatov in characteristic zero.