Degree 3 Cohomology of Function Fields of Surfaces

Abstract

Let |$F$| be the function field of a surface |$X$| over a finite field. Let |$l$| be a prime not equal to the characteristic of |$F$|⁠. Our main result is a local–global principle (with respect to discrete valuations of |$F$|⁠) for a class in |$H^3(F, \mu _l^{\otimes 3})$| to be written as the cup product of a given symbol in |$H^2(F, \mu _l^{\otimes 2})$| by some class in |$H^1(F, \mu _l) \simeq F^*/F^{*l}$|⁠. From this, we deduce if |$F$| contains a primitive |$l{\rm th}$| root of unity, then any element in |$H^3(F, \mu _l^{\otimes 3})$| is represented by a symbol, a result which was previously known only for |$l = 2$|⁠. The local–global principle also enables us to show that if a three-fold |$X$| over a finite field |$\mathbb {F}$| (with char|$(\mathbb {F}) \neq 2$|⁠) admits a conic bundle fibration over a surface, then the unramified cohomology group |$H^3_{nr}(\mathbb {F}(X)/\mathbb {F}, \mathbb {Z}/2)$| vanishes. Combined with earlier works, this gives a proof of the integral Tate conjecture for one-cycles on certain classes of three-fold over |$\mathbb {F}$|⁠. It also leads to a proof that the Brauer–Manin obstruction to existence of a zero-cycle of degree 1 is the only obstruction for a certain classes of surfaces over global fields of positive characteristic.