Autour de la conjecture de Tate `a coefficients Z_l pour les vari'et'es sur les corps finis

TL;DR

This paper explores the Tate conjecture for varieties over finite fields with Z_l-coefficients, highlighting a 1998 result by Schoen that links integral versions of the conjecture for 1-cycles to the conjecture for divisors on surfaces, and derives new results on zero-cycles over function fields.

Contribution

It provides an exposition of Schoen's 1998 theorem and extends its implications to the existence of degree one zero-cycles on certain varieties over function fields.

Findings

Integral Tate conjecture holds for 1-cycles if it holds for divisors on surfaces.

Zero-cycle of degree one exists on certain complete intersections if Tate conjecture for divisors on surfaces is true.

New results on zero-cycles over function fields of varieties extending to proper fibrations.

Abstract

This partly expository paper investigates versions of the Tate conjecture on the cycle map for varieties defined over finite fields with values in 'etale cohomology with Z_\ell-coefficients. The bulk of the paper is an exposition of a 1998 result of C. Schoen which shows that an integral version of the conjecture holds for 1-cycles provided the usual conjecture is true for divisors on surfaces. In a last section we then derive from Schoen's theorem new results on the existence of degree one zero-cycles on varieties defined over the function field of a smooth proper curve C over the algebraic closure of a finite field. In particular, we show that if the variety in question is a smooth projective complete intersection of dimension at least 3 and of degree prime to the characteristic, then a zero-cycle of degree one exists if the Tate conjecture is true for divisors on surfaces and the variety extends to a proper fibration over C all of whose fibres possess a component of multiplicity one.