A rank inequality for the Tate Conjecture over global function fields

Abstract

We present an observation of Ramakrishnan concerning the Tate Conjecture for varieties over a global function field (i.e., the function field of a smooth projecture curve over a finite field), which was pointed out during a lecture given at the AIM's workshop on the Tate Conjecture in July 2007. The result is perhaps “known to the experts,” but we record it here, as it does not appear to be in print elsewhere. We use the global Langlands correspondence for the groups GL_n over global function fields, proved by Lafforgue [Chtoucas de Drinfeld et correspondance de Langlands, Invent. Math. 147 (2002) 1–241], along with an analytic result of Jacquet and Shalika [On Euler products and the classification of automorphic forms. I and II, Amer. J. Math. 103 (1981) 499–558, 777–815] on automorphic L-functions for GL_n. Specifically, we use these to show (see Theorem 2.1 below) that, for a prime ℓ ≠ char k, the dimension of the subspace spanned by the rational cycles of codimension m on our variety in its 2m-th ℓ-adic cohomology group (the so-called algebraic rank) is bounded above by the order of the pole at s=m+1 of the associated L-function (the so-called analytic rank). The interest in this result lies in the fact that, with the exception of some special instances like certain Shimura varieties and abelian varieties which are potentially CM type, the analogous result for varieties over number fields is still unknown in general, even for the case of divisors (m=1).