A sheaf-theoretic reformulation of the Tate conjecture

Abstract

Let p be a prime number. We give a conjecture of a sheaf-theoretic nature which is equivalent to the strong form of the Tate conjecture for smooth, projective varieties X over F_p: for all n>0, the order of pole of the Hasse-Weil zeta function of X at s=n equals the rank of the group of algebraic cycles of codimension n modulo numerical equivalence. Our main result is that this conjecture implies other well-known conjectures in characteristic p, among which: - The (weak) Tate conjecture for smooth, projective varieties X over any finitely generated field of characteristic p: given a prime l different from p, the geometric cycle map from algebraic cycles over X to the Galois invariants of the l-adic cohomology of the geometric fibre of X, tensored by Q_l, is surjective. - For X as above, the algebraicity of the Kunneth components of the diagonal and the hard Lefschetz theorem for cycles modulo numerical equivalence. - For X as above, the existence of a filtration conjectured by Beilinson on the Chow groups of X. - The rational Bass conjecture: for any smooth variety X over F_p, the algebraic K-groups of X have finite rank. - The Bass-Tate conjecture: for F a field of characteristic p, of absolute transcendence degree d, the i-th Milnor K-group of F is torsion for i>d. - Soule's conjecture: given a quasi-projective variety over F_p, the order of the zero of its Hasse-Weil zeta function at an integer n is given by the alternating sum of the ranks of the weight n part of its algebraic K'-groups