A sheaf-theoretic refinement of the Tate conjecture, by Bruno Kahn

Let p be a prime number. In this paper, 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 the field with p elements: for all n>0, the order 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, for all n>0, the geometric cycle map from cycles of codimension n 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.

- The existence of Beilinson's conjectural filtration on Chow groups: for X as above and n>0, there is a separated filtration of length n+1 on the n-th Chow group of X, stable under the action of correspondences and such that, on the associated graded, this action factors through numerical equivalence.

- The rational Bass conjecture: for any smooth variety X over F_p, the algebraic K-groups of X are, after tensoring with Q, finite dimensional vector spaces.

- 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.

- Soulé's conjecture: given a quasi-projective variety X 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.

Bruno Kahn <kahn@math.jussieu.fr>