Weil Conjectures - Grothendieck's Formula For The Zeta Function

Grothendieck's Formula For The Zeta Function

Grothendieck proved an analogue of the Lefschetz fixed point formula for l-adic cohomology theory, and by applying it to the Frobenius automorphism F was able to prove the following formula for the zeta function.

where each polynomial P_i is the determinant of I − TF on the l-adic cohomology group Hi.

The rationality of the zeta function follows immediately. The functional equation for the zeta function follows from Poincaré duality for l-adic cohomology, and the relation with complex Betti numbers of a lift follows from a comparison theorem between l-adic and ordinary cohomology for complex varieties.

More generally, Grothendieck proved a similar formula for the zeta function of a sheaf F₀:

as a product over cohomology groups:

The special case of the constant sheaf gives the usual zeta function.