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 Pi 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.
Read more about this topic: Weil Conjectures
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