Remark on Weil's conjectures

TL;DR

This paper introduces a new cohomology theory for certain projective varieties over finite fields, leading to proofs of key properties of their zeta functions, including rationality and functional equations.

Contribution

It develops a novel cohomology framework derived from C*-algebra traces, enabling proofs of Weil's conjectures for specific varieties.

Findings

Proves the rationality of the zeta function

Establishes the functional equation of the zeta function

Determines the Betti numbers for the varieties

Abstract

We introduce a cohomology theory for a class of projective varieties over a finite field coming from the canonical trace on a C*-algebra attached to the variety. Using the cohomology, we prove the rationality, functional equation and the Betti numbers conjectures for the zeta function of the variety.