On Witt vector cohomology for singular varieties

TL;DR

This paper develops a Witt vector cohomology theory for singular varieties over perfect fields of characteristic p, linking it to rigid cohomology and enabling new point-counting congruences over finite fields.

Contribution

It introduces Witt vector cohomology with compact supports for singular varieties and establishes a canonical link to rigid cohomology, extending classical proper case results.

Findings

Identified the slope < 1 part of rigid cohomology with Witt vector cohomology.

Derived congruences for rational point counts on varieties over finite fields.

Confirmed that point count congruences for theta divisors are independent of divisor choice.

Abstract

Over a perfect field $k$ of characteristic $p > 0$, we construct a ``Witt vector cohomology with compact supports'' for separated $k$-schemes of finite type, extending (after tensorisation with $\mathbb{Q}$) the classical theory for proper $k$-schemes. We define a canonical morphism from rigid cohomology with compact supports to Witt vector cohomology with compact supports, and we prove that it provides an identification between the latter and the slope $< 1$ part of the former. Over a finite field, this allows one to compute congruences for the number of rational points in special examples. In particular, the congruence modulo the cardinality of the finite field of the number of rational points of a theta divisor on an abelian variety does not depend on the choice of the theta divisor. This answers positively a question by J.-P. Serre.