On the characteristic polynomial of the Frobenius on \'etale cohomology

TL;DR

This paper investigates the properties of the characteristic polynomial of the Frobenius acting on the middle-dimensional étale cohomology of smooth proper varieties over finite fields, revealing specific restrictions at a key value.

Contribution

It establishes a new restriction on the value at -1 of the characteristic polynomial of Frobenius on middle-dimensional étale cohomology for even-dimensional varieties over finite fields.

Findings

Restriction on the polynomial's value at -1

Applicable to smooth proper varieties of even dimension

Advances understanding of Frobenius action in étale cohomology

Abstract

Let $X$ be a smooth proper variety of even dimension $d$ over a finite field. We establish a restriction on the value at $(-1)$ of the characteristic polynomial of the Frobenius on the middle-dimensional \'etale cohomology of $X$ with coefficients in ${\mathbb Q}_l(d/2)$.