Some Questions in $l-$adic Cohomology

TL;DR

This paper investigates the independence of Betti numbers from the prime number $l$ for smooth projective varieties over algebraic extensions of finite fields, extending known results over complex numbers.

Contribution

It explores the $l$-independence of Betti numbers in a new setting, generalizing the comparison theorem to varieties over fields of positive characteristic.

Findings

Established conditions for $l$-independence of Betti numbers over algebraic extensions of $\,\mathbb{F}_p$

Extended the comparison theorem to a broader class of varieties

Provided new insights into the relationship between $l$-adic cohomology and algebraic geometry over finite fields

Abstract

The comparison theorem for a smooth projective variety $X$ over $\mathbb{C}$ tells us that the Betti numbers are independent of $l$. We aim to understand the $l$ independence of Betti numbers for smooth projective varieties $X$ over $k$, where $k$ is an algebraic extension of $\mathbb{F}_p$.