Around $\ell$-independence

TL;DR

This paper investigates various forms of ℓ-independence for cohomology and fundamental groups of varieties over finite fields and local fields, establishing new results for unipotent fundamental groups and cohomology in different settings.

Contribution

It proves a strong form of ℓ-independence for unipotent fundamental groups over finite fields and extends weaker forms to local fields and semistable varieties, introducing spreading out techniques.

Findings

Strong ℓ-independence for unipotent fundamental groups over finite fields.

Weaker ℓ-independence results for local fields in the semistable case.

Existence of a Clemens–Schmid exact sequence for formal semistable families.

Abstract

In this article we study various forms of $\ell$-independence (including the case $\ell=p$) for the cohomology and fundamental groups of varieties over finite fields and equicharacteristic local fields. Our first result is a strong form of $\ell$-independence for the unipotent fundamental group of smooth and projective varieties over finite fields, by then proving a certain `spreading out' result we are able to deduce a much weaker form of $\ell$-independence for unipotent fundamental groups over equicharacteristic local fields, at least in the semistable case. In a similar vein, we can also use this to deduce $\ell$-independence results for the cohomology of semistable varieties from the well-known results on $\ell$-independence for smooth and proper varieties over finite fields. As another consequence of this `spreading out' result we are able to deduce the existence of a Clemens--Schmid exact sequence for formal semistable families. Finally, by deforming to characteristic $p$ we show a similar weak version of $\ell$-independence for the unipotent fundamental group of a semistable curve in mixed characteristic.