Density of Arithmetic Representations of Function Fields

TL;DR

This paper introduces a conjecture about the density of arithmetic points in deformation spaces of representations of the étale fundamental group in positive characteristic, with implications for étale cohomology and a proof in a specific case.

Contribution

It proposes a new density conjecture in the context of function fields and proves it for tame degree two on the projective line minus three points.

Findings

Conjecture on density of arithmetic points proposed

Proved in tame degree two case for P^1 minus three points

Implications for étale cohomology and Hard Lefschetz conjecture

Abstract

We propose a conjecture on the density of arithmetic points in the deformation space of representations of the \'etale fundamental group in positive characteristic. This? conjecture has applications to \'etale cohomology theory, for example it implies a Hard Lefschetz conjecture. We prove the density conjecture in tame degree two for the curve $\mathbb{P}^1\setminus \{0,1,\infty\}$. v2: very small typos corrected.v3: final. Publication in Epiga.