On the observability of Galois representations and the Tate conjecture

TL;DR

This paper explores the relationship between the observability of Galois representations and the Tate conjecture, establishing new links and sharpening existing results through the concept of arithmetic monodromy groups.

Contribution

It demonstrates that the observability of arithmetic monodromy groups underpins the implication of Tate classes being algebraic and semisimple Galois representations, extending Moonen's characteristic zero results.

Findings

Unconditional link between observability and Tate conjecture parts

Sharpened results on Galois representations in any characteristic

Discussion on transcendence of p-adic periods

Abstract

The Tate conjecture has two parts: i) Tate classes are linear combination of algebraic classes, ii) semisimplicity of Galois representations (for smooth projective varieties). B. Moonen proved that i) implies ii) in characteristic 0, using $p$-adic Hodge theory. We show that an unconditional result lies behind this implication: the {\it observability} of arithmetic monodromy groups of geometric origin (in any characteristic) - which leads to a sharpening of Moonen's result. We also discuss another aspect of the Tate conjecture related to the transcendence of $p$-adic periods.