A remark on the Tate conjecture

TL;DR

This paper explores the relationship between two parts of the Tate conjecture, demonstrating that in characteristic 0, the algebraicity of Tate classes implies the semisimplicity of Galois representations, with similar results in characteristic p under stronger conditions.

Contribution

It proves that in characteristic 0, the Tate conjecture's assertion on algebraic Tate classes implies the semisimplicity of Galois representations, clarifying their interdependence.

Findings

In characteristic 0, (T) implies (S).

In characteristic p, similar implications hold under stronger assumptions.

Provides theoretical insight into the Tate conjecture's components.

Abstract

The Tate conjecture has two parts: an assertion (S) about semisimplicity of Galois representations, and an assertion (T) which says that every Tate class is algebraic. We show that in characteristic 0, (T) implies (S). In characteristic p an analogous result is true under stronger assumptions.