The abelian part of a compatible system and l-independence of the Tate conjecture

TL;DR

This paper demonstrates that the abelian part of a compatible Galois system remains compatible under certain conditions and establishes l-independence of the Tate conjecture for abelian varieties.

Contribution

It proves the compatibility of the abelian part of Galois systems and shows Tate conjecture independence from l under specific group conditions.

Findings

The abelian subrepresentations form a compatible system.

Tate conjecture for abelian varieties is l-independent under certain conditions.

Conditions include connectedness and Hypothesis A for monodromy groups.

Abstract

Let K be a number field and {V_l} be a rational strictly compatible system of semisimple Galois representations of K arising from geometry. Let G_l and V_l^ab be respectively the algebraic monodromy group and the maximal abelian subrepresentation of V_l for all l. We prove that the system {V_l^ab} is also a rational strictly compatible system under some group theoretic conditions, e.g., when G_l' is connected and satisfies Hypothesis A for some prime l'. As an application, we prove that the Tate conjecture for abelian variety X/K is independent of l if the algebraic monodromy groups of the Galois representations of X satisfy the required conditions.