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

Abstract

© 2018, Springer-Verlag GmbH Germany, part of Springer Nature. Let K be a number field and {Vℓ}ℓ a rational strictly compatible system of semisimple Galois representations of K arising from geometry. Let Gℓ and Vℓab be respectively the algebraic monodromy group and the maximal abelian subrepresentation of Vℓ for all ℓ. We prove that the system {Vℓab}ℓ is also a rational strictly compatible system under some group theoretic conditions, e.g., when Gℓ′ 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.