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

Abstract

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