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

Abstract

Let K be a number field and {V ℓ } ℓ be a rational strictly compatible system of semisimple Galois representations of K arising from geometry. Let G ℓ and V ab ℓ be respectively the algebraic monodromy group and the maximal abelian subrepresentation of V ℓ for all ℓ. We prove that the system {V ab ℓ } ℓ 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.

Monodromy of subrepresentations and irreducibility of low degree automorphic Galois representations

“…Remark Theorem 2.7 has then been used to study the conjectural () under various assumptions on the algebraic monodromy groups $$\mathbf {G}_\lambda$$, for more details, see [37, Theorem 3.21], [39, Theorem 1.2], [41, Theorem 1.6], and the related [40, Theorems 1.5, 1.6].…”

“…Let 𝐹 be a totally real or CM number field and 𝐹 + the maximal totally real subfield of 𝐹. Attached to a regular algebraic, polarized, cuspidal automorphic representation (𝜋, 𝜒) of GL 𝑛 (𝔸 𝐹 ) is a pair of strictly compatible systems ( §2.4.3) defined over a CM field 𝐸 † {𝜌 𝜋,𝜆 ∶ Gal 𝐹 → GL 𝑛 (𝐸 𝜆 )} 𝜆 and {𝜌 𝜒,𝜆 ∶ Gal 𝐹 + → 𝐸 * 𝜆 } 𝜆 (40) such that the conditions (i)-(iv) below are satisfied ([3, §2]).…”

“…Theorem 4.2 [5,Corollary 1.3]. Let 𝐹 be a totally real field and (𝜌 𝜋,𝜆 , 𝜌 𝜒,𝜆 ) the pair of (automorphic) 𝓁-adic Galois representations in (40). If 𝑊 𝜆 is an irreducible factor of 𝜌 𝜋,𝜆 ⊗ ℚ 𝓁 with 𝑊 𝜆 ≅ 𝑊 ∨ 𝜆 ⊗ 𝜖 1−𝑛 𝓁 𝜌 𝜒,𝜆 , then (𝑊 𝜆 , 𝜖 We treat the dim 𝑊 𝜆 ⩽ 2 case next.…”

“…Proposition 4.6. Let 𝐹 be a totally real field and {𝜌 𝜋,𝜆 } 𝜆 the (automorphic) semisimple SCS in (40)…”

“…Let 𝐹 be a totally real (resp. CM) field and 𝜌 𝜋,𝜆 be the 𝓁-adic representation of 𝐹 in (40). Suppose that there is a totally real (resp.…”