-independence for compatible systems of (mod ) representations

Abstract

Abstract. Let K be a number field. For any system of semisimple mod ℓ Galois representations {φ ℓ

Type A images of Galois representations and maximality

“…Then the assertion on composition factors and parts (i) and (ii) follow from [Hu2,Proposition 3.3.3]. Part (iii) follows easily from the definition.…”

“…We define the ℓ-dimension function, dim ℓ , the g-type ℓ-rank function, rk g ℓ , and the total ℓ-rank function, rk ℓ , on finite groups. The definitions of rk g ℓ and rk ℓ can also be found from [Hu2,Definition 14,Remark 3.3.2]. The dimension of an algebraic group G/F as an F -variety is denoted dim G, and we write dim g for dim G for any simple algebraic group G of type g. LetΓ be a finite simple group of Lie type in characteristic ℓ.…”

Invariant dimensions and maximality of geometric monodromy action

“…If K is a number field and X = Spec(K), then Φ ℓ is a Galois representation of K arising from the smooth projective variety Y /K and the largeness of Γ ℓ in G ℓ (Q ℓ ) for ℓ ≫ 0 follows from the remarkable conjectures of Hodge, Grothendieck, Tate, Mumford-Tate, and Serre [Se94,§11], see also [HL15a,§5]. The prototypical result in this direction is due to Serre [Se72], which states that for any non-CM elliptic curve Y , the monodromy Γ ℓ on H 1 is GL 2 (Z ℓ ) for all sufficiently large ℓ, see also [Ri76,Ri85], [Se85],[BGK03, BGK06, BGK10], [Ha11] for certain abelian varieties; [HL15b] for arbitrary abelian varieties; [Se98] for abelian representations; [HL14] for type A representations; and partial results [La95a], [Hu14] for arbitrary varieties. To get large Galois monodromy, one always needs handles on the invariants of V ℓ and Vℓ .…”

Monodromy of Galois representations and equal-rank subalgebra equivalence

“…Remark 1.9. We study ℓ-independence of mod ℓ Galois representations that arise frométale cohomology in a subsequent paper [9] and obtain "mod ℓ" versions of (2) and (3) when ℓ ≫ 0.…”