Type A images of Galois representations and maximality

-independence for compatible systems of (mod ) representations

“…Remark 1.1. As an application of the main results, we prove in [HL14] that Φ ℓ (Gal K ), the ℓ-adic Galois image arising frométale cohomology has certain maximality inside the algebraic monodromy group G ℓ if ℓ is sufficiently large and G ℓ is of type A. This generalizes Serre's open image theorem on non-CM elliptic curves [Se72].…”

On the rationality of certain type A Galois representations

“…The representation Φ ss ℓ and the algebraic monodromy group G ℓ are said to be of type A if every simple factor of g ℓ := Lie(G ℓ × Q ℓ C) is of type A n . The maximality of the monodromy group Γ ℓ inside the ℓ-adic Lie group G ℓ (Q ℓ ) is studied in [14] assuming that G ℓ is of type A. A reductive group H/Q ℓ is said to be unramified if it is quasi-split over Q ℓ and splits over an unramified extension of Q ℓ .…”

“…Since cv ,σ = hv • c σ • fv (Corollary 5.6), Im(cv ,σ ) ⊂ ΩQ ℓ , and hv : ΩQ → ΩQ ℓ is an isomorphism (Lemma 5.5) for allv|ℓ and sufficiently large ℓ, the image of cocycle (c σ ) is contained in ΩQ, i.e., (c σ ) ∈ Z 1 (Q, ΩQ). Hence by the diagram (14), the cocycle (c σ ) maps to the cohomology class [c σ ] ∈ H 1 (Q, OutQG sp Q ) and corresponds to a unique connected reductive quasi-split group G Q over Q by Proposition 4.1 and Theorem 4.2. Let [cv ,σ ] 4 be the cohomology class of the cocycle (cv ,σ ) ∈ Z 1 (Q ℓ , OutQ ℓ G sp Q ℓ ).…”

“…We embed Q ℓ in C and let g ℓ be the Lie algebra of G ℓ × Q ℓ C for all ℓ. The representation Φ ss ℓ and the algebraic monodromy group G ℓ are said to be of type A if every simple factor of g ℓ is equal to A n := sl n+1,C for some n. This definition is independent of the choice of embedding Q ℓ ֒→ C and is equivalent to the one in [14]. Type A representations provide supporting evidence for (2).…”

“…5.5. For any non-Archimedean valuationv onQ extending the ℓ-adic valuation on Q, there is a natural morphism hv of the diagram(14) for (G sp Q , T sp Q ) to the diagram (14) for (G sp Q ℓ , T sp Q ℓ )satisfying the following. (i) The morphism hv is compatible with fv : Gal Q ℓ → Gal Q in the sense of [25, Chapter 1 §2.4], i.e., when we view the diagram (14) for Q as a Gal Q ℓ -diagram via fv, then hv is a Gal Q ℓ -morphism of Gal Q ℓ -diagrams.…”