On the rationality of certain type A Galois representations

Abstract

Abstract. Let X be a complete smooth variety defined over a number field K and i an integer. The absolute Galois group Gal K of K acts on the ithétale cohomology group H í et (XK, Q ℓ ) for all primes ℓ, producing a system of ℓ-adic representations {Φ ℓ } ℓ . The conjectures of Grothendieck, Tate, and Mumford-Tate predict that the identity component of the algebraic monodromy group of Φ ℓ admits a common reductive Q-form for all ℓ if X is projective. Denote by Γ ℓ and G ℓ respectively the monodromy group and the algebraic monodromy group of Φ ss ℓ , the semisimplification of Φ ℓ . Assuming that G ℓ0 satisfies a group theoretic condition for some prime ℓ 0 (Hypothesis A), we construct a connected quasi-split Q-reductive group G Q which is a common Q-form of G• ℓ for all sufficiently large ℓ. Let G sc Q be the universal cover of the derived group of G Q . As an application, we prove that the monodromy group Γ ℓ is big in the sense that Γ sc ℓ ∼ = G sc Q (Z ℓ ) for all sufficiently large ℓ.

Monodromy of subrepresentations and irreducibility of low degree automorphic Galois representations

“…This theorem of Serre is proved by studying the algebraic variety of characteristic polynomials of $$\mathbf {G}_\lambda$$ for every $$\lambda$$, see also [51, §4]. When $$\lbrace \rho _\lambda \rbrace _\lambda$$ is pure of weight $$w$$, Serre also used the method of Frobenius torus to prove the theorem and construct an $$E$$‐torus $$\mathbf {T}$$ such that the base change $$\mathbf {T}_{E_\lambda }$$ is a maximal torus of $$\mathbf {G}_\lambda$$ for all $$\lambda$$ not above some rational prime, see [53, Theorem 1.1] and [39, Theorem 2.6] for a more general case.…”

“…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].…”

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

“…We introduce a geometric viewpoint on the configuration of R in X ⊗ Z R that we learned from [LP90, §1] and developed in [Hu13,§2]. This viewpoint is also the starting point of Theorem 2.26, see [Hu13,§2], [Hu18,§3]. Let X be the character group of the maximal torus T, Z[X] be the group ring, and R be (resp.…”

“…The tautological representation G • ℓ ֒→ GL N,Q ℓ is conjectured to be independent of ℓ [Se94]. For results in this direction, see [Se72,Se85], [Ri76], [Ch92], [Pi98], [LP95] for abelian varieties and [LP92], [Chi04], [Hu13,Hu15,Hu18] for more general cases.…”

“…has non-trivial invariants; to do the same thing for a family {W ℓ , V ℓ } ℓ assuming W ℓ ′ ≤ V ℓ ′ for some ℓ ′ , it suffices to show that the tautological representation of the algebraic monodromy group H ℓ of V ℓ ⊗ W * ℓ is independent of ℓ. Section 2 is devoted to the proofs of the results in this section, which rely on some results of Serre on Galois representations and the techniques we have developed in [Hu13,Hu18]. The appendix studies the criterion on the roots in 1.5(iii) by a geometric configuration of the root system.…”