Deformations of Galois representations and exceptional monodromy

Abstract

Abstract. For any simple algebraic group G of exceptional type, we construct geometric ℓ-adic Galois representations with algebraic monodromy group equal to G, in particular producing the first such examples in types F 4 and E 6 . To do this, we extend to general reductive groups Ravi Ramakrishna's techniques for lifting odd two-dimensional Galois representations to geometric ℓ-adic representations.

Producing geometric deformations of orthogonal and symplectic Galois representations

“…As the condition µ • ρ = ν cuts out a closed subscheme of the universal lifting ring for D ram v , this is a deformation condition we will denote by D ram,ν v . Lemmas 4.10 and 4.11 of [Pat15] show: 3.2. Big Representations.…”

“…The first part of the argument, with only minor technical variation, has also been carried out in [Pat15]. Fix a prime p and finite field k of characteristic p. Let S be a finite set of places of a number field K containing the places above p and the archimedean places, and define Γ S to be the Galois group of the maximal extension of K unramified outside of S. Consider a continuous representation ρ : Γ S → G(k) where G is a smooth affine group scheme over the ring of integers O in a p-adic field such that the identity components of the fibers are reductive.…”

“…To produce lifts, we use a generalization of Ramakrishna's method also used in [Pat15]. It works by establishing a local-to-global result for lifting Galois representations subject to local constraints (Proposition 2.4).…”

“…If K is totally real and ρ is odd, (1.1) holds. There are odd representations for symplectic and orthogonal groups, but no odd representations for GL n when n > 2 (for more details, see [Pat15,§4.5]). These are cases in which we expect geometric lifts, and where Ramakrishna's method generalizes.…”

Deformations of Galois representations and exceptional monodromy, II: raising the level

“…for some subfield k ′ of k; since we will only work with the projectivization ofr, we replace k by k ′ in all that follows. Provided ℓ ≥ 2h G ∨ − 1, we find that (this new) k is the minimal field of definition of each of the (absolutely) irreducible factors in the decomposition of the Lie algebra (see [Pat15,Lemma 7.3]),…”