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.

We study irreducible odd mod p Galois representations ρ : Gal(F/F) → G(F p ), for F a totally real number field and G a general reductive group. For p ≫ G,F 0, we show that any ρ that lifts locally, and at places above p to de Rham and Hodge-Tate regular representations, has a geometric p-adic lift. We also prove non-geometric lifting results without any oddness assumption.

In this paper we prove that a pure, regular, totally odd, polarizable weakly compatible system of l-adic representations is potentially automorphic. The innovation is that we make no irreducibility assumption, but we make a purity assumption instead. For compatible systems coming from geometry, purity is often easier to check than irreducibility. We use Katz's theory of rigid local systems to construct many examples of motives to which our theorem applies. We also show that if F is a CM or totally real field and if π is a polarizable, regular algebraic, cuspidal automorphic representation of GL n (A F ), then for a positive Dirichlet density set of rational primes l, the l-adic representations r l,ı (π) associated to π are irreducible.

No abstract

Let F be a number field. These notes explore Galois-theoretic, automorphic, and motivic analogues and refinements of Tate's basic result that continuous projective representations Gal(F/F) → PGL n (C) lift to GL n (C). We take special interest in the interaction of this result with algebraicity (for automorphic representations) and geometricity (in the sense of Fontaine-Mazur). On the motivic side, we study refinements and generalizations of the classical Kuga-Satake construction. Some auxiliary results touch on: possible infinity-types of algebraic automorphic representations; comparison of the automorphic and Galois "Tannakian formalisms"; monodromy (independenceof-ℓ) questions for abstract Galois representations.