Variations on a Theorem of Tate

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.

Independence of algebraic monodromy groups in compatible systems

“…Let now w 0 be a non-split place of F λ 0 i of residue characteristic 0 . Since F λ 0 i,w 0 does not embed in E i,λ 0 = Q 0 by assumption, we can deduce by an argument analogous to that of [17,Lemma 3.4.13] that ρ i,λ 0 is not regular, which is a contradiction. Therefore, we get that F λ 0 i = F λ 0 ,cm i .…”

“…This gives that σ λ 0 i has Hodge-Tate weights lying in [a, a + 0 − 2]. Since 0 is unramified in F i , and so it is in F i as well, we have that σ λ 0 i is crystalline at each place w 0 of F i lying above 0 by [17,Lemma 2.2.9]. Therefore, σ λ 0 i is potentially diagonalisable at each place w 0 of F i lying above 0 by [1,Lemma 1.4.3].…”

“…. , k. This has been proved by Katz, see [12,Proposition 1], when is the étale fundamental group of a smooth connected affine curve over an algebraically closed field of positive characteristic, and by Patrikis, see [17,Proposition 3.4.1], by adapting Katz's argument, when is the absolute Galois group of a number field. To prove a decomposition result for a compatible system R = {ρ λ } λ∈ of representations of , one would then start from a single λ-adic representation ρ λ in R, decompose it as ρ λ ∼ = ⊕ k i=1 Ind i (σ i,λ ⊗ ω i ), and extend each σ i,λ to a compatible system S i of Lie-irreducible representations.…”

“…By [1, Lemma 5.3.1(2)] each ρ i,λ is Lie-multiplicity free. 8 Therefore, for each λ ∈ , by [17,Lemma 3.4.6(1)] there exist an intermediate field F ⊂ F λ i ⊂ F i , and a rank…”

On symmetric power L-invariants of Iwahori level Hilbert modular forms

“…Next, Arthur's global classification of the discrete spectrum of Sp(2n) ([Art11, Theorem 1.5.2]) implies that there exists a cuspidal automorphic representation τ ′ which is isomorphic to τ at all finite places and the representation τ ′ ∞ is a holomorphic discrete series in the same L-packet at τ ∞ . Finally, Proposition 12.2.2, Corollary 12.2.4 and Proposition 12.3.3 (the fact that π is the symmetric 2n-th power of a cohomological Hilbert modular form implies that hypothesis (2) of this proposition is satisfied) of [Pat12] imply the existence of a regular algebraic cuspidal automorphic representation σ of GSp(2n, A F ) such that if v is either an infinite place or a finite place such that σ v is unramified then σ v | Sp(2n,Fv ) contains τ ′ v . Moreover, the discrete series σ ∞ is holomorphic or else its restriction to Sp(2n, R) would not contain the holomorphic discrete series τ ′ ∞ .…”