The sign of Galois representations attached to automorphic forms for unitary groups

By J. Bellaïche, Gaëtan Chenevier. Compositio Mathematica, 2008

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Abstract Let K be a CM number field and GK its absolute Galois group. A representation of GK is said to be polarized if it is isomorphic to the contragredient of its outer complex conjugate, up to a twist by a power of the cyclotomic character. Absolutely irreducible polarized representations of GK have a sign ±1, generalizing the fact that a self-dual absolutely irreducible representation is either symplectic or orthogonal. If Π is a regular algebraic, polarized, cuspidal automorphic representation of GLn(𝔸K), and if ρ is a p-adic Galois representation attached to Π, then ρ is polarized and we show that all of its polarized irreducible constituents have sign +1 . In particular, we determine the orthogonal/symplectic alternative for the Galois representations associated to the regular algebraic, essentially self-dual, cuspidal automorphic representations of GLn (𝔸F) when F is a totally real number field.

Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/e65f04c5beebfe3628412f96e64eb209ba948b31

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Citations in the corpus

On the infinite fern of Galois representations of unitary type (2009, methodology): “Let be a selfdual, cohomological, and cuspidal, automorphic representation of GLn(AQ), and let : GQ ! GLn(Qp) be an associated p-adic Galois representation, normalized so that ’ (n 1). Recall that is symplectic if n is even and orthogonal otherwise by [ BCh2 ].”
Even Galois representations and the Fontaine–Mazur conjecture (2009, background): “k. For neven, the representations rwill not, in general, be potentially modular (in the sense we are using) over any CM ﬁeld F, because the Bella¨ıche–Chenevier sign of the Galois representation (see [6]) may be −1. Indeed, when n= 2, all representations are self-dual up to twist, and so an even rep”
Irreducibility of automorphic Galois representations of GL(n), n at most 5 (2011, background): “that µ(cv) = 1, as required. We also have the following trivial lemma. Lemma 2.2. If χ: GF → K× is a character, then (χ,χ2) is odd and essentially self-dual. We have the following important result of [BC11]. Theorem 2.3. Let (π,χ) be a RAESDC automorphic representation of GLn(AF), and denote the cor”
Potential automorphy and change of weight (2010, background): “ recall that thanks to the work of many people (we mention in particular Bellaiche, Caraiani, Chenevier, Clozel, Harris, Kottwitz, Labesse, Shin and R.T., and the references [Kot92], [Clo90], [HT01], [BC11], [Shi11], [CHLN11], [Clo10], [Car12a] and [Car12b]) we can associate l-adic representations t”
Serre weights for rank two unitary groups (2011, unknown): “Then by the main result of [BC09], r̄ extends to a representation ρ : GF+ → G2(Fl) with multiplier −1 l .”