On the images of the Galois representations attached to generic automorphic representations of GSp(4)

AbstractIn this paper we generalize the work of Harris–Soudry–Taylor and construct the compatible systems of two-dimensional Galois representations attached to cuspidal automorphic representations of cohomological type on ${\rm GL}_2$ over a CM field with a suitable condition on their central characters. We also prove a local-global compatibility statement, up to semi-simplification.

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On the image of complex conjugation in certain Galois representations

Compositio Mathematica ◽

10.1112/s0010437x16007016 ◽

2016 ◽

Vol 152 (7) ◽

pp. 1476-1488 ◽

Cited By ~ 2

Author(s):

Ana Caraiani ◽

Bao V. Le Hung

Keyword(s):

Symmetric Spaces ◽

Galois Representations ◽

Real Field ◽

Complex Conjugation ◽

Automorphic Representations ◽

Locally Symmetric Spaces ◽

Cuspidal Automorphic Representations ◽

Totally Real ◽

Totally Real Field

We compute the image of any choice of complex conjugation on the Galois representations associated to regular algebraic cuspidal automorphic representations and to torsion classes in the cohomology of locally symmetric spaces for $\text{GL}_{n}$ over a totally real field $F$.

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Galois representations associated to holomorphic limits of discrete series

Compositio Mathematica ◽

10.1112/s0010437x13007355 ◽

2013 ◽

Vol 150 (2) ◽

pp. 191-228 ◽

Cited By ~ 8

Author(s):

Wushi Goldring ◽

Sug Woo Shin

Keyword(s):

Galois Representations ◽

Discrete Series ◽

Unitary Groups ◽

Main Ingredient ◽

Automorphic Representations ◽

Archimedean Component ◽

Cuspidal Automorphic Representations

AbstractGeneralizing previous results of Deligne–Serre and Taylor, Galois representations are attached to cuspidal automorphic representations of unitary groups whose Archimedean component is a holomorphic limit of discrete series. The main ingredient is a construction of congruences, using the Hasse invariant, that is independent of$q$-expansions.

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

Compositio Mathematica ◽

10.1112/s0010437x11005264 ◽

2011 ◽

Vol 147 (5) ◽

pp. 1337-1352 ◽

Cited By ~ 18

Author(s):

Joël Bellaïche ◽

Gaëtan Chenevier

Keyword(s):

Number Field ◽

Galois Representations ◽

Galois Representation ◽

Automorphic Representation ◽

Unitary Groups ◽

Complex Conjugate ◽

Automorphic Representations ◽

Group A ◽

Cuspidal Automorphic Representations ◽

Totally Real

AbstractLet 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.

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Serre weights for U ⁢ ( n ) {U(n)}

Journal für die reine und angewandte Mathematik (Crelles Journal) ◽

10.1515/crelle-2015-0015 ◽

2018 ◽

Vol 2018 (735) ◽

pp. 199-224 ◽

Cited By ~ 3

Author(s):

Thomas Barnet-Lamb ◽

Toby Gee ◽

David Geraghty

Keyword(s):

Galois Representations ◽

Galois Representation ◽

Unitary Groups ◽

Automorphic Representations ◽

Serre's Conjecture ◽

Weight Part ◽

Serre’S Conjecture

Abstract We study the weight part of (a generalisation of) Serre’s conjecture for mod l Galois representations associated to automorphic representations on unitary groups of rank n for odd primes l. Given a modular Galois representation, we use automorphy lifting theorems to prove that it is modular in many other weights. We make no assumptions on the ramification or inertial degrees of l. We give an explicit strengthened result when {n=3} and l splits completely in the underlying CM field.

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