scholarly journals Galois representations associated to holomorphic limits of discrete series

2013 ◽  
Vol 150 (2) ◽  
pp. 191-228 ◽  
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.

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

2011 ◽  
Vol 147 (5) ◽  
pp. 1337-1352 ◽  
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.

scholarly journals On counting cuspidal automorphic representations for GSp(4)

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Manami Roy ◽  
Ralf Schmidt ◽  
Shaoyun Yi
Keyword(s):  
Discrete Series ◽  
Series Representation ◽  
Density Theorem ◽  
Central Character ◽  
Holomorphic Discrete Series ◽  
Automorphic Representations ◽  
Discrete Series Representation ◽  
Archimedean Component ◽  
Cuspidal Automorphic Representations ◽  
Vector Valued

Abstract We find the number s k ⁢ ( p , Ω ) s_{k}(p,\Omega) of cuspidal automorphic representations of GSp ⁢ ( 4 , A Q ) \mathrm{GSp}(4,\mathbb{A}_{\mathbb{Q}}) with trivial central character such that the archimedean component is a holomorphic discrete series representation of weight k ≥ 3 k\geq 3 , and the non-archimedean component at 𝑝 is an Iwahori-spherical representation of type Ω and unramified otherwise. Using the automorphic Plancherel density theorem, we show how a limit version of our formula for s k ⁢ ( p , Ω ) s_{k}(p,\Omega) generalizes to the vector-valued case and a finite number of ramified places.

scholarly journals Galois representations attached to automorphic forms on over fields

2014 ◽  
Vol 150 (4) ◽  
pp. 523-567 ◽  
Author(s):  
Chung Pang Mok
Keyword(s):  
Automorphic Forms ◽  
Galois Representations ◽  
Suitable Condition ◽  
Two Dimensional ◽  
Automorphic Representations ◽  
Cuspidal Automorphic Representations ◽  
Central Characters

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.

scholarly journals On the image of complex conjugation in certain Galois representations

2016 ◽  
Vol 152 (7) ◽  
pp. 1476-1488 ◽  
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 $F$.

Serre weights for U ⁢ ( n ) {U(n)}

2018 ◽  
Vol 2018 (735) ◽  
pp. 199-224 ◽  
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.

Poles ofL-functions and theta liftings for orthogonal groups

2009 ◽  
Vol 8 (4) ◽  
pp. 693-741 ◽  
Author(s):  
David Ginzburg ◽  
Dihua Jiang ◽  
David Soudry
Keyword(s):  
Eisenstein Series ◽  
Orthogonal Group ◽  
Symplectic Groups ◽  
Orthogonal Groups ◽  
Metaplectic Groups ◽  
Automorphic Representations ◽  
Domain Of Holomorphy ◽  
First Occurrence ◽  
Cuspidal Automorphic Representations

AbstractIn this paper, we prove that the first occurrence of global theta liftings from any orthogonal group to either symplectic groups or metaplectic groups can be characterized completely in terms of the location of poles of certain Eisenstein series. This extends the work of Kudla and Rallis and the work of Moeglin to all orthogonal groups. As applications, we obtain results about basic structures of cuspidal automorphic representations and the domain of holomorphy of twisted standardL-functions.

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