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 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 $F$.

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.

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.

scholarly journals A local-global question in automorphic forms

2013 ◽  
Vol 149 (6) ◽  
pp. 959-995 ◽  
Author(s):  
U. K. Anandavardhanan ◽  
Dipendra Prasad
Keyword(s):  
Number Field ◽  
Automorphic Forms ◽  
Quadratic Extension ◽  
Automorphic Representation ◽  
Automorphic Representations ◽  
Formula One ◽  
Distinguished Representations ◽  
Analyze Period ◽  
Cuspidal Automorphic Representations ◽  
Periods Of Automorphic Forms

AbstractIn this paper, we consider the $\mathrm{SL} (2)$ analogue of two well-known theorems about period integrals of automorphic forms on $\mathrm{GL} (2)$: one due to Harder–Langlands–Rapoport about non-vanishing of period integrals on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$ of cuspidal automorphic representations on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{E} )$ where $E$ is a quadratic extension of a number field $F$, and the other due to Waldspurger involving toric periods of automorphic forms on ${\mathrm{GL} }_{2} ({ \mathbb{A} }_{F} )$. In both these cases, now involving $\mathrm{SL} (2)$, we analyze period integrals on global$L$-packets; we prove that under certain conditions, a global automorphic $L$-packet which at each place of a number field has a distinguished representation, contains globally distinguished representations, and further, an automorphic representation which is locally distinguished is globally distinguished.

scholarly journals Nearly ordinary Galois deformations over arbitrary number fields

2008 ◽  
Vol 8 (1) ◽  
pp. 99-177 ◽  
Author(s):  
Frank Calegari ◽  
Barry Mazur
Keyword(s):  
Arbitrary Number ◽  
Quadratic Field ◽  
Galois Representations ◽  
Galois Representation ◽  
Number Fields ◽  
Base Change ◽  
Deformation Spaces ◽  
Totally Real ◽  
Set Of Points ◽  
Dense Set

AbstractLet K be an arbitrary number field, and let ρ : Gal($\math{\bar{K}}$/K) → GL2(E) be a nearly ordinary irreducible geometric Galois representation. In this paper, we study the nearly ordinary deformations of ρ. When K is totally real and ρ is modular, results of Hida imply that the nearly ordinary deformation space associated to ρ contains a Zariski dense set of points corresponding to ‘automorphic’ Galois representations. We conjecture that if K is not totally real, then this is never the case, except in three exceptional cases, corresponding to: (1) ‘base change’, (2) ‘CM’ forms, and (3) ‘even’ representations. The latter case conjecturally can only occur if the image of ρ is finite. Our results come in two flavours. First, we prove a general result for Artin representations, conditional on a strengthening of the Leopoldt Conjecture. Second, when K is an imaginary quadratic field, we prove an unconditional result that implies the existence of ‘many’ positive-dimensional components (of certain deformation spaces) that do not contain infinitely many classical points. Also included are some speculative remarks about ‘p-adic functoriality’, as well as some remarks on how our methods should apply to n-dimensional representations of Gal($\math{\bar{\QQ}}$/ℚ) when n > 2.

scholarly journals Adjoint L-Functions for GL(3) and U2,1

2019 ◽  
Author(s):  
Joseph Hundley ◽  
Qing Zhang
Keyword(s):  
Number Field ◽  
Automorphic Representation ◽  
Base Change ◽  
Unitary Groups ◽  
Finite Part ◽  
Cuspidal Automorphic Representation ◽  
The Relationship

AbstractWe show that the finite part of the adjoint $L$-function (including contributions from all non-archimedean places, including ramified places) is holomorphic in ${\textrm{Re}}(s) \ge 1/2$ for a cuspidal automorphic representation of ${\textrm{GL}}_3$ over a number field. This improves the main result of [21]. We obtain more general results for twisted adjoint $L$-functions of both ${\textrm{GL}}_3$ and quasisplit unitary groups. For unitary groups, we explicate the relationship between poles of twisted adjoint $L$-functions, endoscopy, and the structure of the stable base change lifting.

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 Level-raising and symmetric power functoriality, I

2014 ◽  
Vol 150 (5) ◽  
pp. 729-748 ◽  
Author(s):  
Laurent Clozel ◽  
Jack A. Thorne
Keyword(s):  
Tensor Product ◽  
Recent Result ◽  
Number Field ◽  
Deformation Theory ◽  
Automorphic Representation ◽  
Symmetric Power ◽  
Classical Type ◽  
Initial Representation ◽  
Totally Real ◽  
Langlands Functoriality

AbstractAs the simplest case of Langlands functoriality, one expects the existence of the symmetric power $S^n(\pi )$, where $\pi $ is an automorphic representation of ${\rm GL}(2,{\mathbb{A}})$ and ${\mathbb{A}}$ denotes the adeles of a number field $F$. This should be an automorphic representation of ${\rm GL}(N,{\mathbb{A}})$ ($N=n+1)$. This is known for $n=2,3$ and $4$. In this paper we show how to deduce the general case from a recent result of J.T. on deformation theory for ‘Schur representations’, combined with expected results on level-raising, as well as another case (a particular tensor product) of Langlands functoriality. Our methods assume $F$ totally real, and the initial representation $\pi $ of classical type.

A construction of residues of Eisenstein series and related square-integrable classes in the cohomology of arithmetic groups of low k-rank

2019 ◽  
Vol 31 (5) ◽  
pp. 1225-1263
Author(s):  
Neven Grbac ◽  
Joachim Schwermer
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Reductive Algebraic Group ◽  
Cohomology Of Arithmetic Groups ◽  
Geometric Conditions ◽  
Global Properties ◽  
Eisenstein Cohomology ◽  
Cuspidal Automorphic Representations ◽  
Totally Real ◽  
Low K

AbstractThe cohomology of an arithmetic congruence subgroup of a connected reductive algebraic group defined over a number field is captured in the automorphic cohomology of that group. The residual Eisenstein cohomology is by definition the part of the automorphic cohomology represented by square-integrable residues of Eisenstein series. The existence of residual Eisenstein cohomology classes depends on a subtle combination of geometric conditions (coming from cohomological reasons) and arithmetic conditions in terms of analytic properties of automorphic L-functions (coming from the study of poles of Eisenstein series). Hence, there are almost no unconditional results in the literature regarding the very existence of non-trivial residual Eisenstein cohomology classes. In this paper, we show the existence of certain non-trivial residual cohomology classes in the case of the split symplectic, and odd and even special orthogonal groups of rank two, as well as the exceptional group of type {\mathrm{G}_{2}}, defined over a totally real number field. The construction of cuspidal automorphic representations of {\mathrm{GL}_{2}} with prescribed local and global properties is decisive in this context.

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