scholarly journals A note on spectral analysis in automorphic representation theory for GL2: I

2017 ◽  
Vol 13 (10) ◽  
pp. 2717-2750 ◽  
Author(s):  
Han Wu
Keyword(s):  
Spectral Analysis ◽  
Representation Theory ◽  
Number Field ◽  
Automorphic Representation ◽  
Explicit Computation ◽  
Fourier Inversion ◽  
Intertwining Operator

We establish the Fourier inversion for the smooth vectors in [Formula: see text] over a number field [Formula: see text], using minimal knowledge from automorphic representation theory. We point out a possible way to establish Fourier inversion for larger classes of functions. We also point out the incompleteness of some commonly believed “proof” of Fourier inversion in the literature. Moreover, the explicit computation of the intertwining operator has independent interests.

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 New Zero-Density Results for Automorphic L-Functions of GL(n)

Mathematics ◽  
2021 ◽  
Vol 9 (17) ◽  
pp. 2061
Author(s):  
Wenjing Ding ◽  
Huafeng Liu ◽  
Deyu Zhang
Keyword(s):  
Number Field ◽  
Imaginary Part ◽  
Rational Number ◽  
Upper Bound ◽  
Automorphic Representation ◽  
Density Estimates ◽  
Narrow Strip ◽  
Zero Density ◽  
Detecting Method ◽  
Density Results

Let L(s,π) be an automorphic L-function of GL(n), where π is an automorphic representation of group GL(n) over rational number field Q. In this paper, we study the zero-density estimates for L(s,π). Define Nπ(σ,T1,T2) = ♯ {ρ = β + iγ: L(ρ,π) = 0, σ<β<1, T1≤γ≤T2}, where 0≤σ<1 and T1<T2. We first establish an upper bound for Nπ(σ,T,2T) when σ is close to 1. Then we restrict the imaginary part γ into a narrow strip [T,T+Tα] with 0<α≤1 and prove some new zero-density results on Nπ(σ,T,T+Tα) under specific conditions, which improves previous results when σ near 34 and 1, respectively. The proofs rely on the zero detecting method and the Halász-Montgomery method.

L-Functions for GSp(2) × GL(2): Archimedean Theory and Applications

2009 ◽  
Vol 61 (2) ◽  
pp. 395-426 ◽  
Author(s):  
Tomonori Moriyama
Keyword(s):  
Discrete Series ◽  
Series Representation ◽  
Principal Series ◽  
Algebraic Number Field ◽  
Automorphic Representation ◽  
Explicit Computation ◽  
Principal Series Representation ◽  
Cuspidal Automorphic Representation ◽  
Discrete Series Representation ◽  
Totally Real

Abstract. Let Π be a generic cuspidal automorphic representation of GSp(2) defined over a totally real algebraic number field k whose archimedean type is either a (limit of) large discrete series representation or a certain principal series representation. Through explicit computation of archimedean local zeta integrals, we prove the functional equation of tensor product L-functions L(s,Π × σ) for an arbitrary cuspidal automorphic representation σ of GL(2). We also give an application to the spinor L-function of Π.

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.

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 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 A Specialisation of the Bump–Friedberg L-function

2015 ◽  
Vol 58 (3) ◽  
pp. 580-595
Author(s):  
Nadir Matringe
Keyword(s):  
Integral Representation ◽  
Number Field ◽  
Direct Proof ◽  
Automorphic Representation ◽  
Simple Theory ◽  
Cuspidal Automorphic Representation

AbstractWe study the restriction of Bump–Friedberg integrals to affine lines {(s + α, 2s), s ∊ ℂ}. It has simple theory, very close to that of the Asai L-function. It is an integral representation of the product L(s + α, π)L(2s, Λ2, π), which we denote by Llin(s, π, α) for this abstract, when π is a cuspidal automorphic representation of GL(k, 𝔸) for 𝔸 the adeles of a number field. When k is even, we show that the partial L-function Llin,S(s, π, α) has a pole at 1/2 if and only if π admits a (twisted) global period. This gives a more direct proof of a theorem of Jacquet and Friedberg, asserting that π has a twisted global period if and only if L(α + 1/2, π) ≠ 0 and L(1, Λ2 , π) = ∞. When k is odd, the partial L-function is holmorphic in a neighbourhood of Re(s) ≥ 1/2 when Re(α) is ≥ 0.

scholarly journals THE TWISTED TENSOR L-FUNCTION OF GSp4

2012 ◽  
Vol 08 (02) ◽  
pp. 411-470
Author(s):  
JUSTIN YOUNG
Keyword(s):  
Integral Representation ◽  
Number Field ◽  
Absolute Convergence ◽  
Quadratic Extension ◽  
Automorphic Representation ◽  
Central Character ◽  
Euler Product ◽  
Local Integral ◽  
The Absolute

The author gives an integral representation for the twisted tensor L-function of a cuspidal, globally generic automorphic representation of GSp 4 over a quadratic extension E of a number field F with trivial central character. He proves the Euler product factorization of the global integral; computes the unramified L-factor via explicit branching from GL 4 to Sp 4 and shows it is equal to the normalized unramified local integral; and proves the absolute convergence and nonvanishing of all local integrals.

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