scholarly journals An Integral Representation of Standard AutomorphicLFunctions for Unitary Groups

2007 ◽  
Vol 2007 ◽  
pp. 1-22
Author(s):  
Yujun Qin
Keyword(s):  
Integral Representation ◽  
Eisenstein Series ◽  
Number Field ◽  
Unitary Group ◽  
Theta Series ◽  
Automorphic Representation ◽  
Unitary Groups ◽  
Cusp Forms ◽  
Cuspidal Automorphic Representation ◽  
Type Integral

LetFbe a number field,Ga quasi-split unitary group of rankn. We show that given an irreducible cuspidal automorphic representationπofG(A), its (partial)LfunctionLS(s,π,σ)can be represented by a Rankin-Selberg-type integral involving cusp forms ofπ, Eisenstein series, and theta series.

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 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.

Period relations for cusp forms of GSp4

2018 ◽  
Vol 30 (3) ◽  
pp. 581-598 ◽  
Author(s):  
Harald Grobner ◽  
Ronnie Sebastian
Keyword(s):  
Real Number ◽  
Number Field ◽  
Automorphic Representation ◽  
Cusp Forms ◽  
Special Values ◽  
Large Source ◽  
Cuspidal Automorphic Representation ◽  
Totally Real ◽  
Real Number Field

AbstractLet F be a totally real number field and let π be a cuspidal automorphic representation of {\mathrm{GSp_{4}}(\mathbb{A}_{F})}, which contributes irreducibly to coherent cohomology. If π has a Bessel model, we may attach a period {p(\pi)} to this datum. In the present paper, which is Part I in a series of two, we establish a relation of these Bessel periods {p(\pi)} and all of their twists {p(\pi\otimes\xi)} under arbitrary algebraic Hecke characters ξ. In the appendix, we show that {(\mathfrak{g},K)}-cohomological cusp forms of {\mathrm{GSp_{4}}(\mathbb{A}_{F})} all qualify to be of the above type – providing a large source of examples. We expect that these period relations for {\mathrm{GSp_{4}}(\mathbb{A}_{F})} will allow a conceptual, fine treatment of rationality relations of special values of the spin L-function, which we hope to report on in Part II of this paper.

scholarly journals -ADIC EISENSTEIN SERIES AND -FUNCTIONS OF CERTAIN CUSP FORMS ON DEFINITE UNITARY GROUPS

2014 ◽  
Vol 15 (3) ◽  
pp. 471-510 ◽  
Author(s):  
Ellen Eischen ◽  
Xin Wan
Keyword(s):  
Positive Integer ◽  
Eisenstein Series ◽  
Constant Term ◽  
Unitary Groups ◽  
Cusp Forms ◽  
Imaginary Field

We construct$p$-adic families of Klingen–Eisenstein series and$L$-functions for cusp forms (not necessarily ordinary) unramified at an odd prime$p$on definite unitary groups of signature$(r,0)$(for any positive integer$r$) for a quadratic imaginary field${\mathcal{K}}$split at$p$. When$r=2$, we show that the constant term of the Klingen–Eisenstein family is divisible by a certain$p$-adic$L$-function.

scholarly journals LOCAL BORCHERDS PRODUCTS FOR UNITARY GROUPS

2017 ◽  
Vol 234 ◽  
pp. 139-169
Author(s):  
ERIC HOFMANN
Keyword(s):  
Boundary Component ◽  
Unitary Group ◽  
Picard Group ◽  
Unitary Groups ◽  
Weil Representation ◽  
Cusp Forms ◽  
Modular Variety ◽  
Arithmetic Subgroup ◽  
Torsion Element ◽  
Vector Valued

For the modular variety attached to an arithmetic subgroup of an indefinite unitary group of signature $(1,n+1)$, with $n\geqslant 1$, we study Heegner divisors in the local Picard group over a boundary component of a compactification. For this purpose, we introduce local Borcherds products. We obtain a precise criterion for local Heegner divisors to be torsion elements in the Picard group, and further, as an application, we show that the obstructions to a local Heegner divisor being a torsion element can be described by certain spaces of vector-valued elliptic cusp forms, transforming under a Weil representation.

Poles of Siegel Eisenstein Series on U(n, n)

1999 ◽  
Vol 51 (1) ◽  
pp. 164-175 ◽  
Author(s):  
Victor Tan
Keyword(s):  
Eisenstein Series ◽  
Number Field ◽  
Complex Plane ◽  
Unitary Group

AbstractLet U(n, n) be the rank n quasi-split unitary group over a number field. We show that the normalized Siegel Eisenstein series of U(n, n) has at most simple poles at the integers or half integers in certain strip of the complex plane.

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 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.

scholarly journals Hermitian Theta Series and Maaß Spaces Under the Action of the Maximal Discrete Extension of the Hermitian Modular Group

2020 ◽  
Vol 75 (4) ◽  
Author(s):  
Annalena Wernz
Keyword(s):  
Number Field ◽  
Modular Forms ◽  
Unitary Group ◽  
Modular Group ◽  
Theta Series ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Special Unitary Group ◽  
Imaginary Quadratic Number

AbstractLet $$\Gamma _n(\mathcal {\scriptstyle {O}}_{\mathbb {K}})$$ Γ n ( O K ) denote the Hermitian modular group of degree n over an imaginary quadratic number field $$\mathbb {K}$$ K and $$\Delta _{n,\mathbb {K}}^*$$ Δ n , K ∗ its maximal discrete extension in the special unitary group $$SU(n,n;\mathbb {C})$$ S U ( n , n ; C ) . In this paper we study the action of $$\Delta _{n,\mathbb {K}}^*$$ Δ n , K ∗ on Hermitian theta series and Maaß spaces. For $$n=2$$ n = 2 we will find theta lattices such that the corresponding theta series are modular forms with respect to $$\Delta _{2,\mathbb {K}}^*$$ Δ 2 , K ∗ as well as examples where this is not the case. Our second focus lies on studying two different Maaß spaces. We will see that the new found group $$\Delta _{2,\mathbb {K}}^*$$ Δ 2 , K ∗ consolidates the different definitions of the spaces.

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