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 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.

On a Generalization of a Result of Waldspurger

1996 ◽  
Vol 48 (1) ◽  
pp. 105-142 ◽  
Author(s):  
Jiandong Guo
Keyword(s):  
Number Field ◽  
Trace Formula ◽  
Strong Evidence ◽  
Riemann Hypothesis ◽  
Symmetric Spaces ◽  
Quadratic Extension ◽  
Automorphic Representation ◽  
Cuspidal Representation ◽  
Orbital Integrals ◽  
Langlands Correspondence

AbstractWe consider a generalization of a trace formula identity of Jacquet, in the context of the symmetric spaces GL(2n)/GL(/n) × GL(n) and G′/H′. Here G′ is an inner form of GL(2n) over F with a subgroup H′ isomorphic to GL(n, E) where E/F is a quadratic extension of number field attached to a quadratic idele class character η of F. A consequence of this identity would be the following conjecture: Let π be an automorphic cuspidal representation of GL(2n). If there exists an automorphic representation π′ of G′ which is related to π by the Jacquet-Langlands correspondence, and a vector ø in the space of π′ whose integral over H′ is nonzero, then both L(1/2, π) and L(1/2,π ⊗ η) are nonvanishing. Moreover, we have L(1/2, π)L(1/2, π ⊗ η) > 0. Here the nonvanishing part of the conjecture is a generalization of a result of Waldspurger for GL(2) and the nonnegativity of the product is predicted from the generalized Riemann Hypothesis. In this article, we study the corresponding local orbital integrals for the symmetric spaces. We prove the "fundamental lemma for the unit Hecke functions" which says that unit Hecke functions have "matching" orbital integrals. This serves as the first step toward establishing the trace formula identity and in the same time it provides strong evidence for what we proposed.

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 Higher genus theory

2020 ◽  
Author(s):  
Peter Koymans ◽  
Carlo Pagano
Keyword(s):  
Number Field ◽  
Quadratic Forms ◽  
Class Group ◽  
Quadratic Extension ◽  
Number Fields ◽  
Explicit Description ◽  
Quadratic Number ◽  
Quadratic Number Field ◽  
Genus Theory ◽  
Narrow Class

Abstract In $1801$, Gauss found an explicit description, in the language of binary quadratic forms, for the $2$-torsion of the narrow class group and dual narrow class group of a quadratic number field. This is now known as Gauss’s genus theory. In this paper, we extend Gauss’s work to the setting of multi-quadratic number fields. To this end, we introduce and parametrize the categories of expansion groups and expansion Lie algebras, giving an explicit description for the universal objects of these categories. This description is inspired by the ideas of Smith [ 16] in his recent breakthrough on Goldfeld’s conjecture and the Cohen–Lenstra conjectures. Our main result shows that the maximal unramified multi-quadratic extension $L$ of a multi-quadratic number field $K$ can be reconstructed from the set of generalized governing expansions supported in the set of primes that ramify in $K$. This provides a recursive description for the group $\textrm{Gal}(L/\mathbb{Q})$ and a systematic procedure to construct the field $L$. A special case of our main result gives an upper bound for the size of $\textrm{Cl}^{+}(K)[2]$.

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.

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