Fourier expansion of Arakawa lifting II: Relation with central L-values

Atsushi Murase, Hiroaki Narita

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

This is a continuation of our previous paper [Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts, Israel J. Math. 187 (2012) 317-369]. The aim of the paper here is to study the Fourier coefficients of Arakawa lifts in relation with central values of automorphic L-functions. In the previous paper we provide an explicit formula for the Fourier coefficients in terms of toral integrals of automorphic forms on multiplicative groups of quaternion algebras. In this paper, after studying explicit relations between the toral integrals and the central L-values, we explicitly determine the constant of proportionality relating the square norm of a Fourier coefficient of an Arakawa lift with the central L-value. We can relate the square norm with the central value of some L-function of convolution type attached to the lift and a Hecke character. We also discuss the existence of strictly positive central values of the L-functions in our concern.

Original languageEnglish
Article number1650001
JournalInternational Journal of Mathematics
Volume27
Issue number1
DOIs
Publication statusPublished - 2016 Jan 1
Externally publishedYes

Fingerprint

Fourier Expansion
Fourier coefficients
L-function
Explicit Formula
Norm
Quaternion Algebra
Automorphic Forms
Strictly positive
Continuation
Convolution
Multiplicative

Keywords

  • Central L-values
  • Fourier coefficients
  • quaternion unitary group
  • theta lifts
  • toral integrals

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Fourier expansion of Arakawa lifting II : Relation with central L-values. / Murase, Atsushi; Narita, Hiroaki.

In: International Journal of Mathematics, Vol. 27, No. 1, 1650001, 01.01.2016.

Research output: Contribution to journalArticle

@article{2ac0714035c5448585c5afa0cf8550b2,
title = "Fourier expansion of Arakawa lifting II: Relation with central L-values",
abstract = "This is a continuation of our previous paper [Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts, Israel J. Math. 187 (2012) 317-369]. The aim of the paper here is to study the Fourier coefficients of Arakawa lifts in relation with central values of automorphic L-functions. In the previous paper we provide an explicit formula for the Fourier coefficients in terms of toral integrals of automorphic forms on multiplicative groups of quaternion algebras. In this paper, after studying explicit relations between the toral integrals and the central L-values, we explicitly determine the constant of proportionality relating the square norm of a Fourier coefficient of an Arakawa lift with the central L-value. We can relate the square norm with the central value of some L-function of convolution type attached to the lift and a Hecke character. We also discuss the existence of strictly positive central values of the L-functions in our concern.",
keywords = "Central L-values, Fourier coefficients, quaternion unitary group, theta lifts, toral integrals",
author = "Atsushi Murase and Hiroaki Narita",
year = "2016",
month = "1",
day = "1",
doi = "10.1142/S0129167X16500014",
language = "English",
volume = "27",
journal = "International Journal of Mathematics",
issn = "0129-167X",
publisher = "World Scientific Publishing Co. Pte Ltd",
number = "1",

}

TY - JOUR

T1 - Fourier expansion of Arakawa lifting II

T2 - Relation with central L-values

AU - Murase, Atsushi

AU - Narita, Hiroaki

PY - 2016/1/1

Y1 - 2016/1/1

N2 - This is a continuation of our previous paper [Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts, Israel J. Math. 187 (2012) 317-369]. The aim of the paper here is to study the Fourier coefficients of Arakawa lifts in relation with central values of automorphic L-functions. In the previous paper we provide an explicit formula for the Fourier coefficients in terms of toral integrals of automorphic forms on multiplicative groups of quaternion algebras. In this paper, after studying explicit relations between the toral integrals and the central L-values, we explicitly determine the constant of proportionality relating the square norm of a Fourier coefficient of an Arakawa lift with the central L-value. We can relate the square norm with the central value of some L-function of convolution type attached to the lift and a Hecke character. We also discuss the existence of strictly positive central values of the L-functions in our concern.

AB - This is a continuation of our previous paper [Fourier expansion of Arakawa lifting I: An explicit formula and examples of non-vanishing lifts, Israel J. Math. 187 (2012) 317-369]. The aim of the paper here is to study the Fourier coefficients of Arakawa lifts in relation with central values of automorphic L-functions. In the previous paper we provide an explicit formula for the Fourier coefficients in terms of toral integrals of automorphic forms on multiplicative groups of quaternion algebras. In this paper, after studying explicit relations between the toral integrals and the central L-values, we explicitly determine the constant of proportionality relating the square norm of a Fourier coefficient of an Arakawa lift with the central L-value. We can relate the square norm with the central value of some L-function of convolution type attached to the lift and a Hecke character. We also discuss the existence of strictly positive central values of the L-functions in our concern.

KW - Central L-values

KW - Fourier coefficients

KW - quaternion unitary group

KW - theta lifts

KW - toral integrals

UR - http://www.scopus.com/inward/record.url?scp=84959193496&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84959193496&partnerID=8YFLogxK

U2 - 10.1142/S0129167X16500014

DO - 10.1142/S0129167X16500014

M3 - Article

AN - SCOPUS:84959193496

VL - 27

JO - International Journal of Mathematics

JF - International Journal of Mathematics

SN - 0129-167X

IS - 1

M1 - 1650001

ER -