On the Analytic Properties of the Standard $L$-Function Attached to Siegel-Jacobi Modular Forms

Bouganis, Thanasis; Marzec, Jolanta

doi:10.4171/dm/735

Doc. Math.

2019

DOI: 10.4171/dm/735

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On the Analytic Properties of the Standard $L$-Function Attached to Siegel-Jacobi Modular Forms

Thanasis Bouganis

Jolanta Marzec

Abstract: In this work we study the analytic properties of the standard L-function attached to Siegel-Jacobi modular forms of higher index, generalizing previous results of Arakawa and Murase. Moreover, we obtain results on the analytic properties of Klingen-type Eisenstein series attached to Jacobi groups.

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Cited by 3 publications

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Algebraicity of special L-values attached to Siegel–Jacobi modular forms

Bouganis

Marzec

2020

manuscripta math.

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In this work we obtain algebraicity results on special L-values attached to Siegel–Jacobi modular forms. Our method relies on a generalization of the doubling method to the Jacobi group obtained in our previous work, and on introducing a notion of near holomorphy for Siegel–Jacobi modular forms. Some of our results involve also holomorphic projection, which we obtain by using Siegel–Jacobi Poincaré series of exponential type.

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Algebraicity of special L-values attached to Siegel–Jacobi modular forms

Bouganis

Marzec

2020

manuscripta math.

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On analytic properties of the standard zeta function attached to a vector-valued modular form

Stein

2022

Res. number theory

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We proof a Garrett–Böcherer decomposition of a vector-valued Siegel Eisenstein series $$E_{l,0}^2$$ E l , 0 2 of genus 2 transforming with the Weil representation of $${\text {Sp}}_2({\mathbb {Z}})$$ Sp 2 ( Z ) on the group ring $${\mathbb {C}}[(L'/L)^2]$$ C [ ( L ′ / L ) 2 ] . We show that the standard zeta function associated to a vector-valued common eigenform f for the Weil representation can be meromorphically continued to the whole s-plane and that it satisfies a functional equation. The proof is based on an integral representation of this zeta function in terms of f and $$E_{l,0}^2$$ E l , 0 2 .

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Analytic Properties of Eisenstein Series and Standard -Functions

Stein¹

2020

Nagoya Math. J.

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We prove a functional equation for a vector valued real analytic Eisenstein series transforming with the Weil representation of $\operatorname{Sp}(n,\mathbb{Z})$ on $\mathbb{C}[(L^{\prime }/L)^{n}]$ . By relating such an Eisenstein series with a real analytic Jacobi Eisenstein series of degree $n$ , a functional equation for such an Eisenstein series is proved. Employing a doubling method for Jacobi forms of higher degree established by Arakawa, we transfer the aforementioned functional equation to a zeta function defined by the eigenvalues of a Jacobi eigenform. Finally, we obtain the analytic continuation and a functional equation of the standard $L$ -function attached to a Jacobi eigenform, which was already proved by Murase, however in a different way.

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On the Analytic Properties of the Standard $L$-Function Attached to Siegel-Jacobi Modular Forms

Cited by 3 publications

References 20 publications

Algebraicity of special L-values attached to Siegel–Jacobi modular forms

Algebraicity of special L-values attached to Siegel–Jacobi modular forms

On analytic properties of the standard zeta function attached to a vector-valued modular form

Analytic Properties of Eisenstein Series and Standard -Functions

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