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2012 On the Fourier coefficients of Jacobi forms of index $N$ over totally real number fields
Hisashi Kojima
Tohoku Math. J. (2) 64(3): 361-385 (2012). DOI: 10.2748/tmj/1347369368

Abstract

Skoruppa and Zagier established a bijective correspondence from the space of Jacobi forms $\phi$ of index $m$ to that of elliptic modular forms $f$ of level $m$. Gross, Kohnen and Zagier formulated this correspondence by means of kernel functions. Moreover, they proved that the squares of Fourier coefficients of $\phi$ are essentially equal to the critical values of the zeta functions $L(s,f,\chi)$ of $f$ twisted by a quadratic character $\chi$.

The purpose of this paper is to prove a generalization of such results concerning liftings and Fourier coefficients of Jacobi forms to the case of Jacobi forms of index $N$ over totally real number fields $F$. Using kernel functions associated with the space of quadratic forms, we shall establish the existence of a lifting from the space of Jacobi forms $\phi$ of index $N$ over $F$ to that of Hilbert modular forms $f$ of level $N$ over $F$. Moreover, we determine explicitly the Fourier coefficients of $f$ from those of $\phi$. We prove that an analogue of Waldspurger's theorem in the case of Jacobi forms of index $N$ over $F$ holds.

Citation

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Hisashi Kojima. "On the Fourier coefficients of Jacobi forms of index $N$ over totally real number fields." Tohoku Math. J. (2) 64 (3) 361 - 385, 2012. https://doi.org/10.2748/tmj/1347369368

Information

Published: 2012
First available in Project Euclid: 11 September 2012

zbMATH: 1284.11086
MathSciNet: MR2979287
Digital Object Identifier: 10.2748/tmj/1347369368

Subjects:
Primary: 11F50
Secondary: 11F30, 11F67

Rights: Copyright © 2012 Tohoku University

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Vol.64 • No. 3 • 2012
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