On the Fourier expansions of Jacobi forms

Howard Skogman

doi:10.1155/s0161171204401185

Howard Skogman

Open Access

https://doi.org/10.1155/s0161171204401185

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Abstract

We use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexesp,p2, andpqfor distinct odd primesp,q. Specifically, we show that for such indexes, a Jacobi form is uniquely determined by one of the associated components of the vector-valued modular form. However, in the case of indexes of the formpqorp2, there are restrictions on which of the components will uniquely determine the form. Moreover, for indexes of the formp, this note gives an explicit reconstruction of the entire Jacobi form from a single associated vector-valued modular form component. That is, we show how to start with a single associated vector component and use specific matrices fromSl2(ℤ)to find the other components and hence the entire Jacobi form. These results are used to discuss the possible modular forms of half-integral weight associated to the Jacobi form for different subgroups.

Highlights

ON THE FOURIER EXPANSIONS OF JACOBI FORMSWe use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexes p, p2, and pq for distinct odd primes p, q
Jacobi forms of weight k and index m over the rational numbers possess a Fourier expansion of the form c(n, r )qnξr, n=0 {r ∈Z|4nm−r 2≥0}
The Fourier coefficients break into congruence classes modulo 2m and this is the basis of the connection to vector-valued modular forms

Summary

ON THE FOURIER EXPANSIONS OF JACOBI FORMS

We use the relationship between Jacobi forms and vector-valued modular forms to study the Fourier expansions of Jacobi forms of indexes p, p2, and pq for distinct odd primes p, q. We show that for such indexes, a Jacobi form is uniquely determined by one of the associated components of the vector-valued modular form. For indexes of the form p, this note gives an explicit reconstruction of the entire Jacobi form from a single associated vector-valued modular form component. We show how to start with a single associated vector component and use specific matrices from Sl2(Z) to find the other components and the entire Jacobi form. These results are used to discuss the possible modular forms of half-integral weight associated to the Jacobi form for different subgroups.

Introduction

HOWARD SKOGMAN