Isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level $2$

TL;DR

This paper establishes an isomorphism between Jacobi forms of index D_{2n+1} and elliptic modular forms of level 2, providing explicit Fourier coefficient formulas and constructing related modular forms with shared Hecke eigenvalues.

Contribution

It introduces a novel isomorphism linking Jacobi forms of specific lattice index to elliptic modular forms of level 2, with explicit Fourier coefficient formulas and new modular form constructions.

Findings

Established an explicit isomorphism between Jacobi forms of index D_{2n+1} and elliptic modular forms of level 2.

Derived formulas for Fourier coefficients of Jacobi--Eisenstein series of index D_{2n+1}.

Constructed a holomorphic modular form of weight 3/2 and level 8 from Zagier--Eisenstein series.

Abstract

This paper has three main objectives: (i) To establish an isomorphism between Jacobi forms of index $D_{2n+1}$ (lattice index) and elliptic modular forms of level $2$. (ii) To provide an explicit formula for the Fourier coefficients of Jacobi--Eisenstein series of index $D_{2n+1}$. (iii) To construct a holomorphic modular form of weight $3/2$ and level $8$ (and $4$) from the Zagier--Eisenstein series $\mathscr{F}$ of weight $3/2$ and level $4$. Moreover, we show that the four functions $E^*_2$, $\eta^3$, $\theta^3$ and $\mathscr{F}$ have essentially the same Hecke eigenvalue $1+p$ for any odd prime $p$, where $E^*_2$ is the non-holomorphic Eisenstein series of weight $2$, $\eta$ is the Dedekind eta-function and $\theta$ is the usual theta function. This fact arises as a special case of the isomorphism of (i).