Isomorphism between Jacobi forms of index $D_{2n+1}$ and elliptic modular forms of level $2$
Shuichi Hayashida
TL;DR
This work extends the Jacobi–elliptic modular correspondence to Jacobi forms with lattice index $D_r$ ($r$ odd), proving Mocanu’s conjecture by showing an isomorphism of new Jacobi forms with certain level-2/level-1 elliptic modular forms as Hecke modules. It provides an explicit Fourier-coefficient framework for Jacobi-Eisenstein series of index $D_r$ via Cohen-type Eisenstein data and constructs a holomorphic weight $3/2$ level $8$ form from Zagier’s series, unveiling deep links among $\,\eta^3$, $\theta^3$, $\mathscr F$, and $E^*_2$ through shared Hecke eigenvalues. The paper also develops a robust set of maps from Jacobi forms to elliptic modular forms, employing Ikeda lifting, theta decompositions, and Shimura-type correspondences to translate between half-integral and integral weight settings, yielding consequences for arithmetic functions such as sums of divisors and representations as sums of three squares. The results illuminate a cohesive framework connecting lattice-Jacobi forms, Jacobi-Eisenstein series, and classic modular objects, with clear implications for explicit computations and number-theoretic applications.
Abstract
There are three aims in this paper: (i) We show an isomorphism between Jacobi forms of index $D_{2n+1}$ (lattice index) and elliptic modular forms of level $2$. (ii) We give an explicit formula of Fourier coefficients of Jacobi-Eisenstein series of index $D_{2n+1}$. (iii) We construct a holomorphic modular form of weight $3/2$ of level $8$ from the Zagier Eisenstein series $\mathscr{F}$ of weight $3/2$ of level $4$. Moreover, we show that the four functions $E^*_2$, $η^3$, $θ^3$ 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 $2$, $η$ is the Dedekind eta-function and $θ$ is the usual theta function. This fact follows from a special case of the isomorphism of (i). As an application, we give a formula for a sum of the numbers $r_3(n)$, where $r_3(n)$ is the number of representations of an integer $n \geq 0$ as a sum of $3$ squares.