Jacobi forms with CM and applications

TL;DR

This paper introduces Jacobi forms with complex multiplication, explores their construction from Hecke characters, and demonstrates their applications in elliptic curves, partition functions, and related number theory problems.

Contribution

It defines Jacobi forms with complex multiplication and provides new examples, applications, and connections to elliptic curves and partition congruences.

Findings

Constructed a Jacobi form specializing to η(τ)^26

Developed theta blocks associated with CM elliptic curves

Discovered new congruences and cranks for partition functions

Abstract

We define Jacobi forms with complex multiplication. Analogous to modular forms with complex multiplication, they are constructed from Hecke characters of the associated imaginary quadratic field. From this construction we obtain a Jacobi form which specializes to $\eta(\tau)^{26}$ which we present to highlight an open question of Dyson and Serre. We give other examples and applications of Jacobi forms with complex multiplication including constructing theta blocks associated to elliptic curves with complex multiplication and new families of congruences and cranks for certain partition functions.