On Jacobi–Weierstrass mock modular forms

Abstract

We construct harmonic weak Maass forms that map to cusp forms of weight $k\ge 2$ with rational coefficients under the ξ-operator. This generalizes work of the first author, Griffin, Ono, and Rolen, who constructed distinguished preimages under this differential operator of weight 2 newforms associated to rational elliptic curves using the classical Weierstrass theory of elliptic functions. We extend this theory and construct a vector-valued Jacobi–Weierstrass ζ-function which is a generalization of the classical Weierstrass ζ-function.