Optimal Mock Jacobi Theta Functions

TL;DR

This paper classifies and analyzes the structure of optimal mock Jacobi forms of weight one with rational coefficients, revealing their basis, explicit Fourier coefficient expressions, and connections to Ramanujan's mock theta functions and umbral moonshine.

Contribution

It provides a complete classification of optimal mock Jacobi forms of weight one with rational coefficients and links them to singular moduli and Ramanujan's mock theta functions.

Findings

The space of these forms is thirty-four-dimensional.

Fourier coefficients are explicitly expressed via singular moduli.

All Ramanujan's mock theta functions relate to these optimal forms.

Abstract

We classify the optimal mock Jacobi forms of weight one with rational coefficients. The space they span is thirty-four-dimensional, and admits a distinguished basis parameterized by genus zero groups of isometries of the hyperbolic plane. We show that their Fourier coefficients can be expressed explicitly in terms of singular moduli, and obtain positivity conditions which distinguish the optimal mock Jacobi forms that appear in umbral moonshine. We find that all of Ramanujan's mock theta functions can be expressed simply in terms of the optimal mock Jacobi forms with rational coefficients.