Optimal Mock Jacobi Theta Functions
Number Theory
2017-03-06 v2
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.
Keywords
Cite
@article{arxiv.1605.04480,
title = {Optimal Mock Jacobi Theta Functions},
author = {Miranda C. N. Cheng and John F. R. Duncan},
journal= {arXiv preprint arXiv:1605.04480},
year = {2017}
}
Comments
68 pages; updated introduction, minor edits, and clarification added to proof of Theorem 1.2.1 in v. 2