Algebraic formulas for the coefficients of half-integral weight harmonic weak Maass forms

TL;DR

This paper establishes algebraic formulas for coefficients of certain harmonic Maass forms, linking them to traces of singular moduli and providing explicit formulas for partition numbers within class fields.

Contribution

It constructs a theta lift from weight -2 to weight -1/2 harmonic Maass forms and proves algebraicity of their coefficients, extending the understanding of Maass forms and their arithmetic properties.

Findings

Coefficients are traces of singular moduli for weak Maass forms

Derived a formula for partition function p(n) in class fields

Extended theta lift computations to general weights using Kudla-Millson kernel

Abstract

We prove that the coefficients of certain weight -1/2 harmonic Maass forms are traces of singular moduli for weak Maass forms. To prove this theorem, we construct a theta lift from spaces of weight -2 harmonic weak Maass forms to spaces of weight -1/2 vector-valued harmonic weak Maass forms on Mp_2(Z), a result which is of independent interest. We then prove a general theorem which guarantees (with bounded denominator) when such Maass singular moduli are algebraic. As an example of these results, we derive a formula for the partition function p(n) as a finite sum of algebraic numbers which lie in the usual discriminant -24n+1 ring class field. We indicate how these results extend to general weights. In particular, we illustrate how one can compute theta lifts for general weights by making use of the Kudla-Millson kernel and Maass differential operators.