Computation of harmonic weak Maass forms

TL;DR

This paper introduces a numerical algorithm for computing harmonic weak Maass forms of half-integral weight, providing explicit examples related to elliptic curves and enabling further exploration of their arithmetic properties.

Contribution

The authors develop a new automorphy-based algorithm for the numerical computation of harmonic weak Maass forms of weight 1/2, with explicit examples and extensive data analysis.

Findings

Successfully computed harmonic weak Maass forms for specific elliptic curves

Generated data that may lead to new insights into Fourier coefficient properties

Demonstrated the effectiveness of the automorphy method for these computations

Abstract

Harmonic weak Maass forms of half-integral weight are the subject of many recent works. They are closely related to Ramanujan's mock theta functions, their theta lifts give rise to Arakelov Green functions, and their coefficients are often related to central values and derivatives of Hecke L-functions. We present an algorithm to compute harmonic weak Maass forms numerically, based on the automorphy method due to Hejhal and Stark. As explicit examples we consider harmonic weak Maass forms of weight 1/2 associated to the elliptic curves 11a1, 37a1, 37b1. We made extensive numerical computations and the data we obtained is presented in the final section of the paper. We expect that experiments based on our data will lead to a better understanding of the arithmetic properties of the Fourier coefficients.