Experimental Mathematics

Computation of Harmonic Weak Maass Forms

Jan H. Bruinier and Fredrik Strömberg

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Abstract

Harmonic weak Maass forms of half-integral weight have been the subject of much recent work. They are closely related to Ramanujan’s mock theta functions, and 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 automorphymethod 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 have made extensive numerical computations, and the data we obtained are presented in this paper. We expect that experiments based on our data will lead to a better understanding of the arithmetic properties of the Fourier coefficients of harmonic weak Maass forms of half-integral weight.

Article information

Source
Experiment. Math., Volume 21, Issue 2 (2012), 117-131.

Dates
First available in Project Euclid: 31 May 2012

Permanent link to this document
https://projecteuclid.org/euclid.em/1338430825

Mathematical Reviews number (MathSciNet)
MR2931309

Zentralblatt MATH identifier
1304.11028

Subjects
Primary: 11Y35: Analytic computations 11Y40: Algebraic number theory computations 11F30: Fourier coefficients of automorphic forms 11G05: Elliptic curves over global fields [See also 14H52]

Keywords
Modular form harmonic Maass form elliptic curve L-function Weil representation

Citation

Bruinier, Jan H.; Strömberg, Fredrik. Computation of Harmonic Weak Maass Forms. Experiment. Math. 21 (2012), no. 2, 117--131. https://projecteuclid.org/euclid.em/1338430825


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