Mathematical Research Letters

Volume 18 (2011)

Number 5

A Maass Lifting of $\Theta^3$ and Class Numbers of Real and imaginary Quadratic Fields

Pages: 1001 – 1012

DOI: http://dx.doi.org/10.4310/MRL.2011.v18.n5.a17

Authors

Robert C. Rhoades (Department of Mathematics, Stanford University, Bldg 380, Stanford, CA 94305, USA)

Matthias Waldherr (Mathematical Institute, University of Cologne, Weyertal 86-90, 50931 Cologne, Germany)

Abstract

We give an explicit construct of a harmonic weak Maass form $F_{\Theta}$ that is a “lift” of $\Theta^3$, where $\Theta$ is the classical Jacobi theta function. Just as the Fourier coefficients of $\Theta^3$ are related to class numbers of imaginary quadratic fields, the Fourier coefficients of the “holomorphic part” of $F_{\Theta}$ are associated to class numbers of real quadratic fields.

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