Integral traces of singular values of weak Maass forms

Duke, William; Jenkins, Paul

doi:10.2140/ant.2008.2.573

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Integral traces of singular values of weak Maass forms

William Duke and Paul Jenkins

Vol. 2 (2008), No. 5, 573–593

DOI: 10.2140/ant.2008.2.573

Abstract

We define traces associated to a weakly holomorphic modular form $f$ of arbitrary negative even integral weight and show that these traces appear as coefficients of certain weakly holomorphic forms of half-integral weight. If the coefficients of $f$ are integral, then these traces are integral as well. We obtain a negative weight analogue of the classical Shintani lift and give an application to a generalization of the Shimura lift.

Keywords

weak Maass forms, weakly holomorphic modular forms, traces of singular moduli

Mathematical Subject Classification 2000

Primary: 11F30

Secondary: 11F37

Milestones

Received: 3 December 2007

Revised: 15 May 2008

Accepted: 13 June 2008

Published: 4 July 2008

Authors

William Duke
UCLA Mathematics Department Box 951555 Los Angeles, CA 90095-1555 United States
http://www.math.ucla.edu/~wdduke/
Paul Jenkins
UCLA Mathematics Department Box 951555 Los Angeles, CA 90095-1555 United States
http://www.math.ucla.edu/~jenkins/

Vol. 2, No. 5, 2008