Integral traces of singular values of weak Maass forms

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.