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In this paper, we construct Shintani lifts from integral weight weaklyholomorphic modular forms to half-integral weight weakly holomorphic modularforms. Although defined by different methods, these coincide with the classicalShintani lifts when restricted to the space of cusp forms. As a side effect,this gives the coefficients of the classical Shintani lifts as new cycleintegrals. This yields new formulas for the $L$-values of Hecke eigenforms.When restricted to the space of weakly holomorphic modular forms orthogonal tocusp forms, the Shintani lifts introduce a definition of weakly holomorphicHecke eigenforms. Along the way, auxiliary lifts are constructed from the spaceof harmonic weak Maass forms which yield a "fractional derivative" from thespace of half-integral weight harmonic weak Maass forms to half-integral weightweakly holomorphic modular forms. This fractional derivative complements theusual $\xi$-operator introduced by Bruinier and Funke.

Shintani lifts and fractional derivatives for harmonic weak Maass forms