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arXiv · 1405.4590

Shintani lifts and fractional derivatives for harmonic weak Maass forms

Kathrin Bringmann·Pavel Guerzhoy·Ben Kane
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Abstract

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

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Kathrin Bringmann, Pavel Guerzhoy, Ben Kane. 2014-05-19. Shintani lifts and fractional derivatives for harmonic weak Maass forms. https://arxiv.org/abs/1405.4590

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