Shintani lifts and fractional derivatives for harmonic weak Maass forms

TL;DR

This paper develops new lifts from weakly holomorphic modular forms to half-integral weight forms, introduces a fractional derivative for harmonic weak Maass forms, and derives new formulas for L-values of Hecke eigenforms.

Contribution

It constructs novel Shintani lifts for weakly holomorphic forms and defines a fractional derivative operator extending the Bruinier-Funke $\xi$-operator.

Findings

Shintani lifts coincide with classical lifts on cusp forms

Coefficients of classical Shintani lifts expressed as cycle integrals

New formulas for L-values of Hecke eigenforms

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.