Shintani lifts and fractional derivatives for harmonic weak Maass forms
Shintani lifts and fractional derivatives for harmonic weak Maass forms
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DOI：
10.1016/j.aim.2014.01.015
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发表时间：
2014-04
影响因子：
1.7
通讯作者：
K. Bringmann;P. Guerzhoy;B. Kane
中科院分区：
文献类型：
--
作者：
K. Bringmann;P. Guerzhoy;B. Kane
本文构造了整权弱全纯模形式到半整权弱全纯模形式的Shintani提升。虽然由不同的方法定义，这些符合经典Shintani电梯时，限制在尖点形式的空间。作为副作用，这给出了经典Shintani升降机的系数作为新的循环积分。这就产生了Hecke特征形L值的新公式。当限制到弱全纯模形式的空间正交于尖点形式时，Shintani提升引入了弱全纯Hecke特征形的定义。沿着的方式，辅助电梯构造从调和弱Maass形式的空间产生的“分数阶导数”从半整数重量调和弱Maass形式的空间半整数重量弱全纯模形式。这个分数阶导数补充了Bruinier和Funke引入的通常的算子。
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 theL-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 usualξ-operator introduced by Bruinier and Funke.