Construction of newform theory for modular forms of half-integral weight

我们研究的目的是建立一种新的半整数权理论。我们从2002年4月到2005年3月对这一课题进行了调查。首先，我们证明了任意偶数能级和任意偶数导体情形下扭曲Hecke算子的迹恒等式。正如我们已经猜想的那样，扭曲Hecke算子的迹是由Hecke算子迹和Atkin-Lehner整权算子迹的线性组合表示的。其次，我们成功地建立了半整权的新形式理论，为了得到新形式的半整权理论，我们需要完整地描述旧形式的空间，然后必须得到半整权尖点形式的傅立叶系数的某种非零性。我们利用2的模幂商环上的亚普勒群的表示理论得到了这种非零性。我们的最终目的是得到任意水平N的半整权的新形式。为此，我们必须将上述非零性推广到模N的任意剩余类的环上。我们现在正在研究这种非零性。

The purpose of our research is to construct a theory of newforms of half-integral weight. We investigated this subject from April 2002 to March 2005. And we have the following results.First, we proved trace identities of the twisted Hecke operators in the case of arbitrary even levels and any even conductors. As we already conjectures, the traces of the twisted Hecke operators are represented by linear combinations of traces of Hecke operators and Atkin-Lehner operators of integral weight.Next, we successfully established a theory of newform of half-integral weight in the case that levels are powers of 2.In order to get a theory of newforms of half-integral weight, we need to completely describe spaces of oldforms, and then for that, we must obtain a certain non-vanishing property of Fourier coefficients of cusp forms of half-integral weight. We got such non-vanishing properties by using representation theory of Metaplectic groups over quotient rings modulo powers of 2.Our final purpose is to get a theory of newform of half-integral weight for arbitrary levels N. For that, we must extend the above non-vanishing properties for arbitrary rings of residue classes modulo N. We are now investigating such non-vanishing properties.