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1998年4月到2001年3月对这一问题进行了调查。并且我们有以下两个结果。第一，我们对以前的工作进行了改进。我们已经得到了Kohnen空间的新形式理论。然而，在某些情况下，非纯新形式出现在Kohnen空间的新形式空间中。我们在一定的周条件下成功地去除了这种非纯新形式。其主要工具是利用四次Dirichlet特征标的扭转算子。其次，我们研究了半整数权和水平可被32整除的尖点形式。在这种情况下，还没有新形式的理论。我们引入了与以前完全不同的扭转算子。通过使用这些算子，我们可以在几种情况下建立新形式的理论。此外，我们与A.Murase教授合作研究了迹关系。迹关系对于新理论的构建是最重要的，但我们还不明白为什么会存在迹关系，我们对此进行了数值实验。因此，我们认为Shintani Lift似乎发挥了重要作用。现在我们正在对这一观察结果进行调查。

The purpose of our research is to construct a theory of newforms of half-integral weight. We investigated this subject from April 1998 to March 2001. And we have the following two results.First, we made a refinement of the previous work. We had already got a theory of newforms for Kohnen spaces. However, in some cases, non-pure newforms occur in the space of newforms of Kohnen spaces. We succeeded in removing such non-pure newforms under a certain week condition. The main tool of this is to use twisting operators of quartic Dirichlet characters.Next, we investigated cusp forms of half-integral weight and of level divisible by 32. In such cases, there is not a theory of newforms yet.We introduced completely different twisting operators from the previous one.By using these operators, we could make a theory of newforms in several cases.Moreover, we collaborated with Professor A.Murase for investigation of trace relations. Trace relations are most important for contruction of theory of newforms.But, we do not understand why trace relations exist yet.We made an numerical experiment on this subject. As a result, we consider that Shintani lift seems to play an important role. Now we are investigating this observation.