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

本文的研究目的是建立半积分权的新形式理论。我们从1998年4月到2001年3月对这个问题进行了调查。我们得到了以下两个结果。首先，我们对以前的工作做了改进。我们已经有了柯南空间的新形式理论。然而，在某些情况下，非纯新形式出现在柯南空间的新形式的空间中。在一定的周条件下，我们成功地去除了这些非纯粹的新形式。这种方法的主要工具是使用四次狄利克雷字符的扭转算子。其次，我们研究了半积分权值和能被32整除的阶的尖形。在这种情况下，还没有新形式理论。我们引入了与之前完全不同的扭转算子。通过使用这些运算符，我们可以在几种情况下建立新形式理论。此外，我们还与A.Murase教授合作进行了痕量关系的研究。迹关系对新形态理论的构建具有重要意义。但是，我们还不明白为什么存在痕量关系。我们对这个问题做了一个数值实验。因此，我们认为新谷抬升似乎起着重要的作用。现在我们正在调查这一观察结果。

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.