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

半积分权模形式新形式理论的构建

• 批准号: 10640023
• 负责人: UEDA Masaru
• 金额: $ 1.6万
• 依托单位: Nara Women's University
• 依托单位国家: 日本
• 项目类别: Grant-in-Aid for Scientific Research (C)
• 财政年份: 1998
• 资助国家: 日本
• 起止时间: 1998 至 2000
• 项目状态: 已结题
• 来源: https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-10640023/
• 关键词: modular forms; half-integral weight; trace formula; twisting operator

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.

我们的研究目的是构造一种新的半整权理论。我们从1998年4月到2001年3月调查了这个问题。我们得到了以下两个结果：第一，我们对以前的工作做了改进。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。在一定的弱条件下，我们成功地去除了这种不纯的新形态。其主要工具是利用四次Dirichlet特征标的扭算子。其次，我们研究了半整权的尖点形式和水平可被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曲算子。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教授合作研究迹关系。迹关系对于新形式理论的建立是非常重要的，但是，我们还不清楚为什么存在迹关系，我们对这个问题做了一个数值实验。因此，我们认为新谷电梯似乎发挥了重要作用。现在我们正在调查这一观察结果。

项目成果

Masaru Ueda: "On twisting operators and newforms of half-integrel weight III"Commentarii Mathematica University St.Pauli. (次号に掲載決定). (2001)

Masaru Ueda：“关于扭转算子和半积分权的新形式 III”Commentarii Mathematica University St.Pauli（下一期出版）。

Masaru Ueda: "On twisting operators and newforms of half-integral weight III"Commentarii Mathematici University St.Pauli. (in printing). (2001)

Masaru Ueda：“关于扭曲算子和半积分权的新形式 III”评论圣保利数学大学。

Masaru Ueda: "On twisting operators and new forms of half-integral weight III"Commentarii Mathematici University St.Pauli. (次号に掲載決定印刷中). (2001)

Masaru Ueda：“关于扭曲算子和半积分权的新形式 III”Commentarii Mathematici University St.Pauli（下一期出版，目前正在印刷）。