一个里埃-雅各比-迪里克莱特序列,用于直角组上的尖端形式
1Department of Mathematical Sciences, Durham University, South. Rd., Durham, DH1 3LE U.K.
Research in number theory
|September 22, 2025
概括
这项研究探讨了Dirichlet数列,使用对直角群的 cusp 形式的福里埃-雅各比系数. 它将这些数列连接到L函数,并揭示了特定情况下的欧勒乘积表达式.
科学领域:
- 数学理论 数学理论
- 代表理论 代表理论
- 自体形的形式 自体形的形式
背景情况:
- 形是数论中的基本对象,与模块化形式有着深厚的联系.
- 迪里克莱序列和L函数编码算术信息,在分析数论中至关重要.
- 直角群及其表示在数学各个领域发挥着重要作用.
研究的目的:
- 为了研究与富里埃-雅各比系数相关的迪里克莱序列,用于直角群的尖端形式.
- 为了建立这个迪里克莱序列和标准L函数之间的连接.
- 在特定条件下,为迪里克莱序列推导欧勒乘积表达式.
主要方法:
- 使用里埃-雅各比系数的尖端形式F和G.
- 分析涉及这些系数的迪里克莱特数列,用于正交的签名组 (2,n+2).
- 应用自变形形式和L函数理论中的技术.
主要成果:
- 建立了狄里克莱序列和标准L函数之间的连接,用于Hecke自形F和Maass升降器G.
- 对于某些直角群的迪里克莱序列,导出了明确的欧勒乘积式.
- 恢复了西格尔模块化形式的经典结果,并引入了新的例子.
结论:
- 这项研究为与尖端形状和直角组相关的迪里克莱序列结构提供了新的见解.
- 这些发现扩大了对L函数及其性质的现有知识.
- 这项研究为探索与其他类型的模块化形式 (包括paramodular,Hermitic和quaternionic形式) 的联系开辟了道路.
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