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保型L関数の特殊値

自同构L函数的特殊值

基本信息

批准号：
22KF0214
负责人：
市野篤史
金额：
$ 1.41万
依托单位：
Kyoto University
依托单位国家：
日本
项目类别：
Grant-in-Aid for JSPS Fellows
财政年份：
2023
资助国家：
日本
起止时间：
2023-03-08 至 2025-03-31
项目状态：
未结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-22KF0214/
关键词：
Special L-values Deligne's conjecture Betti-Whittaker periods

项目摘要

Our research mainly focuses on the study of algebraicity of critical values of automorphic L-functions. In the literature, most algebraicity results were proved using the integral representation. Our current research project is to provide yet another approach to tackle the algebraicity problem when integral representation is not available. We consider ratios of Rankin-Selberg L-functions. In the previous result of G. Harder and A. Raghuram, the authors consider the ratios for a fixed Rankin-Selberg L-function. On the contrary, we fix a critical point and vary the automorphic representations. In this case, we conjectured that the ratios belong to the rationality field of the automorphic representations.Assuming the validity of the conjecture, we prove the conjectures proposed by D. Blasius and P. Deligne on the tensor product L-functions and the symmetric power L-functions of modular forms where the algebraicity is expressed in terms of product of motivic periods associated to the modular forms. For the conjecture on the algebraicity of ratios, we can prove it under some parity and regularity conditions on the archimedean component of the automorphic representations.We also proved a period relation between the Betti-Whittaker periods associated to a regular algebraic cuspidal automorphic representation and its contragredient. This is an automorphic analogue of a relation between the period invariants of motives under duality. As a consequence, we prove the algebraicity of the ratios of successive critical L-values for GSpin(2n) x GL(n').

我们的研究主要集中于自同构L函数临界值的代数性研究。在文献中，大多数代数性结果都是使用积分表示来证明的。我们当前的研究项目是提供另一种方法来解决积分表示不可用时的代数问题。我们考虑 Rankin-Selberg L 函数的比率。在 G. Harder 和 A. Raghuram 之前的结果中，作者考虑了固定 Rankin-Selberg L 函数的比率。相反，我们固定一个临界点并改变自同构表示。在这种情况下，我们推测这些比率属于自守表示的理性域。假设该猜想成立，我们证明了 D. Blasius 和 P. Deligne 提出的关于模形式的张量积 L 函数和对称幂 L 函数的猜想，其中代数性用与模形式相关的动机周期的乘积来表示。对于比率的代数性猜想，我们可以在自守表示的阿基米德分量上在一定的宇称性和正则性条件下证明它。我们还证明了与正则代数尖形自守表示相关的Betti-Whittaker周期与其对立关系。这是二元性下动机的周期不变量之间关系的自守类比。因此，我们证明了 GSpin(2n) x GL(n') 的连续临界 L 值之比的代数性。