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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').

本文主要研究自同构L-函数临界值的代数性。在文献中，大多数代数性结果都是用积分表示来证明的。我们目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性问题。我们考虑了Rankin-Selberg L-函数的比率。在G.Harder和A.Raghuram的先前结果中，作者考虑了固定的Rankin-Selberg L函数的比率。相反，我们确定一个临界点并改变自同构表示。在这种情况下，我们猜想这些比属于自同构表示的有理性域，并假设猜想的正确性，证明了D.Blasius和P.Deligne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，在一定的奇偶性和正则性条件下，我们可以证明它是关于自同构表示的阿基米德分量的，并证明了与正则代数尖点自同构表示相关的Betti-Whittaker周期与其逆序的周期关系。这是对偶性下动机的周期不变量之间关系的自同构类比。作为结果，我们证明了GSpin(2n)x GL(n‘)的连续临界L值之比的代数性。