保型L関数の特殊値

本文主要研究自守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证明的。我们目前的研究项目是提供另一种方法来解决代数性问题时，积分表示不可用。我们考虑Rankin-Selberg L-函数的比。在G. Harder和A. Raghuram，作者考虑固定Rankin-Selberg L-函数的比率。相反，我们固定一个临界点，并改变自守表示。在这种情况下，我们证明了比属于自守表示的合理性域，并在假设猜想成立的前提下，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，其中代数性用与模形式相关的motivic周期的乘积表示。对于比的代数性猜想，我们可以在自守表示的阿基米德分支上的某些奇偶性和正则性条件下证明它，并证明了正则代数尖点自守表示的Betti-Whittaker周期与其逆分支之间的周期关系。这是一个自守的类比之间的关系期间不变量的动机下的对偶。作为结果，我们证明了GSpin（2n）× GL（n '）的连续临界L值之比的代数性.

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').