保型L関数の特殊値

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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').