| 研究実績の概要 |
In the literature, most known results on Deligne's conjecture or its automorphic analogue were obtained by cohomological interpretation of integral representation of the L-functions. The purpose of this research project is to investigate another approach to Deligne's conjecture. We consider ratios of Rankin-Selberg L-functions of algebraic automorphic representations of general linear groups. Under some regularity conditions, we prove that these ratios are algebraic at critical points. As applications, we prove new cases of Deligne's conjecture for symmetric power L-functions and tensor product L-functions of elliptic modular forms, which are previously known only for small degree. One technical difficulty in the proof of our main result is the non-vanishing of certain archimedean pairing. We extend the non-vanishing result of B. Sun to non-unitary cohomologically induced representations. The results of this research have been compiled into a paper and published as a preprint.
We also prove the trivialness of the relative period associated to a regular algebraic cuspidal automorphic representation of GL(2n) of orthogonal type. Together with the result of G. Harder and A. Raghuram, this implies the algebraicity of the ratios of successive critical L-values for GSpin(2n) x GL(n’). The results of this research were compiled into a paper and will be published by the International Mathematics Research Notices.
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