Research on automorphic representations and L-functions

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In late 70's, Saito and Kurokawa independently conjectured that there should be a lifting from elliptic modular forms to Siegel modular forms of degree 2. This conjecture was proved by Maass, Andrianov, Eichler, Zagier, Piatetsky-Shapiro and others in early 8O's. This is now called the Saito-Kurokawa lifts. In this research project, the author generalized the Saito-Kurokawa lifts to higher degrees, and gave an explicit Fourier coefficient formula. Moreover, the author proved an analogous lifting in hermitian modular case, and obtained a Fourier coefficient formula.The pullback of the Siegel modular form we constructed and be thought of a kernel function, and the author constructed another lifting by means of this kernel function. This lifting is now called the Miyawaki lifting.The author calculated the L-function of the Miyawaki lifting and gave a conjecture on the inner product of the Miyawaki lifting.The triple product L-function is investigated by many authors after Garrett's discovery of the integral expression.The special values of the triple product L-function has a dichotomy, the definite case and the indefinite case.The indefinite case is more difficult, and the result of Harris and Kudla seems the only result about the indefinite case.In author's joint work with Atsushi Ichino, it was shown that some indefinite special value of the triple product L-function can be expressed as the inner product of a hermitian lifting and the Saito-Kurokawa lifting.This result is compatible with the Gross-Prasad conjecture.The author is investigating a refinement of the Gross-Prasad conjecture in a joint work with Ichino.