Study on automorphic forms, automorphic L-functions and Shintani functions.

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1.Sintani functions and construction of L-functionsWe investigated local Shintani functions on the unitary groups of quaternion skew hermitian forms. Moreover, to construct automorphic L-functions, we calculated a functional equation of Eisenstein series explicitly.2.Automorphic forms on unitary groups of degree three (joint work with A.Murase)(1)Fourier-Jacobi expansion of Kudla lift : Let f be a holomorphic cusp form on U(1,1). S.Kudla constructed a holomorphic cusp form L(f) on U(2,1). We obtained a refined Fourier-Jacobi expansion of L(f) by primitive theta functions. As an application, we gave a criterion for non-vanishing of L(f) in terms of U(1)-period of f.(2)Siegel-Weil formula : We proved Siegel-Weil formula for the pair ((U(2,1),U(2,2)). We do not need regularization process.(3)Inner product formula : The ratio <L(f),L(f)>/<f,f> of Petersson inner prodects of L(f) and f is calculated. It is written by the special value L(f;1) and local data at ramified primes. Consequently we gave another criterion for the non-vanising of L(f).3.Kudla lift from Jacobi formsWe considered Kudla lift as a lifting from Jacobi forms to U(2,1). When the calss number of the quadratic number fields is one, it is compatible with Jacobi Hecke action. As an application of Jacobi form version, we gave another simple proof of Resnikoff-Tai's result on the structure of thegraded algebra of automorphic forms on SU(2,1) in Gaussian field case.