Arithmetic studies on Abelian surfaces

点击翻译按钮获取中文摘要

(1) The classical correspondence between Jacobi forms and Sigel cusp forms of half-integral weights is studied from representation theoretic point of view. The basic tool is Weil representation. The results are published on "On Siegel modular forms of half-integral weights and Jacobi forms" (Trans. A.M.S.351 (1999), pp.735-780).(2) Weil's generalized Poisson summation formula, which is valid only for theta group, is extended to the general paramodular groups. As applications ; 1) a representation theoretic proof of the transformation formula of Riemann's theta series, and 2) the transformation formula of theta series associated with a integral quadratic form with harmonic polynomials. The results will be published on the paper "On an extension of generalized Poisson summation formuls of Weil and its applications" (to appear on Commentarii Math. Univ. Sancti Pauli).(3) Hermite polynomials of multi-variables are defined in two ways through a detailed study of the irreducible decompositio … More n of the Weil representation of Sp (n, R) restricted to the dual pair (U (n), U (1)). As K-type vectors for K=U (n), we will get products of the classical (one-variable) Hermite polynomials which give a complete system of the solutions of the Schrodinger equation of n-dimennsional harmonic ascillator. On the other hand, as K-type vectors for K=U (1), we will get another complete system of the solution of the Schrodinger equation which is not of separated variables. The results will be published on the paper "K-type vectors of Weil representation and generalized Hermite polynomials".(4) Simple proofs for the uniform convergence and the boundedness of the trace of a reproducing kernel of a space of automorphic forms associated with an irreducible integrable unitary representation of a semi-simple linear real Lie group. These results will be published on the paper "On cenvergence and boundedness of reproducing kernel for automorphic forms associated with an integrable unitary representation". Less