Arithmetic studies on Abelian surfaces

(1)从表示理论的角度研究了半积分权值的Jacobi形式与Sigel尖形的经典对应关系。基本工具是Weil表示法。结果发表在“关于半积分权值的Siegel模形式和Jacobi形式”上。A.M.S.351(1999), pp.735 - 780)。(2)将仅对θ群有效的Weil广义泊松求和公式推广到一般的旁模群。作为应用程序；1)黎曼级数的变换公式的表示理论证明，2)与调和多项式的积分二次型相关的变换公式。研究结果将发表在论文“关于Weil的广义泊松求和公式的推广及其应用”上（发表在《数学评论》上）。圣保利大学)。(3)通过对不可约分解的详细研究，用两种方法定义了多变量的Hermite多项式。Sp （n， R）的Weil表示限制在对偶对(U （n)， U(1)）。作为K=U (n)的K型向量，我们将得到经典（单变量）埃尔米特多项式的积，它给出了n维谐振子薛定谔方程解的完整系统。另一方面，作为K=U(1)的K型向量，我们将得到另一个不含分离变量的薛定谔方程解的完整系统。研究结果将发表在论文“Weil表示的k型向量和广义Hermite多项式”上。(4)与半简单线性实李群的不可约可积酉表示相关的自同构形式空间的再现核的迹的一致收敛性和有界性的简单证明。这些结果将发表在论文“关于可积酉表示的自同构形式的再生核的收敛性和有界性”上。少

(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