Arithmetic studies on Abelian surfaces

阿贝尔曲面的算术研究

基本信息

批准号：
11640006
负责人：
TAKASE Koichi
金额：
$ 1.28万
依托单位：
Miyagi University of Education
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
1999
资助国家：
日本
起止时间：
1999 至 2000
项目状态：
已结题

项目摘要

(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

(1)从表示论的角度研究了半积分权重的雅可比形式和西格尔尖点形式之间的经典对应关系，其结果发表在《论半积分权重的西格尔模形式》上。 Jacobi 形式”（Trans. A.M.S.351 (1999), pp.735-780）。（2）Weil 的广义泊松求和公式，即仅对theta群有效，扩展到一般的参数模群作为应用；1）黎曼theta级数变换公式的表示理论证明，2）与调和多项式的积分二次形式相关的theta级数变换公式。研究结果将发表在论文《On an extension ofgeneralized Poisson summation Formula of Weil and its applications》（发表在 Commentarii Math. Univ. Sancti）上。 (3) 通过详细研究 Sp (n, R) 的 Weil 表示形式限制于对偶 (U (n)) 的不可约分解，可以用两种方式定义多变量 Hermite 多项式，作为 K=U (n) 的 K 型向量，我们将得到经典（单变量）埃尔米特多项式的乘积，它给出了薛定谔方程的完整解系。另一方面，作为 K=U (1) 的 K 型向量，我们将得到另一个完整的非分离变量薛定谔方程组的解。论文“Weil 表示和广义 Hermite 多项式的 K 型向量”。(4) 相关自同构空间的再生核迹的一致收敛性和有界性的简单证明具有半简单线性实李群的不可约可积酉表示，这些结果将发表在论文“On cvergence and有界性的自同构形式与可积酉表示相关”上。