2000 Fiscal Year Final Research Report Summary
Arithmetic studies on Abelian surfaces
Project/Area Number 
11640006

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Miyagi University of Education 
Principal Investigator 
TAKASE Koichi Miyagi University of Education, Department of Mathematics, Aossiciated Professor > 宮城教育大学, 教育学部, 助教授 (60197093)

CoInvestigator(Kenkyūbuntansha) 
YAMADA Haruki Miyagi University of Education, Department of Mathematics, Professor, 教育学部, 教授 (00092578)
TAKEMOTO Hideo Miyagi University of Education, Department of Mathematics, Professor, 教育学部, 教授 (00004408)
SHIRAI Susumu Miyagi University of Education, Department of Mathematics, Professor, 教育学部, 教授 (30115175)

Project Period (FY) 
1999 – 2000

Keywords  Number Theory / Automorphic Forms / Unitary Representation / Theta Series / Weil Representation / Abelian Varieties / Jacobi Forms / PreHomogeneous Vector Space 
Research Abstract 
(1) The classical correspondence between Jacobi forms and Sigel cusp forms of halfintegral 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 halfintegral weights and Jacobi forms" (Trans. A.M.S.351 (1999), pp.735780). (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 multivariables are defined in two ways through a detailed study of the irreducible decompositio
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n of the Weil representation of Sp (n, R) restricted to the dual pair (U (n), U (1)). As Ktype vectors for K=U (n), we will get products of the classical (onevariable) Hermite polynomials which give a complete system of the solutions of the Schrodinger equation of ndimennsional harmonic ascillator. On the other hand, as Ktype 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 "Ktype 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 semisimple 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

Research Products
(12 results)