2000 Fiscal Year Final Research Report Summary
Moduli of Kummer varieties and its applications to number theory.
Project/Area Number 
10640043

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Nihon University 
Principal Investigator 
SASAKI Ryuji Nihon University Sci. and Tech., Math., Professor, 理工学部, 教授 (50120465)

Project Period (FY) 
1998 – 2000

Keywords  modular variety / theta function / hermitian theta function / derivative formula / modular equation / hermitian Jacobi form / SaitoKurokawa Conjecture 
Research Abstract 
The aim of this study is to clarify the structure of the modular variety, which is, in the case of genus 1, is the Zariskiclosure of the set {(j(γ), j(nγ))γ's are the points of the upperhalf plane}. Here j(γ) is the jinvariant. When the genus is equal to 2, the invariants are known but rather complicated. So we take the Riemann theta constant θ_<mn>(γ), where m, n are halfintegral vectors and γ is a point in the Siegel upperhalf space, in place of the invariants. Classically the ratio of the θ_<mn>(γ)^2 is called the moduli of the abelian variety associated with γ. We consider the map Θ : S_g → IP^N(N = 2^g  1), Θ(γ) = (… : Θ_<aO>(2γ) : …)(a ∈ 1/2Z^g/Z^g). The image Θ(γ) is essentially nothing but the moduli of γ.Let p be an odd prime. The Zariski closure of the set {(Θ(γ), Θ(pγ))γ ∈ S_g} ⊂ IP^N × IP^N is called the modular variety of degree p and level (2,4). The defining eqations of the modular variety of degree 3 are known. In our study, we get the defining equations of modular variety of degree 7. In the course of the above study, we reach to hermitian theta functions which is closely connected to the moduli of Kummer surfaces. For these hermitian theta functions, we get the derivative formula, which is an analogue of the wellknown Jacobi's derivative formula and Rosenhain's one. After that we begin to study hermitian modular forms and hermitian Jacobi forms and give an analogue of SaitoKurokawa conjecture for hermitian Jacobi forms of degree 1. Our original study is to investigate modular varieties of genus g 【greater than or equal】 2. We have some small results about the structure of these varieties, and will try to study these objects more deeply.

Research Products
(8 results)