Moduli of Kummer varieties and its applications to number theory.

本研究的目的是阐明模变种的结构，在属1的情况下，它是集合{(j（γ), j(nγ)）的zariski闭包，|γ′是上半平面的点}。这里j（γ）是j不变量。当属等于2时，不变量是已知的，但相当复杂。我们取黎曼常数θ_<mn>(γ)其中m和n是半积分向量γ是西格尔上半空间中的一个点，代替不变量。经典地，θ_<mn>(γ)^2的比值称为与γ相关的阿贝尔变的模。我们认为地图Θ:S_g→IP ^ N (N = 2 ^ g - 1),Θ(γ ) = (... : Θ_ < aO >(2γ):…)(∈1/2Z Z ^ ^ g / g)。图像Θ（γ）本质上就是γ的模。设p是奇素数。集合{（Θ（γ）， Θ（pγ））|γ∈S_g}∧IP^N × IP^N的Zariski闭包称为度p和水平（2,4）的模变。已知3次模变的定义方程。在我们的研究中，我们得到了7次模变的定义方程。在上述研究过程中，我们得到了与Kummer曲面模密切相关的厄米函数。对于这些厄米函数，我们得到导数公式，它类似于著名的雅可比导数公式和罗森海恩导数公式。然后开始研究厄米模形式和厄米雅可比形式，并给出了1次厄米雅可比形式的Saito-Kurokawa猜想的类比。我们最初的研究是研究g【大于或等于】2属的模变种。我们对这些品种的结构有了一些小的研究结果，并将尝试更深入地研究这些物体。

The aim of this study is to clarify the structure of the modular variety, which is, in the case of genus 1, is the Zariski-closure of the set {(j(γ), j(nγ))|γ's are the points of the upper-half plane}. Here j(γ) is the j-invariant. 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 half-integral vectors and γ is a point in the Siegel upper-half 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 well-known 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 Saito-Kurokawa 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.