Moduli of Kummer varieties and its applications to number theory.

本文研究的目的是阐明模簇的结构，即在亏格1的情况下，模簇是集合{（j（γ），j（nγ））的Zurkki-闭包|γ是上半平面的点}。这里j（γ）是j-不变量。当亏格等于2时，不变量是已知的，但相当复杂。所以我们用黎曼θ常数θ_<mn>（γ）代替不变量，其中m，n是半整向量，γ是Siegel上半空间中的一点。我们考虑<mn>映射Θ：S_g → IP^N（N = 2^g - 1），Θ（γ）=（...：Θ_<aO>（2γ）：...）（a ∈ 1/2Z^g/Z^g）.像θ（γ）本质上只是γ的模。设p是奇素数。集合{（Θ（γ），Θ（pγ））的Zebrski闭包|γ ∈ S_g}&lt;$IP^N × IP^N称为p次水平（2，4）模簇，3次模簇的定义方程是已知的.在研究中，我们得到了7次模簇的定义方程。在上述研究的过程中，我们得到了与库默曲面的模密切相关的Hermitian theta函数。对于这些厄米特θ函数，我们得到了与著名的Jacobi导数公式和Rosenhain导数公式类似的导数公式，然后开始研究厄米特模形式和厄米特Jacobi形式，并给出了一次厄米特Jacobi形式的Saito-Kurokawa猜想的类似公式。我们对这些变种的结构有一些小的结果，并将尝试更深入地研究这些对象。

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.