moduli of algebraic curves and its applications to numbers theory

本文的目的是在主偏极阿贝尔曲面的模空间的基础上，建立阿贝尔函数的算术理论.我们的课题之一是理解上述模空间，另一个是研究阿贝尔曲面的扭点.现在我们将解释我们的结果.首先，我们研究了亏格为2的具有水平（2，4）结构的超椭圆曲线的模空间与交换曲面的模空间之间的关系。我们证明了这些空间自然地被认为是SO_3（C）的子集。这里θ常数平方的9个正交分量构成一个特殊的正交矩阵。下面我们将解释第二个结果。设τ是Siegel上半空间的2次点。我们用A（tau）和K（tau）表示与tau相关的阿贝尔曲面和库默曲面。假设主极化阿贝尔曲面A（tau）是亏格为2的超椭圆曲线的雅可比簇。设jalpha（tau）= rheta α 0（2 tau）/rheta_<oo>（2 tau）（alpha*1/2ZETA^2/ZETA^2），则库默体表面K（tau）定义在场F（tau）=Q（jalpha（tau））上。由K（τ）的p-扭点之比产生的场记为F_p（τ）：F_p（τ）=Q（reta_<alphao>（2 τ| 2（tau，1）h）/rheta_<oo>（2tau| 2（tau，1）h）; h1/pZETA^4/ZETA^4）。定理设p为奇数正整数. F_p（tau）是F（tau）的Galois扩张.若τ是一般点，则F_p/F的伽罗瓦群同构于下列群：

The aim of our study is to develope an arithmetic theory of abelian functions on the basis of the moduli space of principally polarized abelian surfaces with level (2, 4).One of our subjects is to understand the above moduli space, and the other is to study of torsion points of abeliar surfaces.Now we shall explain our results. First, we invesigate the relation between the moduli apace of hyperelliptic curves, of genus 2, with level (2, 4) structure and that of abelian surfaces. We showed that these spaces are considered as subsets of SO_3 (C), naturally. Here the nine quotients of the square of theta constants form a special orthogonal matrix.Next we shall explain the second result.Let tau be a point of the Siegel upper-half space of degree 2. We denote by A (tau) and K (tau), the abelian surface and the kummer surface associated to tau. Assume that the principally polarized abelian surface A (tau) is the Jacobian variety of a hyperelliptic curve of genus 2. Put jalpha (tau) =rhetaalphaO (2tau) /rheta_<oo> (2tau) (alpha*1/2ZETA^2/ZETA^2), then the kummer surface K (tau) is defined over the field F (tau) =Q (jalpha(tau)). The field generated by the ratio of p-torsion points of K (tau) will be denoted by F_p (tau) : F_p (tau) =Q (rheta_<alphao>(2tau|2(tau, 1)h) /rheta_<oo>(2tau|2(tau, 1)h) ; h1/pZETA^4/ZETA^4). Then we have the following :Theorem Let p be an odd positive integer.1. F_p (tau) is a Galois extension of F (tau).2. If tau is a general point, then the Galois group of F_p/F is isomorphic to the following group :