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moduli of algebraic curves and its applications to numbers theory

Research Project

Project/Area Number 08640066
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Algebra
Research InstitutionNihon University

Principal Investigator

SASAKI Ryuji  Nihon University, College of Science and Technology, Professor, 理工学部, 教授 (50120465)

Project Period (FY) 1996 – 1997
Project Status Completed (Fiscal Year 1997)
Budget Amount *help
¥700,000 (Direct Cost: ¥700,000)
Fiscal Year 1997: ¥400,000 (Direct Cost: ¥400,000)
Fiscal Year 1996: ¥300,000 (Direct Cost: ¥300,000)
KeywordsTheta functions / moduli / abelian surface / Kummer surface / hyperelliptic curve / Galois gronp / データ函数 / 主偏極アーベル多様体 / テ-タ函数 / j不変量 / 閉リーマン面
Research Abstract

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 :

Report

(3 results)
  • 1997 Annual Research Report   Final Research Report Summary
  • 1996 Annual Research Report

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Published: 1996-04-01   Modified: 2016-04-21  

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