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2000 Fiscal Year Final Research Report Summary

Moduli of Kummer varieties and its applications to number theory.

Research Project

Project/Area Number 10640043
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 Sci. and Tech., Math., Professor, 理工学部, 教授 (50120465)

Project Period (FY) 1998 – 2000
Keywordsmodular variety / theta function / hermitian theta function / derivative formula / modular equation / hermitian Jacobi form / Saito-Kurokawa 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 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.

  • Research Products

    (8 results)

All Other

All Publications (8 results)

  • [Publications] Ryuji SASAKI: "Moduli space of hyperelliptic curves of genus two with level (2, 4) structures and the special orthogonal group of degree three."Kyushu Journal of Mathematics. 53. 333-361 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Ryuji SASAKI: "An arithemtic of modular function fields of degree two."Acta Mathematica et Informatica Univ.Ostraviensis. 7. 79-105 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Ryuji SASAKI: "Derivative formulas for hermitian theta functions of degree two."Japanese Journal of Mathematics. 27. (2001)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Ryuji SASAKI: "S.Kanemitsu and K.Gyory (eds) Numler Theory and Its applications"Kluwer A cademic Publishers. 291-302 (1999)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] Ryuji SASAKI: "Moduli space of hyperelliphic curves of genes two with level (2,4) structures and the special orthogonal group of degree three"Kyushu Journal of Math.. 53. 336-361 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ryuji SASAKI: "An arithmetic of modular function fields of degree two"Acta Math.et Infor.Univ.Ostraviensis. 7. 79-105 (1999)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ryuji SASAKI: "Derivative formulas for hermitian theta functions of degess two."Japanese Journal of Math.. 27. (2001)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] Ryuji SASAKI: "S.Kanemitsu and K.Gyory (eds) Number Theory and Its applications"Kburer Academic Publishers. 291-302 (1999)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2002-03-26  

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