Project/Area Number 
13640002

Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Hokkaido University 
Principal Investigator 
MATSUMOTO Keiji Hokkaido Univ., Grad.School of Sci., Asso.Prof., 大学院・理学研究科, 助教授 (30229546)

CoInvestigator(Kenkyūbuntansha) 
SHIMADA Ichiro Hokkaido Univ., Grad.School of Sci., Asso.Prof., 大学院・理学研究科, 助教授 (10235616)
SAITO Mutsumi Hokkaido Univ., Grad.School of Sci., Asso.Prof., 大学院・理学研究科, 助教授 (70215565)
MAEDA Yoshitaka Hokkaido Univ., Grad.School of Sci., Asso.Prof., 大学院・理学研究科, 助教授 (60173720)

Project Period (FY) 
2001 – 2003

Project Status 
Completed (Fiscal Year 2003)

Budget Amount *help 
¥3,300,000 (Direct Cost: ¥3,300,000)
Fiscal Year 2003: ¥800,000 (Direct Cost: ¥800,000)
Fiscal Year 2002: ¥1,000,000 (Direct Cost: ¥1,000,000)
Fiscal Year 2001: ¥1,500,000 (Direct Cost: ¥1,500,000)

Keywords  Moduli Space / Theta Function / Hypergeometric Function / Period Map / Configuration Space / Automorphic Form / Fundamental Group / テータ関数 
Research Abstract 
The head of investigator MATSUMOTO Keiji constructed period maps and automorphic forms derived from the inverses of period maps for certain families of algebraic varieties by using Prym varieties of algebraic curves. In fact, it was shown that the period map for the family of smooth cubic surfaces could be expressed in terms of periods of the Prym varieties for curves of genus 10. Automorphic forms on the 4dimensional complex ball giving the inverse of this period map were expressed by theta constants associated to the Prym varieties. For the family of the 4fold coverings of the complex projective line branching at eight points, the period map from this family to the 5dimensional complex ball was constructed by using the Prym varieties of these curves. Automorphic forms on the 5dimensional complex ball giving the inverse of this period map were expressed by theta constants associated to the Prym varieties. SAITO Mutsumi showed that the ring of differential operators on affine tone varieties and the algebra of symmetries of the system of Ahypergeometric differential equations were antiisomorphic, and classified systems of Ahypergeometric differential equations combinatonally under these symmetries. He studied the condition that the graded ring gr(D(R_A)) was finitely generated, and gave the composition factors of the ring R_A of functions on any tone variety as a D(R_A)module. SHIMADA Ichiro showed that if the singularity of each singular fiber was not bad for an algebraic fiber space, the boundary homomorphism from the second homotopy group of the base space to the fundamental group of any general fiber could be constructed. He showed that the fundamental group of the complement of a resultant hypersurface was commutative. He also showed that any supersingular K3 surface could be expressed as a branched double cover of the projective plane.
