Project/Area Number  09640047 
Research Category 
GrantinAid for Scientific Research (C)

Allocation Type  Singleyear Grants 
Section  一般 
Research Field 
Algebra

Research Institution  Saga University 
Principal Investigator 
ICHIKAWA Takashi Saga University, Faculty of Science and Engneering, Professor, 理工学部, 教授 (20201923)

CoInvestigator(Kenkyūbuntansha) 
HIROSE Susumu Saga University, Faculty of Science and Engneering, Lecturer, 理工学部, 講師 (10264144)
UEHARA Tsuyoshi Saga University, Faculty of Science and Engneering, Professor, 理工学部, 教授 (80093970)
MITOMA Itaru Saga University, Faculty of Science and Engneering, Professor, 理工学部, 教授 (40112289)
NAKAHARA Toru Saga University, Faculty of Science and Engneering, Professor, 理工学部, 教授 (50039278)
TANAKA Tatsuji Saga University, Faculty of Science and Engneering, Professor, 理工学部, 教授 (80039370)

Project Period (FY) 
1997 – 1998

Project Status 
Completed(Fiscal Year 1998)

Budget Amount *help 
¥2,700,000 (Direct Cost : ¥2,700,000)
Fiscal Year 1998 : ¥1,100,000 (Direct Cost : ¥1,100,000)
Fiscal Year 1997 : ¥1,600,000 (Direct Cost : ¥1,600,000)

Keywords  Algebraic curve / Period integral / Theta function / Soliton equation / Moduli space / Automophic form / Teichmueller groupoid / Galois action / 一意化 / フュージング・ムーヴ / リーマン面 / ショットキー群 / p進解析 / テタ関数 
Research Abstract 
1. We extended the ShottkyMumford unifonnization theory on algebraic curves to the case that the base ring consists of formal power series over the rational integer ring, and constructed concretely a deformation of any degenerate curve over this power series ring. Further, we gave a method to caluculate differential forms and period integrals of this deformation. 2. We constructed analytic curves of infinite genus over local fields as uniformizations of infinitely generated Schottky groups, and gave an expression of differential forms and period integrals of these curves. As its application, we showed that the theta functions of padic analytic curves of infinite genus generate solutions of the KP equation which is one of soliton equations. 3. Using the result in 1, we showed the finitelygeneratedness of the ring of automorphic forms over the rational integer ring on the moduli space of algebraic curves ( : automorphic functions on the Teichniueller space), and described the structure of this ring by the ring of Siegel modular forms in the genus 2 and 3 cases. 4. Using the result in 1, we compared the parameters attached to different degenerations of a given curve in the category of formal geometry over the rational integer ring. As its application to Grothendieck's conjecture, we constructed a natural base set of the arithmetic Teichmueller groupoid, and gave a description of Ahe Galois action on this groupoid.
