| Co-Investigator(Kenkyū-buntansha) |
KAWAUCHI Akio Osaka City University, Graduate School of Science, Professor, 大学院・理学研究科, 教授 (00112524)
KOMORI Youhei Osaka City University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (70264794)
NOGUCHI Junjiro The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (20033920)
MATSUMOTO Yukio The University of Tokyo, Graduate School of Mathematical Sciences, Professor, 大学院・数理科学研究科, 教授 (20011637)
SHIGA Hiroshige Tokyo Institute of Technology, Graduate School of Science and Engineering, Professor, 大学院・理工学研究科, 教授 (10154189)
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| Research Abstract |
Applying Teichmuller space, we studied Diophantine problem over function fields and holomorphic familes of Riemann surfaces, and we obtained the following results :
(1)We determined all the holomorphic sections of holomorphic families of closed Riemann surfaces of genus 2 induced by certain Kodaira surfaces. Using elliptic functions, we got defining equations of these families, and so obtained all the solutions of the Diophantine problem for these defining equations.
(2)For a hyperbolic Riemann surface S of type (g,n), let B={(x,y)∈S×S|x≠y}, M={(x,y,z)∈S×S×S|x≠y,y≠z,z≠x}, and π:M→B the canonical projection. We determined completely types of Bers for elements of monodromy of the holomorphic family (M,π,B).
(3)For a holomorphic family (M,π,R) over a Riemann surface R, we studied complex analytic properties of the universal covering space of M.
(4)For a given pseudo-periodic map f of negative type, we constructed a holomorphic family (M,π,Δ^*) over the punctured unit disc Δ^* with monodromy f. This is an alternative proof for a theorem due to Matsumoto and Montesinos, and gives a systematic method to construct these holomorphic families.
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