Moduli space of algebraic curves and automorphic forms

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1. We extended the Shottky-Mumford 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 p-adic 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 finitely-generatedness 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.