Study on Diophantine problem over function fields and Teichmuller spaces

利用TeichMuller空间，我们研究了函数场上的丢番图问题和黎曼曲面的全纯族，得到了以下结果：(1)我们确定了亏格为2的闭黎曼曲面由某些Kodaira曲面诱导的全纯族的所有全纯截面。(2)对于(g，n)型双曲黎曼曲面S，设B={(x，y)∈S×S|x≠y}，M={(x，y)∈S×S×S|x≠y，y≠z，z≠x}，π：M→B为正则投影。(3)对于黎曼曲面R上的一个全纯族(M，π，R)，我们研究了M的泛覆盖空间的复分析性质。(4)对于给定的负型伪周期映射f，我们在具有单纯性f的穿孔单位圆盘上构造了一个全纯族(M，π^*)。这是松本和π，Δ的一个定理的另一种证明，并给出了构造这些全纯族的一个系统的方法。

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.