We study the Diophantine geometry of algebraic curves on relative moduli of special linear rank two local systems over surfaces. We prove that the set of integral points on any nondegenerately embedded algebraic curve can be effectively determined. Under natural hypotheses on the embedding in relation to mapping class group dynamics of the moduli space, the set of all imaginary quadratic integral points on the curve is shown to be finite. Our ingredients include a boundedness result for nonarchimedean systoles of local systems and Baker's theory. We also derive a structure theorem for morphisms from the affine line into the moduli space.

中文翻译：

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局部系统模量曲线的算术

我们研究了表面上特殊线性秩为二的局部系统的相对模的代数曲线的丢番图几何。我们证明可以有效地确定任何非退化嵌入代数曲线上的积分点集。在与模空间的映射类群动力学相关的嵌入的自然假设下，曲线上所有虚二次积分点的集合被证明是有限的。我们的成分包括局部系统的非阿基米德收缩的有界结果和贝克的理论。我们还推导出了从仿射线到模空间的态射的结构定理。