Mathematical Sciences: Arakelov Theory

张 调查员打算研究阿拉克洛夫理论。本文提出了以下几个问题：（a）利用Poincare度量的稳定曲线及其模空间的Arakelov理论;（B）小点等分布定理在高维交换簇的Serre开象型定理和Chinberg和Rumely容量理论中的应用;（c）证明Hecke代数和Shimura交换簇的高权CM-圈上的高度的两个定理; d）研究Arakelov zeta函数，并将其应用于Mordell猜想有效版本的概率版本。 这是数论和代数几何交叉领域的研究。 在它的起源，数论研究的是多项式方程的整数解，而代数几何处理的是可以由多项式方程定义的图形。如今，这些领域不仅使用代数方法，而且还使用分析、拓扑学和微分几何方法，相反，这些方法在这些领域以及物理学、理论计算机科学和机器人学中也得到了应用。

Zhang The investigator intends to study Arakelov theory. The following are some proposed problems: a) Arakelov theory of stable curves and their moduli spaces using Poincare metric; b) Application of the equidistribution theorem of small points to Serre's open image type theorem for high dimensional abelian varieties, and to Chinberg and Rumely's capacity theory; c) Verification of two conjectures on Hecke algebras and heights on CM-cycles of high weight for Shimura abelian varieties; d) Study the Arakelov zeta function and apply it to a probabilistic version of the effective version of Mordell conjecture. This is research in the intersection of the fields of number theory and algebraic geometry. In its origin, number theory studied integral solutions of polynomial equations, while algebraic geometry treated figures that could be defined by polynomial equations. Nowadays the fields make use of methods not only from algebra, but also from analysis, topology, and differential geometry, and conversely are finding application in those fields as well as in physics, theoretical computer science, and robotics.