Mathematical Sciences: Arakelov Theory

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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.