Mathematical Sciences: RUI: Factorization of Modular Invariants and Heights of Heegner Points

本研究将集中于代数几何中的两个问题。第一种方法是对与特定代数曲线相关的某个不变量进行因式分解，称为N级的经典模曲线，其中N是正整数。第二个问题是Drinfeld模曲线上称为Heegner点的高度特殊点的计算。研究具有超奇异约化的二阶Drinfeld模的自同态环和正定四元数代数的阶。这个项目是在代数几何的一般领域。代数几何研究自然产生的曲线和曲面，作为一个或多个代数方程的解的集合。

This research will focus on two problems in algebraic geometry. The first is to produce a factorization of a certain invariant associated to a particular algebraic curve called the classical modular curve of level N where N is a positive integer. The second problem is the computation of the heights special points, called Heegner points, on Drinfeld modular curves. This investigation will entail the study of the endomorphism rings of rank 2 Drinfeld modules having supersingular reduction and orders in definite quaternion algebras. This project is in the general area of algebraic geometry. Algebraic geometry concerns the study of the curves and surfaces that arise naturally as the sets of solutions of one or more algebraic equations.