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

本文主要研究代数几何中的两个问题。第一种方法是对某一不变量进行因式分解，该不变量与N级经典模曲线相关，其中N为正整数。第二个问题是计算德林菲尔德模曲线上的高度特殊点，称为Heegner点。本文研究了确定四元数代数中具有超奇异约化和阶的2阶Drinfeld模的自同态环。这个项目属于代数几何的一般领域。代数几何涉及对曲线和曲面的研究，这些曲线和曲面是作为一个或多个代数方程的解的集合而自然出现的。

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.