Mathematical Sciences: Height Functions and Arithmetic Geometry

本研究涉及等差几何一般领域的问题，这些问题涉及或应用了高度函数理论。研究将包括证明在某些变种上高度小于给定界的点的数目的公式，证明椭圆曲线代数族中点的Neron-Tate高度变化的精确公式，以及证明某些阿贝尔变种上点的Neron-Tate高度的下界。这项研究是在代数算术几何的一般领域。这是一门涉及代数曲面上的积分点或积分坐标的学科。特别地，本研究涉及到计算2属曲线上积分点的数目。

This research concerns questions in the general area of arithmetic geometry which involve or are applications of the theory of height functions. The research will involve proofs of formulas for the number of points having height less than a given bound on certain varieties, a precise formula for the variation of the Neron-Tate height of points in an algebraic family of elliptic curves, and a lower bound for the Neron-Tate height of points on certain abelian varieties. This research is in the general area of algebraic arithmetic geometry. This is a subject which concerns the integral points or integral coordinates which lie on algebraic surfaces. In particular, this research involves counting the number of integral points on curves of genus two.