Mathematical Sciences: Investigations in the Arithmetic of Curves and Surfaces

该项目支持对一些问题的研究 关于曲线和曲面的算术几何，以及 作为可能的高维推广。 研究 详细研究了具有无限自同构的K3曲面 组，围绕在这样的标准高度定义为中心 表面。 该项目将研究p-adic 高度椭圆表面上，扩展最近的结果， 经典高度，可能应用于L系列。 的 研究还将涉及对各种一般立场的研究 代数几何中的定理，数的定量界 亏格至少为2的曲线上的有理点， 发展一种新的计算正则高度的算法， 椭圆曲线，不需要完全分解 判别式。 这个项目是在数论的一般领域， 代数几何 这是一个研究领域， 解集，它是曲线或曲面在较低的 代数多项式族的维数。

This project supports research on a number of problems concerning the arithmetic geometry of curves and surfaces, as well as possible higher-dimensional generalizations. The research involves a detailed study of K3 surfaces with infinite automorphism group, centered around the canonical heights defined on such surfaces. The project will study the variation of the p-adic height on elliptic surfaces, extending recent results for the classical height, with possible applications to L-series. The research will also involve the study of various general position theorems in algebraic geometry, quantitative bounds for the number of rational points on curves of genus at least two, and the development of a new algorithm for computing canonical heights on elliptic curves which does not require a complete factorization of the discriminant. This project is in the general area of number theory and algebraic geometry. This is an area of research that focuses on the solutions sets, which are curves or surfaces in lower dimensions, of families of algebraic polynomials.