Mathematical Sciences: Analytic Number Theory on Elliptic Curves

该奖项支持哥伦比亚大学Dorian Goldfeld教授在自同构和模形式的解析数论方面的研究。Goldfeld博士计划研究亚纯模函数系数的增长，并希望根据Kloosterman和推导出这些函数的渐近公式。Goldfeld教授的一个相关项目是将模椭圆曲线的算法与与该曲线相关的权值2的Hecke顶点形式的零点之间的最小双曲距离联系起来。非欧几里得平面几何开始于19世纪初，当时只是数学上的一种好奇，但到19世纪末，数学家们已经意识到，许多具有根本重要性的物体在其基本性质上都是非欧几里得的。对非欧几里得平面几何的详细研究产生了现代数学的几个分支，其中模和自同构形式的研究是最活跃的分支之一。这个领域主要关注关于整数的问题，但在使用几何和分析时，它保留了与历史根源的联系。

This award supports the research in the analytic number theory of automorphic and modular forms of Professor Dorian Goldfeld of Columbia University. Dr. Goldfeld plans to investigate the growth of the coefficients of meromorphic modular functions and hopes to derive asymptotic formulas for these in terms of Kloosterman sums. A related project of Professor Goldfeld is to relate the arithmetic of a modular elliptic curve to the minimal hyperbolic distance between the zeros of the Hecke cusp form of weight two associated to this curve. Non-Euclidean plane geometry began in the early nineteenth century as a mathematical curiosity, but by the end of that century, mathematicians had realized that many objects of fundamental importance are non-Euclidean in their basic nature. The detailed study of non-Euclidean plane geometries has given rise to several branches of modern mathematics, of which the study of modular and automorphic forms is one of the most active. This field is principally concerned with questions about the whole numbers, but in its use of geometry and analysis, it retains connection to its historical roots.