Mathematical Sciences: Arithmetic of Elliptic Curves and Related Topics

这个奖项是为了支持哈佛大学诺姆·埃尔基斯博士在算术代数几何方面的研究。Elkies博士建议在有限常数场的函数场上继续研究超奇异曲线的算法；通过仔细观察，他已经在某些高维欧几里得空间中发现了比之前已知的密度更大的球体填充。此外，埃尔基斯博士计划从计算的角度研究显式模块方程，并将其与抽象的高度理论联系起来。这是算术代数几何领域的研究，是一门结合了代数几何和数论技术的学科。在其最初的公式中，代数几何处理的图形可以用最简单的方程，即多项式在平面上定义。数论始于整数和整数能否被整数整除之类的问题。这两个看似相距甚远的主题，实际上从最早的时代起就相互影响，但在过去的四分之一世纪中，相互影响大大增加。算术代数几何领域现在使用了所有现代数学的技术，并在其本身的边界之外产生了相应的影响。

This award is for support of the research in arithmetic algebraic geometry of Dr. Noam Elkies of Harvard University. Dr. Elkies proposes to continue his studies of the arithmetic of supersingular curves over function fields with finite constant field; by looking closely at these, he has already discovered sphere packings in certain high-dimensional Euclidean spaces that are more dense than any previously known. In addition, Dr. Elkies plans to investigate explicit modular equations from a computational viewpoint and connect these with the abstract theory of heights. This is research in the field of arithmetic algebraic geometry, a subject that combines the techniques of algebraic geometry and number theory. In its original formulation, algebraic geometry treated figures that could be defined in the plane by the simplest equations, namely polynomials. Number theory started with the whole numbers and such questions as divisibility of one whole number by another. These two subjects, seemingly so far apart, have in fact influenced each other from the earliest times, but in the past quarter century the mutual influence has increased greatly. The field of arithmetic algebraic geometry now uses techniques from all of modern mathematics, and is having corresponding influence beyond its own borders.