Mathematical Sciences: Galois Representations and Applications

本文研究了椭圆曲线和模曲线的伽罗瓦表示。几年的研究已经产生了大量的这些表现的类别和对其性质的欣赏。PI建议在这些类中链接表示。如果成功实施，这将为Serre-Mazur- Ribet方案提供一个替代方案，该方案绕过了Shimura-Taniyama-Weil猜想。这是数论领域的研究。数论从整数和诸如整数能被整数整除之类的问题开始。它是数学中最古老的领域之一，最初纯粹出于审美原因而追求它。然而，在过去的半个世纪里，它已经成为开发计算机科学新算法和电子学新纠错码的重要工具。

This proposal deals with the study of Galois representations associated to elliptic curves and modular curves. Several years of study have yielded a large number of classes of these representations and an appreciation for their properties. The PI proposes to link representations in these classes. If carried out successfully, this will provide an alternative to the Serre-Mazur- Ribet program that gets around the Shimura-Taniyama-Weil conjecture. This is research in the field of number theory. Number theory starts with the whole numbers and questions such as the divisibility of one whole number by another. It is among the oldest fields of mathematics and it was originally pursued for purely aesthetic reasons. However, within the last half century, it has become an essential tool in developing new algorithms for computer science and new error correcting codes for electronics.