Modular Forms and Galois Representations

小行星9711005 在完成了他的工作谷山志村猜想半稳定椭圆曲线，主要研究人员已将注意力转向研究可约表示。这里的问题是要表明，普通的2维伽罗瓦表示与可约剩余表示必然来自模形式。 这个项目福尔斯属于算术几何的一般领域-一个融合了两个最古老的数学领域：数论和几何的主题。 事实证明，这种结合非常富有成效--最近解决了几代人都无法解决的问题。 在它的许多后果是新的纠错码。 这种代码对于现代计算机（硬盘）和光盘都是必不可少的。

9711005 Wiles After completing his work on the Taniyama-Shimura conjecture for semistable elliptic curves, the principal investigator has turned his attention to the study of reducible representations. Here the problem is to show that ordinary 2-dimensional Galois representations with reducible residual representation necessarily come from modular forms. This project falls into the general area of arithmetic geometry -a subject that blends two of the oldest areas of mathematics: number theory and geometry. This combination has proved extraordinarily fruitful - having recently solved problems that withstood generations. Among its many consequences are new error correcting codes. Such codes are essential for both modern computers (hard disks) and compact disks.