Modular Forms and Galois Representations

9711005怀尔斯在完成关于半稳定椭圆曲线的谷山-志村猜想的工作后，首席研究员将他的注意力转向了可约表示的研究。这里的问题是证明具有可约残差表示的普通二维伽罗瓦表示必然来自模形式。这个项目属于算术几何的一般领域，这门学科融合了数学中两个最古老的领域：数论和几何。事实证明，这一组合非常富有成效——最近解决了几代人经受住了考验的问题。其诸多后果之一是新的纠错码。这些代码对现代计算机（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.