Modular Forms and Galois Representations

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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.