ABSTRACT Fred Diamond Rutgers University 98 01497 Eichler Shimura theory and its generalizations associate objects from algebraic geometry, such as elliptic curves and Galois representations, to modular forms. The Shimura-Taniyama-Weil conjecture predicts that all elliptic curves defined over the rational numbers occur through such an association. This means that the arithmetic of elliptic curves can be studied using tools from modular forms. There has been a substantial progress very recently building on A. Wiles celebrated proof that the Shimura-Taniyama-Weil conjecture holds for a large class of elliptic curves. Professor Diamond will continue his successful investigations along these lines. He will attempt to prove certain Galois representations arise from modular forms and to study related questions about special values of L-functions and congruences between modular forms. This is research in the area of mathematics known as number theory, and in particular the study of elliptic curves. Many important problems in number theory involve finding the solutions to certain equations using only ratios of integers. These equations can be associated with their graphs and thought of as curves. Geometrically curves can be classified by a number called the genus. The simplest curves have genus zero and their rational solutions have been studied since ancient times, to the point where mathematicians consider them well understood. About 15 years ago, G. Faltings showed that any curve with genus greater than one could have only finitely many rational solutions. The remaining curves have genus one and are called elliptic. The collection of rational solutions to an elliptic curve can be finite or infinite, and always has a interesting mathematical structure in its own right. In his recent proof of Fermat's Last Theorem, Andrew Wiles also gave mathematics a powerful new tool for the study of the arithmetic of elliptic curves. In this project Professor Diamond will further develop this new method.
