Arithmetic of Modular Forms

Eichler Shimura理论及其推广将代数几何中的对象，如椭圆曲线和伽罗瓦表示，与模形式联系起来。Shimura-Taniyama-Weil猜想预测，所有在有理数上定义的椭圆曲线都是通过这种关联发生的。这意味着椭圆曲线的算法可以用模形式的工具来研究。最近在a . Wiles著名的证明上有了实质性的进展，证明了Shimura-Taniyama-Weil猜想适用于一大类椭圆曲线。戴蒙德教授将沿着这条路线继续他成功的研究。他将尝试证明由模形式产生的伽罗瓦表示，并研究l函数的特殊值和模形式之间的同余的相关问题。这是数论领域的数学研究，特别是对椭圆曲线的研究。数论中的许多重要问题都涉及到只使用整数的比率来求某些方程的解。这些方程可以与它们的图形联系起来，并被认为是曲线。几何上，曲线可以用一个叫做“属”的数字来分类。最简单的曲线属零，它们的有理解自古以来就被研究过，数学家们认为它们很好地理解了。大约15年前，G. Faltings证明了任何大于1的属曲线只能有有限多个有理解。其余的曲线属1，称为椭圆曲线。椭圆曲线的有理解的集合可以是有限的，也可以是无限的，并且总是有一个有趣的数学结构。在他最近对费马大定理的证明中，安德鲁·怀尔斯也为研究椭圆曲线的算法提供了一个强大的新工具。在这个项目中，戴蒙德教授将进一步发展这种新方法。

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.