Arithmetic of Modular Forms

摘要 弗雷德·戴蒙德罗格斯大学98 01497 艾希勒·志村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。 志村-谷山-韦伊猜想预言所有定义在有理数上的椭圆曲线都是通过这样的关联出现的。 这意味着可以使用模形式的工具来研究椭圆曲线的算术。最近在A.怀尔斯庆祝证明志村-谷山-韦伊猜想适用于一大类椭圆曲线。 戴蒙德教授将沿着这些路线继续他成功的研究。 他将试图证明某些伽罗瓦表示所产生的模块化形式和研究有关问题的特殊价值的L-功能和同余之间的模块化形式。 这是数学领域的研究，称为数论，特别是椭圆曲线的研究。 数论中的许多重要问题都涉及到仅使用整数比来求解某些方程。这些方程可以与它们的图形相关联，并被认为是曲线。 在几何学上，曲线可以用一个叫做亏格的数字来分类。 最简单的曲线的亏格为零，它们的有理解自古以来就被研究，数学家认为它们已经被很好地理解了。 大约15年前，G。Faltings证明了亏格大于1的曲线只能有1/2个有理解。 其余的曲线有亏格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。在他最近的费马大定理的证明中，安德鲁·怀尔斯也给了数学一个强大的新工具，用于研究椭圆曲线的算术。 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.