Modular Forms and Number Theory

在过去的几十年里，模形式得到了深入的研究。它们自然地作为生成函数出现，用于许多不同的算术对象，例如l值（在算术几何和其他地方）、正定二次型的表示数和分区。最突出的是，模形式在怀尔斯最近对费马大定理的证明中发挥了关键作用。在这个建议中，研究者提出研究各种算术对象，其生成函数具有模性质。例如，他期望研究分割的剩余类的同余性质和分布，模l函数中心临界值的代数部分，奇异模，高斯超几何函数的值，数域的类数，椭圆曲线和动机上的某些不变量。在做这个项目时，他期望使用，例如，p进技术和涉及伽罗瓦表示和组合的技术。提议的研究领域是数论。数论是一门经典学科；欧拉和高斯也研究了研究者感兴趣的一些对象。配分函数就是这样一个对象，定义起来特别简单：它计算将一个正整数写成若干个更小的正整数的方法的个数。然而，正如数论中经常出现的情况一样，关于配分函数的算术的许多自然问题已被证明是相当困难的。我们对这一功能的理解的主要进展依赖于模块化形式的复杂现代技术，例如拟议研究中提出的那些技术。

DMS 0600400Matthew BoylanIn the last few decades, modular forms have been studied intensely. They naturally occur as generating functions for many diverse objects of arithmetic interest such as L-values (in arithmetic geometry and elsewhere), representation numbers of positive definite quadratic forms, and partitions. Most prominently, modular forms were key players in Wiles' recent proof of Fermat's Last Theorem. In this proposal, the investigator proposes to study a variety of arithmetic objects whose generating functions have modular properties. For example, he expects to study the congruence properties and distribution in residue classes of partitions, algebraic parts of central critical values of modular L-functions, singular moduli, values of Gaussian hypergeometric functions, class numbers of number fields, and certain invariants attached to elliptic curves and motives. In doing this project, he expects to use, for example, p-adic techniques and techniques involving Galois representations and combinatorics.The area of the proposed research is in number theory. Number theory is a classical subject; some of the objects of interest to the investigator were also studied by Euler and Gauss. One such object, the partition function, is particularly simple to define: it counts the number of ways to write a positive integer as a sum of smaller positive integers. However, as is often the case in number theory, many of the natural questions on the arithmetic of the partition function have proved to be rather difficult. Major advances in our understanding of this function rely on sophisticated modern techniques in modular forms, such as those suggested by the proposed research.