In 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.
