Belmont, E., Lee, H., Musat, A., & Trebat-Leder, S. (2014). l-adic properties of partition functions. Monatshefte für Mathematik, 173(1), 1-34.
Work done at 2011 Number theory REU at Emory University with Ken Ono
Summary: Partition functions occupy the juncture between number theory and the theory of modular functions, and serve as a testing ground for deep conjectures relating these two areas. Defined as the number of ways to divide n objects into groups, the partition function p(n) satisfies divisibility relations that are most easily explained using modular functions: for instance, p(5n+4) is always divisible by 5.
Building upon Prof. Ken Ono’s work, in 2011 I used the theory of l-adic modular forms to prove congruences for a whole family of related partition functions, namely, powers of the partition generating function and Andrews’s smallest parts (spt) function. In particular, I developed a suitable modification of Ono’s techniques that work even though the spt function is not directly related to a modular form, but rather to a function with weaker properties called a Maass form. Furthermore, by using l-adic modular forms, I give a conceptual explanation of congruences that were only previous proven on a case-by-case basis (for instance, Andrew’s congruences modulo 5, 7, and 13), and generalize them to other prime powers.