MATH Seminar

Title: The partition function modulo prime powers
Seminar: Algebra and Number Theory
Speaker: Matthew Boylan of University of South Carolina
Contact: Zachary A. Kent, kent@mathcs.emory.edu
Date: 2011-08-25 at 3:00PM
Venue: MSC E406
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Abstract: Recently, Folsom, Kent, and Ono discovered surprising general arithmetic properties of values of p(n), the ordinary partition function, modulo prime powers. More precisely, let l \textgreater 3 be prime, and let m be a positive integer. Their work implies systematic linear relations modulo l\^{}m among values of p(n) in certain arithmetic progressions modulo l\^{}b for all odd b \textgreater b (l,m), a constant depending on l and m. In this talk, we prove a refined upper bound on b(l,m). Our bound is sharp in all computed cases. Abstractly, b(l,m) measures the stabilization rate of a certain sequence of modules of modular forms with coefficients reduced modulo l\^{}m. To define these modules, Folsom, Kent, and Ono introduce a new operator, D(l). We obtain our bound by carefully studying how D(l) effects filtrations of the relevant modular forms. This is joint work with John Webb.