Date of Award

1-1-2011

Document Type

Campus Access Dissertation

Department

Mathematics

First Advisor

Matthew G Boylan

Abstract

Let n be a positive integer. A partition of n is a sequence of non-increasing positive integers whose sum is n. The partition function, p(n), counts the number of partitions of n. Although p(n) is easy to define, questions on the arithmetic of its values lie much deeper. Ninety years ago, Ramanujan proved the celebrated congruences for p(n) which bear his name. For example, he proved, for all n, that p(5n + 4) == 0 mod 5. Work of Ono and others over the last ten years reveals the extent to which p(n) satisfies congruence phenomena of this type. The unifying framework that explains such properties of p(n) comes from the fact that its values arise naturally as theFourier coefficients of a modular form. Using the theory of modular forms, we study the behavior of p(n) and related partition statistics and relate partition values to other objects of interest in the theory of automorphic forms, the central critical values of L-functions.

Recent work of Folsom, Kent, and Ono reveals, for primes l > 3, that p(n) is l-adically fractal. They show that the generating functions for partition values on particular arithmetic progressions of n are eventually self-similar when viewed reduced modulo powers of l, in effect generalizing Ramanujan's congruences for every power of every prime l &ge 5. Boylan and the author extend this work by closely examining the structure of modules of modular forms associated to these generating functions. A sharp refinement of the "zoom rate", the rate at which the generating functions become self-similar, is proven. The generating functions are also shown to be periodic. New examples of congruences for all primes 13 &le l < 1300 are given in the appendix.

For l in {5,7,11}, we prove congruences modulo l between ratios of partition values and ratios of central critical values of L-functions associated to certain modular forms. The proof uses deep theorems of Shimura and Waldspurger which give a connection between the Fourier coefficients of a half-integer weight newforms and the central critical values of L-functions attached to twists of an integer weight Hecke eigenform. The result can be used to show that these central critical values are non-zero if p(n) is not divisible by l2 for a specific n.

Let k be a positive integer. We say that a partition of n is k-regular if none of its parts is divisible by k. Let b13(n) denote the number of 13-regular partitions of n. Calkin, et al. conjectured that for all integers n and t with &ge 0 and t &ge 2, we haveb13( 3t n + (5* 3t -1 - 1)/2) == 0 mod 3 . We confirm this conjecture by relating the generating function for b13(3n+1) to a modular form and then studying its image under certain operators.

We say that a partition of n is t-core if none of the hook lengths in the Ferrers-Young diagram are divisible by t. We prove a wealth of new congruences for 2t-core partitions modulo 2 by examining the nilpotency of Hecke operators on level 1 cusp forms. Based on extensive calculations we give a conjecture for the structure of Tp-cyclic subspaces for these forms.

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