## LSU Doctoral Dissertations

#### Degree

Doctor of Philosophy (PhD)

Mathematics

Dissertation

#### Abstract

Ramanujan in $1920$s discovered remarkable congruence properties of the partition function $p(n)$. Later, Watson and Atkin proved these congruences using the theory of modular forms. Atkin, Gordon, and Hughes extended these works to $k$-colored partition functions. In $2010$, Folsom-Kent-Ono and Boylan-Webb proved the congruences of $p(n)$ by studying a $\ell$-adic module associated with a certain sequence of modular functions which are related to $p(n)$.

Primary goal of this thesis is to generalize the work of Atkin, Gordon, Hughes, Folsom-Kent-Ono, and Boylan-Webb about the partition function to a larger class of partition functions. For this purpose we study a closely related two parameters family of related functions $p_{[1^c\ell^d]}(n)$ for arbitrary integers $c,d$. We can define it in the following way: $\sum_{n=0}^{\infty}p_{[1^c{\ell}^d]}(n)q^n:=\prod_{n=1}^{\infty}\dfrac{1}{(1-q^n)^c(1-q^{\ell n})^d}.$

In this dissertation we prove an infinite family of congruences for the function $p_{[1^c\ell^d]}(n)$ for $\ell=5,7,11,13,$ and $17$. Then we use it to find congruences for $\ell$-regular partitions, $\ell$-core partitions, $\ell$-colored generalized Frobenius partitions.\\

Next, we study the $\ell$-adic module structures related to $p_{[1^c\ell^d]}(n)$. Then we prove an upper bound for the rank of a $\ell$-adic module associated with the partition function $p_{[1^c\ell^d]}(n)$ and use that to discuss $\ell$-adic properties of $p_{[1^c\ell^d]}(n)$.

11-10-2021

Tu, Fang-Ting