Theses and Dissertations

Spring 2022

Document Type

Open Access Dissertation

Mathematics

Matthew Boylan

Abstract

This dissertation considers two topics. In the first part of the dissertation, we prove the existence of fourteen congruences for the $p$-core partition function of the form given by Garvan in \cite{G1}. Different from the congruences given by Garvan, each of the congruences we give yield infinitely many congruences of the form $$a_p(\ell\cdot p^{t+1} \cdot n + p^t \cdot k - \delta_p) \equiv 0 \pmod \ell.$$ For example, if $t \geq 0$ and $\sfrac{m}{n}$ is the Jacobi symbol, then we prove $$a_7(7^t \cdot n - 2) \equiv 0 \pmod 5, \text{ \ \ if \bfrac{n}{5} = 1 and \bfrac{n}{7} = -1}.$$ It follows that for all natural numbers $n$ and for $k \in \{6,19,24,26,31,34\}$, $$a_7(5\cdot7^{t+1}\cdot n + 7^t\cdot k - 2) \equiv 0 \pmod 5.$$

In the second part of the dissertation, we give results on where Hecke operators map spaces of modular forms which arise as multiples of eta-quotients. Let ${N\in \{1, 2, 3, 4, 5, 6, 8, 9\}}$ and let $f(z)$ be a level $N$ holomorphic eta quotient with integer weight. Then we precisely describe how $T_n$ with $\gcd(n, 6) = 1$ permute subspaces of the form $$\{f(Dz)F(Dz) : F(z) \in M_w(\Gamma_0(N), \chi)\}.$$ Subspaces of this type play a significant role in recent works \cite{A1, A2, B, BB, G2, Y1, Y2, ZZ}, primarily for $N = 1$ and with applications, for example, to congruences for partition functions.

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