#### Date of Award

Spring 2022

#### Document Type

Open Access Dissertation

#### Department

Mathematics

#### First Advisor

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.

#### Recommended Citation

Warnock, C. D.(2022). *Some Properties and Applications of Spaces of Modular Forms With ETA-Multiplier.* (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/6853