Graduate Thesis Or Dissertation
 

Existence of Eta-Quotients for Squarefree Levels and Modular Type Supercongruences for Hypergeometric Functions

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  • In this dissertation, we consider two problems in number theory, both relating to modular forms. First we consider when a given modular form can be expressed as a quotient in Dedekind's $\eta$ function. Rouse and Webb \cite{RW} have determined the integers $N \leq 500$ such that the graded ring of modular forms $M(\Gamma_0(N))$ is generated by $\eta$-quotients. Arnold-Roksandich, James, and Keaton \cite{ARJK} provide counts of the number of linearly independent $\eta$-quotients in $M_k(\Gamma_0(p))$ for prime levels $p$. The author, Anderson, Hamakiotes, Oltsik, and Swisher \cite{AAHOS} give necessary and sufficient conditions for the space $M_k(\Gamma_1(N))$ to contain $\eta$-quotients when $N$ is relatively prime to $6$ and is either prime or a product of two distinct primes. In this dissertation we generalize these necessary conditions for all levels $N$ which are relatively prime to $6$. We then show that these conditions are sufficient as well for a large family of squarefree levels $N$ coprime to 6. In the latter chapters of the dissertation we consider congruences between hypergeometric series and Fourier coefficients of modular forms. In 2003, Rodriguez Villegas \cite{RV} conjectured 14 supercongruences between hypergeometric functions arising as periods of certain families of rigid Calabi-Yau threefolds and the Fourier coefficients of weight 4 modular forms. Uniform proofs of these supercongruences were given in 2019 by Long, Tu, Yui, and Zudilin \cite{LTYZ}. Using p-adic techniques of Dwork \cite{Dwork}, they reduce the original supercongruences to related congruences which involve only the hypergeometric series. We generalize their techniques to consider six further supercongruences recently conjectured by Long \cite{Long20}. In particular we prove an analogous version of Long, Tu, Yui, and Zudilin's reduced congruences for each of these six cases. We also conjecture a generalization of Dwork's work which has been observed computationally and which would, together with a proof of modularity for Galois representations associated to our hypergeometric data, yield a full proof of Long's conjectures.
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