Eisenstein Series, Eta Quotients and Their Applications in Number Theory
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We use tools from the theory of modular forms and the interplay between Eisenstein series and eta quotients to deal with some number theoretic curiosities. We describe three of them below. Let $k\geq 2$ be an integer and $j$ an integer satisfying $1\leq j \leq 4k-5$. We define a family $\{ C_{j,k}(z) \}_{1\leq j \leq 4k-5} $ of eta quotients, and prove that this family constitute a basis for the space $S_{2k} (\Gamma_0 (12))$. We then use this basis together with certain properties of modular forms at their cusps to prove an extension of the Ramanujan-Mordell formula. We express the newforms in $S_2(\Gamma_0(N))$ for various $N$ as linear combinations of Eisenstein series and eta quotients, and list their corresponding strong Weil curves. We use modularity theorem to give generating functions for the order of $E (\zz_p)$ for these strong Weil curves. We then use our generating functions to deduce congruence relations for the order of $E (\zz_p)$. We determine all the eta quotients in $M_2(\Gamma_0(N))$ for $N \leq 100$. We then determine the Fourier coefficients of four classes of those eta quotients.
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Copyright © 2016 the author(s). Theses may be used for non-commercial research, educational, or related academic purposes only. Such uses include personal study, research, scholarship, and teaching. Theses may only be shared by linking to Carleton University Institutional Repository and no part may be used without proper attribution to the author. No part may be used for commercial purposes directly or indirectly via a for-profit platform; no adaptation or derivative works are permitted without consent from the copyright owner.
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- 2016
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