Eisenstein Series, Eta Quotients and Their Applications in Number Theory

Abstract

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.