Elliptic curves, etaquotients, and Weierstrass mock modular forms Open Access
Clemm, Amanda Susan (2016)
Published
Abstract
The relationship between elliptic curves and modular forms informs many modern mathematical discussions, including the solution of Fermat's Last Theorem and the Birch and SwinnertonDyer Conjecture. In this thesis we explore properties of elliptic curves, a particular family of modular forms called etaquotients and the relationships between them. We begin by discussing elliptic curves, specifically considering the question of which quadratic fields have elliptic curves with everywhere good reduction. By revisiting work of Setzer, we expand on congruence conditions that determine the real and imaginary quadratic fields with elliptic curves of everywhere good reduction and rational $j$invariant. Using this, we determine the density of such real and imaginary fields. In the next chapter, we begin investigating the properties of etaquotients and use this theory to prove a conjecture of Han related to the vanishing of coefficients of certain combinatorial functions. We prove the original conjecture that relates the vanishing of the hook lengths of partitions and the number of 3core partitions to the coefficients of a third series by proving a general theorem about this phenomenon. Lastly, we will see how these etaquotients relate to the Weierstrass mock modular forms associated with certain elliptic curves. Alfes, Griffin, Ono, and Rolen have shown that the harmonic Maass forms arising from Weierstrass $\zeta$functions associated to modular elliptic curves ``encode'' the vanishing and nonvanishing for central values and derivatives of twisted HasseWeil $L$functions for elliptic curves. We construct a canonical harmonic Maass form for the five curves proven by Martin and Ono to have weight 2 newforms with complex multiplication that are etaquotients. The holomorphic part of this harmonic Maass form is referred to as the Weierstrass mock modular form. We prove that the derivative of the Weierstrass mock modular form for these five curves is itself an etaquotient or a twist of one.
Table of Contents
 Introduction
 Elliptic curves with everywhere good reduction
 3cores and modular forms
 Weierstrass mock modular forms and etaquotients
 Background
 Good Reduction
 Harmonic Maass forms
 Etaquotients
 Newforms
 Newforms with complex multiplication
 Weight 2 newforms
 Elliptic curves with everywhere good reduction
 Introduction
 Consructing EGR_Q Quadratic Fields
 Finding Lower Bounds
 Examples
 A conjecture of Han on 3cores and modular forms
 Introdution
 Proof of Theorem 4.1.1
 Relating A(z) and C(z)
 Weierstrass mock modular forms
 Introduction
 Weierstrass mock modular forms
 Etaquotients
 Examples
 N_E=27
 N_E=32
 N_E=36
 N_E=64
 N_E=144
 Proof of Theorem 5.1.1 and Theorem 5.1.2
 Bibliography
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