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
Swinnerton-Dyer Conjecture. In this thesis we
explore properties of elliptic curves, a particular family of
modular forms called eta-quotients 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 eta-quotients 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
3-core partitions to the coefficients of a third series by proving
a general theorem about this phenomenon. Lastly, we will see how
these eta-quotients 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 Hasse-Weil
$L$-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
eta-quotients. 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 eta-quotient or a twist of one.
Table of Contents
- Introduction
- Elliptic curves with everywhere good reduction
- 3-cores and modular forms
- Weierstrass mock modular forms and eta-quotients
- Background
- Good Reduction
- Harmonic Maass forms
- Eta-quotients
- 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 3-cores 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
- Eta-quotients
- 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
About this Dissertation
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