Elliptic curves, eta-quotients, and Weierstrass mock modular forms 公开
Clemm, Amanda Susan (2016)
Permanent URL: https://etd.library.emory.edu/concern/etds/xs55md034?locale=zh 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 andSwinnerton-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 ofSetzer, 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 certaincombinatorialfunctions. 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, andRolenhave shown that the harmonicMaassforms arising from Weierstrass$\zeta$-functions associated to modular elliptic
curves ``encode'' the vanishing andnonvanishingfor central values and derivatives
of twistedHasse-Weil$L$-functions for elliptic curves. We construct
a canonical harmonicMaassform for the
five curves proven by Martin and Ono to have weight 2newformswith complex multiplication that are
eta-quotients. Theholomorphicpart of
this harmonicMaassform 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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