Doctoral Dissertations

5-5-2017

5-2-2017

Keywords

Torsion, Elliptic Curves, Abelian, Quartic, Galois, Number Fields

Álvaro Lozano-Robledo

Liang Xiao

Mathematics

Degree

Doctor of Philosophy

Open Access

Abstract

Let $E$ be an elliptic curve defined over $\mathbb{Q}$. We investigate $E(K)_{\text{tors}}$ for various abelian extensions $K$ of $\mathbb{Q}$. For number fields, a theorem of Merel implies a uniform bound on the size of the torsion subgroup based on the degree of the number field. We discuss a number of results that classify torsion subgroups of elliptic curves over number fields of a fixed degree. We prove a classification of torsion subgroups for elliptic curves $E/\mathbb{Q}$ base extended to quartic Galois number fields. For infinite extensions of $\mathbb{Q}$, the Mordell-Weil theorem no longer applies, and so the torsion subgroup of $E/\mathbb{Q}$ is a priori not even finite. We prove that when base extended to $\mathbb{Q}^{ab}$, the size of the torsion subgroup of an elliptic curve $E/\mathbb{Q}$ is uniformly bounded. Moreover, we classify all groups that arise as $E(\mathbb{Q}^{ab})_{\text{tors}}$ for elliptic curve $E/\Q$.

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