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\contentsline {section}{\tocsection {}{1}{Introduction and Background}}{2}
\contentsline {subsection}{\tocsubsection {}{1.1}{Elliptic Curve Definitions}}{2}
\contentsline {section}{\tocsection {}{2}{Formal Groups}}{3}
\contentsline {subsection}{\tocsubsection {}{2.1}{The Basics}}{3}
\contentsline {subsection}{\tocsubsection {}{2.2}{So where are the elements?}}{6}
\contentsline {subsection}{\tocsubsection {}{2.3}{More Theory of Formal Groups}}{6}
\contentsline {section}{\tocsection {}{3}{Application to Elliptic Curves: the Finitude of $E(K)_\textrm {tor}$}}{7}
\contentsline {subsection}{\tocsubsection {}{3.1}{An Elliptic Curve's Formal Group}}{7}
\contentsline {subsection}{\tocsubsection {}{3.2}{Reduction on Elliptic Curves}}{8}
\contentsline {section}{\tocsection {}{4}{Statements of Two Theorems About Torsion}}{11}
\contentsline {section}{\tocsection {}{5}{Torsion Points on Elliptic Curves: Algorithmic Questions}}{12}
\contentsline {subsection}{\tocsubsection {}{5.1}{The Nagell-Lutz Theorem}}{12}
\contentsline {subsection}{\tocsubsection {}{5.2}{Doud's Algorithm}}{13}
\contentsline {section}{\tocsection {}{6}{Background on Modular Abelian Varieties}}{14}
\contentsline {subsection}{\tocsubsection {}{6.1}{Modular Groups \& Modularity}}{14}
\contentsline {subsection}{\tocsubsection {}{6.2}{Hecke Operators and Modular Symbols}}{16}
\contentsline {section}{\tocsection {}{7}{Computations with Modular Abelian Varieties}}{17}
\contentsline {subsection}{\tocsubsection {}{7.1}{Finding the Quotients of $J_0(N)$}}{17}
\contentsline {subsection}{\tocsubsection {}{7.2}{The Torsion Multiple}}{17}
\contentsline {section}{\tocsection {}{8}{Examples}}{18}
\contentsline {subsection}{\tocsubsection {}{8.1}{Bounds using Good Reduction}}{20}
\contentsline {subsection}{\tocsubsection {}{8.2}{Conjectures}}{21}
\contentsline {section}{\tocsection {}{9}{Acknowledgments}}{22}
\contentsline {section}{\tocsection {}{}{References}}{23}