\contentsline {section}{\tocsection {}{1}{Introduction}}{2}{section.1}
\contentsline {section}{\tocsection {}{2}{Background}}{2}{section.2}
\contentsline {section}{\tocsection {}{3}{Derivatives of $L$-functions}}{3}{section.3}
\contentsline {subsection}{\tocsubsection {}{3.1}{Gross-Zagier}}{3}{subsection.3.1}
\contentsline {subsection}{\tocsubsection {}{3.2}{The Index of the Heegner point}}{4}{subsection.3.2}
\contentsline {subsection}{\tocsubsection {}{3.3}{Precision}}{4}{subsection.3.3}
\contentsline {section}{\tocsection {}{4}{Kolyvagin's Method and Consequences}}{5}{section.4}
\contentsline {subsection}{\tocsubsection {}{4.1}{Kolyvagin's approach to ${\unhbox \voidb@x \hbox {{\fontencoding {OT2}\fontfamily {wncyr}\fontseries {m}\fontshape {n}\selectfont Sh}}}_\textrm {tors}$}}{5}{subsection.4.1}
\contentsline {subsection}{\tocsubsection {}{4.2}{Computational Difficulties}}{6}{subsection.4.2}
\contentsline {section}{\tocsection {}{5}{Weakening the Hypotheses of Kolyvagin's Method}}{6}{section.5}
\contentsline {subsection}{\tocsubsection {}{5.1}{Irreducibility of the mod $p$ representation}}{6}{subsection.5.1}
\contentsline {subsection}{\tocsubsection {}{5.2}{Reducing the hypotheses to statements about torsion}}{7}{subsection.5.2}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.1}{The mod $p$ representation}}{7}{subsubsection.5.2.1}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.2}{Vanishing of Cohomology Groups I}}{8}{subsubsection.5.2.2}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.3}{Vanishing of Cohomology Groups II}}{9}{subsubsection.5.2.3}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.4}{Existence of K}}{10}{subsubsection.5.2.4}
\contentsline {section}{\tocsection {}{6}{Elliptic Curves with Complex Multiplication}}{11}{section.6}
\contentsline {section}{\tocsection {}{7}{Kato's Theorem}}{12}{section.7}
\contentsline {section}{\tocsection {}{8}{Algorithm to Bound ${\unhbox \voidb@x \hbox {{\fontencoding {OT2}\fontfamily {wncyr}\fontseries {m}\fontshape {n}\selectfont Sh}}}$}}{12}{section.8}
\contentsline {section}{\tocsection {}{9}{Other Algorithms}}{13}{section.9}
\contentsline {subsection}{\tocsubsection {}{9.1}{Computing the Mordell-Weil group}}{13}{subsection.9.1}
\contentsline {subsection}{\tocsubsection {}{9.2}{p-descents}}{14}{subsection.9.2}
\contentsline {subsection}{\tocsubsection {}{9.3}{Finding Isogenies}}{14}{subsection.9.3}
\contentsline {subsection}{\tocsubsection {}{9.4}{Root Number}}{14}{subsection.9.4}
\contentsline {section}{\tocsection {}{10}{Results of Computations}}{14}{section.10}
\contentsline {subsection}{\tocsubsection {}{10.1}{Introduction}}{14}{subsection.10.1}
\contentsline {subsection}{\tocsubsection {}{10.2}{The Plan}}{15}{subsection.10.2}
\contentsline {subsection}{\tocsubsection {}{10.3}{Status}}{16}{subsection.10.3}
\contentsline {subsection}{\tocsubsection {}{10.4}{Analysis}}{16}{subsection.10.4}
\contentsline {subsection}{\tocsubsection {}{10.5}{A Potentially Serious Obstruction}}{20}{subsection.10.5}
\contentsline {subsection}{\tocsubsection {}{10.6}{Some Other Questions (for Dick Gross)}}{22}{subsection.10.6}
\contentsline {section}{\tocsection {}{}{References}}{23}{section*.2}