\relax \bibstyle{amsalpha} \ifx\hyper@anchor\@undefined \global \let \oldcontentsline\contentsline \gdef \contentsline#1#2#3#4{\oldcontentsline{#1}{#2}{#3}} \global \let \oldnewlabel\newlabel \gdef \newlabel#1#2{\newlabelxx{#1}#2} \gdef \newlabelxx#1#2#3#4#5#6{\oldnewlabel{#1}{{#2}{#3}}} \AtEndDocument{\let \contentsline\oldcontentsline \let \newlabel\oldnewlabel} \else \global \let \hyper@last\relax \fi \citation{MR31:3420} \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}{section.1}} \citation{gross-zagier} \@writefile{toc}{\contentsline {section}{\numberline {2}Derivatives of $L$-functions}{2}{section.2}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Gross-Zagier}{2}{subsection.2.1}} \citation{mccallum} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Index of Heegner points}{3}{subsection.2.2}} \newlabel{bsd-heegner}{{2.3}{3}{Index of Heegner points\relax }{theorem.2.3}{}} \citation{cremona-algorithms} \citation{cremona-mwrank} \citation{cremona-algorithms} \citation{cremona-algorithms} \citation{kolyvagin:euler_systems} \citation{gross:kolyvagin} \@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Efficient Computations}{4}{subsection.2.3}} \citation{serre:propgal} \@writefile{toc}{\contentsline {section}{\numberline {3}Kolyvagin's Method and Consequences}{5}{section.3}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Kolyvagin's approach to ${\unhbox \voidb@x \hbox {{\fontencoding {OT2}\fontfamily {wncyr}\fontseries {m}\fontshape {n}\selectfont Sh}}}_\textrm {tors}$}{5}{subsection.3.1}} \citation{gross:kolyvagin} \citation{grigor} \citation{cha:kolyvagin} \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Computational Difficulties}{6}{subsection.3.2}} \@writefile{toc}{\contentsline {section}{\numberline {4}Weakening the Hypotheses of Kolyvagin's Method}{6}{section.4}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Irreducibility of the mod $p$ representation}{6}{subsection.4.1}} \citation{kolyvagin:euler_systems} \citation{mazur:rational} \citation{serre:propgal} \@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Reducing the hypotheses to statements about torsion}{8}{subsection.4.2}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {4.2.1}The mod $p$ representation}{8}{subsubsection.4.2.1}} \newlabel{weil-pairing-lemma}{{4.2}{8}{The mod $p$ representation\relax }{theorem.4.2}{}} \newlabel{lemma1}{{4.3}{8}{The mod $p$ representation\relax }{theorem.4.3}{}} \newlabel{lemma2}{{4.4}{8}{The mod $p$ representation\relax }{theorem.4.4}{}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {4.2.2}Rational Torsion Points and Rational Isogenies}{9}{subsubsection.4.2.2}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {4.2.3}$K$-rational Torsion Points}{10}{subsubsection.4.2.3}} \newlabel{lemma:stein}{{4.9}{11}{$K$-rational Torsion Points\relax }{theorem.4.9}{}} \citation{gross:kolyvagin} \citation{MR92e:11050} \citation{MR92a:11058} \citation{MR87g:11061b} \citation{MR2020572} \citation{MR81i:12014} \citation{MR2000a:11077} \citation{MR2002a:11051} \@writefile{toc}{\contentsline {subsubsection}{\numberline {4.2.4}General Strategy}{12}{subsubsection.4.2.4}} \@writefile{toc}{\contentsline {subsubsection}{\numberline {4.2.5}Existence of K}{12}{subsubsection.4.2.5}} \citation{silverman:aec} \citation{rubin:main-conjectures} \citation{perrin-riou} \citation{perrin-riou:thesis} \citation{rubin:main-conjectures} \@writefile{toc}{\contentsline {section}{\numberline {5}Elliptic Curves with Complex Multiplication}{13}{section.5}} \@writefile{toc}{\contentsline {section}{\numberline {6}Computations}{14}{section.6}} \citation{cremona:algs} \@writefile{toc}{\contentsline {section}{\numberline {7}Algorithms}{15}{section.7}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.1}Computing the Mordell-Weil group}{15}{subsection.7.1}} \citation{MR2004g:11045} \citation{cremona:algorithms} \bibdata{biblio} \@writefile{toc}{\contentsline {subsection}{\numberline {7.2}p-descents}{16}{subsection.7.2}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.3}Finding Isogenies}{16}{subsection.7.3}} \@writefile{toc}{\contentsline {subsection}{\numberline {7.4}Root Number}{16}{subsection.7.4}}