\relax \@writefile{toc}{\contentsline {chapter}{\tocchapter {Chapter}{3}{Heegner points on $X_0(N)$}}{29}} \@writefile{lof}{\addvspace {10\p@ }} \@writefile{lot}{\addvspace {10\p@ }} \newlabel{chap:chap3}{{3}{29}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3.1}{Complex multiplication}}{29}} \newlabel{sec:cxmult}{{3.1}{29}} \citation{serre_cm} \newlabel{eqn:normalise_end}{{3.2}{30}} \citation{cassels_frolich} \newlabel{eqn:action_pic}{{3.4}{31}} \newlabel{thm:cft}{{3.3}{31}} \newlabel{eqn:defzgamma}{{3.6}{32}} \newlabel{eqn:deftauprime}{{3.7}{33}} \newlabel{eqn:tautauprime}{{3.8}{33}} \newlabel{thm:main_cm}{{3.5}{33}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3.2}{Heegner points}}{33}} \newlabel{sec:hp}{{3.2}{33}} \newlabel{thm:cm_precise}{{3.6}{33}} \citation{cremona} \newlabel{thm:shimura_reciprocity}{{3.7}{34}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3.3}{Numerical examples}}{34}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3.4}{Properties of Heegner points}}{35}} \newlabel{prop:heegner_cond}{{3.8}{35}} \newlabel{prop:norm_compatibilities}{{3.10}{35}} \newlabel{eqn:tracehecke1}{{3.10}{35}} \citation{gross_kolyvagin} \citation{gross_x0n} \citation{gross_x0n} \newlabel{eqn:tracehecke2}{{3.11}{36}} \newlabel{prop:action_tau}{{3.11}{36}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3.5}{Heegner systems}}{36}} \newlabel{sec:heegner_systems}{{3.5}{36}} \newlabel{def:heegsys}{{3.12}{36}} \newlabel{thm:heegsysheeg}{{3.13}{36}} \newlabel{lemma:finite_torsion}{{3.14}{36}} \citation{gross_x0n} \@writefile{toc}{\contentsline {section}{\tocsection {}{3.6}{Relation with the Birch and Swinnerton-Dyer conjecture}}{37}} \newlabel{eqn:twist}{{3.12}{37}} \newlabel{eqn:local_fact}{{3.13}{37}} \newlabel{thm:rankin_method}{{3.15}{37}} \citation{jacquet} \citation{gross_zagier} \citation{gross_x0n} \citation{jacquet} \citation{rohrlich} \newlabel{eqn:feKchi}{{3.14}{38}} \newlabel{eqn:prod_formula}{{3.15}{38}} \newlabel{conj:heegsys}{{3.16}{38}} \newlabel{thm:sign_formula}{{3.17}{38}} \newlabel{conj:sign_formula}{{3.19}{39}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3.7}{The Gross-Zagier formula}}{39}} \newlabel{sec:gross_zagier}{{3.7}{39}} \citation{gross_zagier} \citation{zhang_asian} \citation{waldspurger} \citation{murty_murty_book} \citation{bfh} \citation{murty_murty_annals} \newlabel{thm:gross_zagier}{{3.20}{40}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3.8}{Kolyvagin's theorem}}{40}} \newlabel{sec:kolyvagin}{{3.8}{40}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3.9}{Proof of the Gross-Zagier-Kolyvagin theorem}}{40}} \citation{cassels_frolich} \citation{cox} \citation{gross_zagier} \citation{gross_kohnen_zagier} \citation{gross_x0n} \citation{zagier1111} \citation{zhang_annals} \citation{zhang_asian} \citation{gross_kolyvagin} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{Further results}}{41}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{Exercises}}{42}} \newlabel{ex3:pico}{{1}{42}} \newlabel{ex3:jintegral}{{2}{42}} \newlabel{ex3:cm_precise}{{4}{42}} \newlabel{ex3:checkaction}{{5}{42}} \newlabel{ex3:commutes}{{6}{42}} \newlabel{ex3:inseperable}{{7}{42}} \citation{mazur_swinnerton-dyer} \newlabel{ex3:compute_heegner}{{8}{43}} \newlabel{ex3:5077}{{9}{43}} \newlabel{ex3:integral_cm}{{10}{43}} \newlabel{ex3:integral_converse}{{11}{43}} \newlabel{ex3:integral_class}{{12}{43}} \newlabel{ex3:twist}{{15}{43}} \newlabel{ex3:lfactors}{{16}{43}} \newlabel{ex3:factorLEH}{{17}{44}} \newlabel{ex3:finitesha}{{18}{44}} \@setckpt{chap3}{ \setcounter{page}{45} \setcounter{equation}{18} \setcounter{enumi}{18} \setcounter{enumii}{3} \setcounter{enumiii}{0} \setcounter{enumiv}{0} \setcounter{footnote}{0} \setcounter{mpfootnote}{0} \setcounter{parentequation}{0} \setcounter{part}{0} \setcounter{chapter}{3} \setcounter{section}{9} \setcounter{subsection}{0} \setcounter{subsubsection}{0} \setcounter{paragraph}{0} \setcounter{subparagraph}{0} \setcounter{figure}{0} \setcounter{table}{0} \setcounter{theorem}{23} }