\relax 
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}}
\newlabel{eqn:bsd}{{1}{1}}
\newlabel{thm:gzk}{{1.1}{1}}
\citation{zhang}
\citation{yuan-zhang-zhang}
\citation{longo}
\citation{bd-mc}
\citation{DaLo}
\newlabel{thm:zhang}{{1.2}{2}}
\newlabel{thm:longo}{{1.3}{2}}
\newlabel{eqn:rank-parity}{{2}{2}}
\citation{DaLo}
\citation{DaLo}
\citation{DaLo}
\citation{Ri}
\citation{yuan-zhang-zhang}
\citation{tian-zhang}
\citation{shimura-book}
\citation{oda-book}
\citation{DaLo}
\newlabel{conj:main}{{1.4}{3}}
\newlabel{thm:main}{{1.5}{3}}
\citation{wiles}
\citation{taylor-wiles}
\citation{bcdt}
\citation{SkWi}
\citation{Fu}
\citation{JM}
\citation{zhang}
\citation{Ge}
\citation{Pa}
\citation{Ro}
\citation{Ro2}
\citation{Pa}
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Background}}{4}}
\newlabel{sec:background}{{2}{4}}
\newlabel{sec:gzkz}{{2.1}{4}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.1}{The Birch and Swinnerton-Dyer conjecture in low analytic rank}}{4}}
\newlabel{eqn:mod-par-Q}{{3}{4}}
\newlabel{eqn:mod-par-F}{{4}{4}}
\citation{gross-zagier}
\citation{zhang}
\citation{kolyvagin}
\citation{KL2}
\citation{zhang}
\citation{tian-zhang}
\citation{bfh}
\citation{murty-murty}
\citation{bfh}
\citation{Wa}
\citation{DaLo}
\newlabel{sec:atr-cycles}{{2.2}{5}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.2}{Oda's period relations and ATR points}}{5}}
\citation{gaertner}
\citation{oda-book}
\citation{oda-book}
\newlabel{eqn:XC}{{6}{6}}
\newlabel{conj:oda}{{2.1}{6}}
\newlabel{eqn:AJf}{{7}{6}}
\newlabel{eqn:defAJf}{{8}{6}}
\newlabel{conj:atr}{{2.2}{7}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{The Birch and Swinnerton-Dyer conjecture for $\@mathbb {Q}$-curves}}{7}}
\newlabel{sec:bsdQ}{{3}{7}}
\newlabel{Qcurves}{{3.1}{7}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{Review of $\@mathbb {Q}$-curves}}{7}}
\citation{KW}
\citation{Ri}
\citation{Zhao}
\citation{GQ}
\citation{Ri}
\citation{Mi}
\citation{GG}
\newlabel{eqn:mod-par-Qcurves}{{10}{8}}
\newlabel{Ldecomposition}{{11}{8}}
\newlabel{parametrisation}{{12}{8}}
\newlabel{Kfmodule}{{13}{8}}
\newlabel{Weil}{{16}{8}}
\newlabel{conductor}{{17}{8}}
\citation{KL}
\citation{Ri}
\citation{Cremona}
\newlabel{factorisation}{{19}{9}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{The main result}}{9}}
\newlabel{eqn:plain-square}{{20}{9}}
\newlabel{eqn:field-diag}{{21}{10}}
\newlabel{prop:all-about-M}{{3.2}{10}}
\newlabel{thm:yuan-zhang-zhang}{{3.3}{10}}
\newlabel{trivialproductcharacter}{{22}{10}}
\citation{tian-zhang}
\citation{yuan-zhang-zhang}
\citation{tian-zhang}
\citation{yuan-zhang-zhang}
\citation{zhang2}
\citation{shimura-book}
\citation{PARI}
\citation{Cremona}
\citation{pinch}
\newlabel{thm:main-bis}{{3.4}{11}}
\newlabel{eqn:fromEtof}{{23}{11}}
\newlabel{eqn:fromFtoK}{{24}{11}}
\newlabel{eqn:prodOfL}{{25}{11}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Heegner points on Shimura's elliptic curves}}{11}}
\newlabel{sec:heegner}{{4}{11}}
\gdef \LT@i {\LT@entry 
    {1}{19.12497pt}\LT@entry 
    {1}{25.00005pt}\LT@entry 
    {1}{25.00005pt}\LT@entry 
    {1}{25.00005pt}\LT@entry 
    {1}{25.00005pt}\LT@entry 
    {1}{25.00005pt}\LT@entry 
    {1}{30.00006pt}\LT@entry 
    {1}{30.00006pt}\LT@entry 
    {1}{30.00006pt}\LT@entry 
    {1}{30.00006pt}\LT@entry 
    {1}{30.00006pt}\LT@entry 
    {1}{30.00006pt}\LT@entry 
    {1}{25.00005pt}}
\citation{Co}
\newlabel{sec:explicit}{{4.1}{12}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{4.1}{An explicit Heegner point construction}}{12}}
\newlabel{eqn:field-diag-complete}{{26}{12}}
\newlabel{c}{{4.3}{13}}
\newlabel{CM}{{4.4}{13}}
\newlabel{reclaw}{{27}{14}}
\newlabel{f1}{{28}{14}}
\newlabel{f3}{{29}{14}}
\newlabel{equal}{{4.5}{14}}
\newlabel{thm:1}{{4.6}{15}}
\newlabel{zM}{{30}{15}}
\newlabel{Wpar}{{31}{15}}
\citation{DaLo}
\citation{yuan-zhang-zhang}
\citation{Shiota}
\citation{Cremona}
\citation{PARI}
\citation{Shiota}
\citation{Cremona}
\citation{DaLo}
\citation{DaLo}
\citation{DaLo}
\citation{DaLo}
\citation{DaLo}
\citation{DaLo}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{4.2}{Heegner points and ATR cycles}}{16}}
\newlabel{conj:main-oda}{{4.7}{16}}
\newlabel{eqn:main-conj}{{32}{16}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{4.3}{Numerical examples}}{16}}
\gdef \LT@ii {\LT@entry 
    {1}{59.3467pt}\LT@entry 
    {1}{38.57924pt}\LT@entry 
    {1}{30.58777pt}\LT@entry 
    {1}{40.21796pt}\LT@entry 
    {1}{193.42139pt}\LT@entry 
    {1}{19.44434pt}}
\gdef \LT@iii {\LT@entry 
    {1}{63.0133pt}\LT@entry 
    {1}{45.91246pt}\LT@entry 
    {1}{30.58777pt}\LT@entry 
    {1}{40.21796pt}\LT@entry 
    {1}{149.0065pt}\LT@entry 
    {1}{19.44434pt}}
\newlabel{table1}{{2}{17}}
\@writefile{lot}{\contentsline {table}{\numberline {2}{ATR extensions of $\@mathbb {Q}(\sqrt  {29})$ and ATR points on $E_{29}$}}{17}}
\newlabel{table3}{{3}{17}}
\@writefile{lot}{\contentsline {table}{\numberline {3}{ATR extensions of $\@mathbb {Q}(\sqrt  {37})$ and ATR points on $E_{37}$}}{17}}
\gdef \LT@iv {\LT@entry 
    {1}{63.0133pt}\LT@entry 
    {1}{48.02357pt}\LT@entry 
    {1}{30.58777pt}\LT@entry 
    {1}{40.21796pt}\LT@entry 
    {1}{233.86359pt}\LT@entry 
    {1}{19.44434pt}}
\citation{Hecke}
\citation{Daberkow}
\citation{Mi2}
\citation{Cox}
\newlabel{table5}{{4}{18}}
\@writefile{lot}{\contentsline {table}{\numberline {4}{ATR extensions of $\@mathbb {Q}(\sqrt  {41})$ and ATR points on $E'_{41}$}}{18}}
\newlabel{Sc}{{4.4}{18}}
\@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{4.4}{Proof of Proposition 4.3\hbox {}.}}{18}}
\newlabel{disc}{{4.8}{18}}
\newlabel{c-decrease}{{4.9}{18}}
\citation{HSW}
\citation{Hilbert}
\newlabel{openGroup1}{{33}{19}}
\newlabel{openGroup2}{{34}{19}}
\newlabel{c-firstpart}{{4.10}{19}}
\citation{Mi2}
\newlabel{enough}{{4.11}{20}}
\newlabel{uniqueq}{{4.12}{20}}
\bibcite{bcdt}{1}
\bibcite{bd-mc}{2}
\bibcite{bfh}{3}
\bibcite{Cox}{4}
\bibcite{Co}{5}
\bibcite{Cremona}{6}
\bibcite{Daberkow}{7}
\bibcite{DaLo}{8}
\bibcite{ES}{9}
\bibcite{Fu}{10}
\bibcite{gaertner}{11}
\bibcite{Ge}{12}
\bibcite{GG}{13}
\bibcite{GQ}{14}
\bibcite{gross-zagier}{15}
\bibcite{Hecke}{16}
\bibcite{Hilbert}{17}
\bibcite{HSW}{18}
\bibcite{JM}{19}
\bibcite{kolyvagin}{20}
\bibcite{KL}{21}
\bibcite{KL2}{22}
\bibcite{KW}{23}
\bibcite{longo}{24}
\bibcite{Mi}{25}
\bibcite{Mi2}{26}
\bibcite{murty-murty}{27}
\newlabel{rayisring}{{35}{21}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{21}}
\bibcite{oda-book}{28}
\bibcite{PARI}{29}
\bibcite{Pa}{30}
\bibcite{pinch}{31}
\bibcite{Ri}{32}
\bibcite{Ro}{33}
\bibcite{Ro2}{34}
\bibcite{shimura-book}{35}
\bibcite{Shiota}{36}
\bibcite{SkWi}{37}
\bibcite{taylor-wiles}{38}
\bibcite{tian-zhang}{39}
\bibcite{Wa}{40}
\bibcite{wiles}{41}
\bibcite{yuan-zhang-zhang}{42}
\bibcite{zhang}{43}
\bibcite{zhang2}{44}
\bibcite{Zhao}{45}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{12.7778pt}
\newlabel{tocindent1}{17.77782pt}
\newlabel{tocindent2}{29.38873pt}
\newlabel{tocindent3}{0pt}