\contentsline {section}{\tocsection {}{1}{Introduction}}{1}
\contentsline {section}{\tocsection {}{2}{Background}}{4}
\contentsline {subsection}{\tocsubsection {}{2.1}{The Birch and Swinnerton-Dyer conjecture in low analytic rank}}{4}
\contentsline {subsection}{\tocsubsection {}{2.2}{Oda's period relations and ATR points}}{5}
\contentsline {section}{\tocsection {}{3}{The Birch and Swinnerton-Dyer conjecture for $\@mathbb {Q}$-curves}}{7}
\contentsline {subsection}{\tocsubsection {}{3.1}{Review of $\@mathbb {Q}$-curves}}{7}
\contentsline {subsection}{\tocsubsection {}{3.2}{The main result}}{9}
\contentsline {section}{\tocsection {}{4}{Heegner points on Shimura's elliptic curves}}{11}
\contentsline {subsection}{\tocsubsection {}{4.1}{An explicit Heegner point construction}}{12}
\contentsline {subsection}{\tocsubsection {}{4.2}{Heegner points and ATR cycles}}{16}
\contentsline {subsection}{\tocsubsection {}{4.3}{Numerical examples}}{16}
\contentsline {subsection}{\tocsubsection {}{4.4}{Proof of Proposition 4.3\hbox {}.}}{18}
\contentsline {section}{\tocsection {}{}{References}}{21}