\contentsline {section}{\tocsection {}{}{Introduction}}{1}
\contentsline {section}{\tocsection {}{1}{Modular forms and Hida families}}{4}
\contentsline {subsection}{\tocsubsection {}{1.1}{Modular symbols}}{4}
\contentsline {subsection}{\tocsubsection {}{1.2}{Hida theory}}{6}
\contentsline {subsection}{\tocsubsection {}{1.3}{Measure-valued modular symbols}}{8}
\contentsline {subsection}{\tocsubsection {}{1.4}{The Mazur-Kitagawa $p$-adic $L$-functions}}{11}
\contentsline {section}{\tocsection {}{2}{Modular forms on quaternion algebras}}{12}
\contentsline {subsection}{\tocsubsection {}{2.1}{Basic definitions}}{12}
\contentsline {subsection}{\tocsubsection {}{2.2}{The Bruhat-Tits tree}}{16}
\contentsline {subsection}{\tocsubsection {}{2.3}{The Jacquet-Langlands correspondence}}{18}
\contentsline {subsection}{\tocsubsection {}{2.4}{Measure-valued forms}}{19}
\contentsline {subsection}{\tocsubsection {}{2.5}{Forms of weight two and $p$-adic boundary measures}}{21}
\contentsline {subsection}{\tocsubsection {}{2.6}{Indefinite integrals}}{23}
\contentsline {subsection}{\tocsubsection {}{2.7}{Elliptic curves and the $\@mathcal {L}$-invariant}}{26}
\contentsline {section}{\tocsection {}{3}{Hida $p$-adic $L$-functions attached to imaginary quadratic fields}}{27}
\contentsline {subsection}{\tocsubsection {}{3.1}{Optimal embeddings and special values}}{27}
\contentsline {subsection}{\tocsubsection {}{3.2}{Two-variable $p$-adic $L$-functions}}{30}
\contentsline {subsection}{\tocsubsection {}{3.3}{The interpolation formula}}{32}
\contentsline {subsection}{\tocsubsection {}{3.4}{Calculation of derivatives}}{34}
\contentsline {section}{\tocsection {}{4}{Heegner points on Shimura curves}}{35}
\contentsline {subsection}{\tocsubsection {}{4.1}{Shimura curves}}{36}
\contentsline {subsection}{\tocsubsection {}{4.2}{Modular parametrizations}}{36}
\contentsline {subsection}{\tocsubsection {}{4.3}{Heegner points}}{37}
\contentsline {subsection}{\tocsubsection {}{4.4}{Heegner points and derivatives of Hida $L$-functions}}{39}
\contentsline {section}{\tocsection {}{5}{The main result}}{39}
\contentsline {subsection}{\tocsubsection {}{5.1}{A factorisation of $p$-adic $L$-functions }}{39}
\contentsline {subsection}{\tocsubsection {}{5.2}{Proof of the main result}}{41}
\contentsline {section}{\tocsection {}{}{References}}{42}