\contentsline {section}{\tocsection {}{}{Introduction}}{1}
\contentsline {section}{\tocsection {}{1}{Hecke characters and periods}}{4}
\contentsline {subsection}{\tocsubsection {}{1.1}{Algebraic Hecke characters}}{4}
\contentsline {subsection}{\tocsubsection {}{1.2}{Abelian varieties associated to characters of type $(1,0)$}}{5}
\contentsline {subsection}{\tocsubsection {}{1.3}{Complex periods and special values of $L$-functions}}{8}
\contentsline {subsection}{\tocsubsection {}{1.4}{$p$-adic periods}}{9}
\contentsline {section}{\tocsection {}{2}{$p$-adic $L$-functions and Rubin's formula}}{10}
\contentsline {subsection}{\tocsubsection {}{2.1}{The Katz $p$-adic $L$-function}}{10}
\contentsline {subsection}{\tocsubsection {}{2.2}{$p$-adic Rankin $L$-series}}{12}
\contentsline {subsection}{\tocsubsection {}{2.3}{A $p$-adic Gross-Zagier formula}}{14}
\contentsline {subsection}{\tocsubsection {}{2.4}{A factorisation of the $p$-adic Rankin $L$-series}}{16}
\contentsline {subsection}{\tocsubsection {}{2.5}{Proof of Rubin's Theorem}}{17}
\contentsline {subsection}{\tocsubsection {}{2.6}{Elliptic curves with complex multiplication}}{22}
\contentsline {subsection}{\tocsubsection {}{2.7}{A special case}}{24}
\contentsline {section}{\tocsection {}{}{References}}{25}