\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}