\contentsline {section}{\tocsection {}{}{Introduction}}{1}
\contentsline {section}{\tocsection {}{1}{Hecke characters and periods}}{9}
\contentsline {subsection}{\tocsubsection {}{1.1}{Algebraic Hecke characters}}{9}
\contentsline {subsection}{\tocsubsection {}{1.2}{Abelian varieties associated to characters of type $(1,0)$}}{10}
\contentsline {subsection}{\tocsubsection {}{1.3}{Complex periods and special values of $L$-functions}}{13}
\contentsline {subsection}{\tocsubsection {}{1.4}{$p$-adic periods}}{14}
\contentsline {section}{\tocsection {}{2}{$p$-adic $L$-functions and Rubin's formula}}{15}
\contentsline {subsection}{\tocsubsection {}{2.1}{The Katz $p$-adic $L$-function}}{15}
\contentsline {subsection}{\tocsubsection {}{2.2}{$p$-adic Rankin $L$-series}}{17}
\contentsline {subsection}{\tocsubsection {}{2.3}{A $p$-adic Gross-Zagier formula}}{19}
\contentsline {subsection}{\tocsubsection {}{2.4}{A factorisation of the $p$-adic Rankin $L$-series}}{21}
\contentsline {subsection}{\tocsubsection {}{2.5}{Proof of Rubin's Theorem}}{22}
\contentsline {subsection}{\tocsubsection {}{2.6}{Elliptic curves with complex multiplication}}{27}
\contentsline {subsection}{\tocsubsection {}{2.7}{A special case}}{29}
\contentsline {section}{\tocsection {}{3}{Chow-Heegner points}}{30}
\contentsline {subsection}{\tocsubsection {}{3.1}{Motives for rational and homological equivalence}}{30}
\contentsline {subsection}{\tocsubsection {}{3.2}{The motive of a Hecke character}}{31}
\contentsline {subsection}{\tocsubsection {}{3.3}{Deligne-Scholl motives}}{32}
\contentsline {subsection}{\tocsubsection {}{3.4}{Modular parametrisations attached to CM forms}}{33}
\contentsline {subsection}{\tocsubsection {}{3.5}{Generalised Heegner cycles and Chow-Heegner points}}{36}
\contentsline {subsection}{\tocsubsection {}{3.6}{A special case}}{37}
\contentsline {section}{\tocsection {}{4}{Chow-Heegner points over $\@mathbb {C}_p$}}{38}
\contentsline {subsection}{\tocsubsection {}{4.1}{The $p$-adic Abel-Jacobi map}}{38}
\contentsline {subsection}{\tocsubsection {}{4.2}{Rationality of Chow-Heegner points over $\@mathbb {C}_p$}}{39}
\contentsline {subsection}{\tocsubsection {}{4.3}{A special case}}{39}
\contentsline {section}{\tocsection {}{5}{Chow-Heegner points over $\@mathbb {C}$}}{42}
\contentsline {subsection}{\tocsubsection {}{5.1}{The complex Abel-Jacobi map}}{42}
\contentsline {subsection}{\tocsubsection {}{5.2}{Numerical experiments}}{43}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.1}{Chow-Heegner points of level $1$}}{44}
\contentsline {subsubsection}{\tocsubsubsection {}{5.2.2}{Chow-Heegner points of prime level}}{45}
\contentsline {section}{\tocsection {}{}{References}}{47}