\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}