\contentsline {section}{\tocsection {}{}{Introduction}}{2}
\contentsline {section}{\tocsection {}{1}{Preliminaries}}{6}
\contentsline {subsection}{\tocsubsection {}{1.1}{Algebraic modular forms}}{6}
\contentsline {subsection}{\tocsubsection {}{1.2}{Modular forms over $\@mathbf {C}$}}{9}
\contentsline {subsection}{\tocsubsection {}{1.3}{$p$-adic modular forms}}{11}
\contentsline {subsection}{\tocsubsection {}{1.4}{Elliptic curves with complex multiplication}}{12}
\contentsline {subsubsection}{\tocsubsubsection {}{}{Cohomology}}{12}
\contentsline {subsubsection}{\tocsubsubsection {}{}{Isogenies}}{13}
\contentsline {subsection}{\tocsubsection {}{1.5}{Values of modular forms at CM points}}{14}
\contentsline {section}{\tocsection {}{2}{Generalised Heegner cycles}}{15}
\contentsline {subsection}{\tocsubsection {}{2.1}{Kuga-Sato varieties}}{15}
\contentsline {subsection}{\tocsubsection {}{2.2}{The variety $X_r$ and its cohomology}}{18}
\contentsline {subsection}{\tocsubsection {}{2.3}{Definition of the cycles}}{19}
\contentsline {subsection}{\tocsubsection {}{2.4}{Relation with Heegner cycles and $L$-series}}{19}
\contentsline {section}{\tocsection {}{3}{$p$-adic Abel-Jacobi maps}}{20}
\contentsline {subsection}{\tocsubsection {}{3.1}{The \'etale Abel-Jacobi map}}{20}
\contentsline {subsection}{\tocsubsection {}{3.2}{The comparison isomorphism}}{21}
\contentsline {subsection}{\tocsubsection {}{3.3}{Extensions of filtered Frobenius modules}}{22}
\contentsline {subsection}{\tocsubsection {}{3.4}{The $p$-adic Abel-Jacobi map}}{23}
\contentsline {subsection}{\tocsubsection {}{3.5}{de Rham cohomology over $p$-adic fields}}{24}
\contentsline {subsection}{\tocsubsection {}{3.6}{The Coleman primitive}}{27}
\contentsline {subsection}{\tocsubsection {}{3.7}{$p$-adic integration and the $p$-adic Abel-Jacobi map }}{28}
\contentsline {subsection}{\tocsubsection {}{3.8}{Calculation of the Coleman primitive}}{31}
\contentsline {section}{\tocsection {}{4}{Period integrals and central values of Rankin-Selberg $L$-functions}}{33}
\contentsline {subsection}{\tocsubsection {}{4.1}{Rankin $L$-series and their special values}}{33}
\contentsline {subsection}{\tocsubsection {}{4.2}{Differential operators}}{38}
\contentsline {subsection}{\tocsubsection {}{4.3}{Period integrals and values at CM points}}{39}
\contentsline {subsection}{\tocsubsection {}{4.4}{Explicit theta lifts}}{42}
\contentsline {subsection}{\tocsubsection {}{4.5}{Seesaw duality and the Siegel-Weil formula}}{49}
\contentsline {subsection}{\tocsubsection {}{4.6}{Local zeta integrals}}{51}
\contentsline {subsubsection}{\tocsubsubsection {}{4.6.1}{Case I: $q \nmid cN d_K $.}}{52}
\contentsline {subsubsection}{\tocsubsubsection {}{4.6.2}{Case II: $q\mid c$. }}{52}
\contentsline {subsubsection}{\tocsubsubsection {}{4.6.3}{Case III: $q^n || N$ with $n\ge 2$.}}{53}
\contentsline {subsubsection}{\tocsubsubsection {}{4.6.4}{ Case IV: $q || N$, $q\nmid d_K$.}}{54}
\contentsline {subsubsection}{\tocsubsubsection {}{4.6.5}{Case V: $q || N$, and $q\mid d_K$.}}{55}
\contentsline {subsubsection}{\tocsubsubsection {}{4.6.6}{Case VI: $q\mid d_K$, $q\nmid N$. }}{56}
\contentsline {subsection}{\tocsubsection {}{4.7}{The explicit form of Waldspurger's formula}}{56}
\contentsline {section}{\tocsection {}{5}{Anticyclotomic $p$-adic $L$-functions}}{56}
\contentsline {subsection}{\tocsubsection {}{5.1}{Periods and algebraicity}}{56}
\contentsline {subsection}{\tocsubsection {}{5.2}{$p$-adic interpolation}}{60}
\contentsline {subsection}{\tocsubsection {}{5.3}{The main theorem}}{62}
\contentsline {section}{\tocsection {Appendix}{A}{Kuga-Sato schemes\\ by Brian Conrad}}{64}
\contentsline {section}{\tocsection {Appendix}{}{References}}{67}