\contentsline {section}{\tocsection {}{1}{Introduction}}{2}
\contentsline {section}{\tocsection {}{2}{Cohomology and modular forms}}{7}
\contentsline {subsection}{\tocsubsection {}{2.1}{The de Rham cohomology of curves over $p$-adic rings}}{7}
\contentsline {subsection}{\tocsubsection {}{2.2}{$p$-adic modular forms}}{10}
\contentsline {subsection}{\tocsubsection {}{2.3}{Nearly holomorphic modular forms and the Shimura-Maass derivative}}{14}
\contentsline {subsection}{\tocsubsection {}{2.4}{Nearly overconvergent modular forms and the $d$ operator}}{16}
\contentsline {subsection}{\tocsubsection {}{2.5}{Periods of modular forms}}{20}
\contentsline {subsection}{\tocsubsection {}{2.6}{Hida families }}{22}
\contentsline {section}{\tocsection {}{3}{Diagonal cycles and $p$-adic Abel-Jacobi maps}}{26}
\contentsline {subsection}{\tocsubsection {}{3.1}{Generalised Gross-Kudla-Schoen cycles}}{26}
\contentsline {subsection}{\tocsubsection {}{3.2}{A formula for ${\@mathrm {AJ}}_p(\Delta _{k,\ell ,m})$ in terms of Coleman integration}}{28}
\contentsline {subsection}{\tocsubsection {}{3.3}{Proof of the formula: Besser's finite polynomial cohomology}}{31}
\contentsline {subsection}{\tocsubsection {}{3.4}{A formula for ${\@mathrm {AJ}}_p(\Delta _{k,\ell ,m})$ in terms of $p$-adic modular forms}}{34}
\contentsline {section}{\tocsection {}{4}{Garrett-Rankin triple product $L$-functions}}{39}
\contentsline {subsection}{\tocsubsection {}{4.1}{Classical $L$-functions}}{39}
\contentsline {subsection}{\tocsubsection {}{4.2}{$p$-adic $L$-functions}}{40}
\contentsline {section}{\tocsection {}{5}{The $p$-adic Gross-Zagier formula}}{45}
\contentsline {section}{\tocsection {}{}{References}}{46}