\contentsline {section}{\tocsection {}{1}{Introduction}}{1}
\contentsline {section}{\tocsection {}{2}{Rankin $L$-series}}{3}
\contentsline {subsection}{\tocsubsection {}{2.1}{Complex $L$-series}}{3}
\contentsline {subsubsection}{\tocsubsubsection {}{2.1.1}{Rankin's method}}{4}
\contentsline {subsubsection}{\tocsubsubsection {}{2.1.2}{Critical values}}{4}
\contentsline {subsubsection}{\tocsubsubsection {}{2.1.3}{Algebraicity and Deligne's conjecture}}{5}
\contentsline {subsubsection}{\tocsubsubsection {}{2.1.4}{The value at the near central point}}{6}
\contentsline {subsection}{\tocsubsection {}{2.2}{$p$-adic $L$-series}}{7}
\contentsline {subsubsection}{\tocsubsubsection {}{2.2.1}{Ordinary projections}}{7}
\contentsline {subsubsection}{\tocsubsubsection {}{2.2.2}{Hida's $p$-adic L-series}}{8}
\contentsline {section}{\tocsection {}{3}{Beilinson-Flach elements}}{9}
\contentsline {subsection}{\tocsubsection {}{3.1}{Definition and basic properties}}{9}
\contentsline {subsection}{\tocsubsection {}{3.2}{Complex regulators}}{10}
\contentsline {subsection}{\tocsubsection {}{3.3}{$p$-adic regulators}}{10}
\contentsline {section}{\tocsection {}{4}{The Beilinson formula}}{15}
\contentsline {subsection}{\tocsubsection {}{4.1}{The complex setting}}{15}
\contentsline {subsection}{\tocsubsection {}{4.2}{The $p$-adic setting}}{16}
\contentsline {section}{\tocsection {}{}{References}}{17}