\contentsline {section}{\tocsection {}{}{Introduction}}{1}
\contentsline {section}{\tocsection {}{1}{Strategy of proof}}{3}
\contentsline {section}{\tocsection {}{2}{A $p$-adic Beilinson formula }}{6}
\contentsline {subsection}{\tocsubsection {}{2.1}{Siegel units and Eisenstein series}}{6}
\contentsline {subsection}{\tocsubsection {}{2.2}{Beilinson-Flach elements}}{7}
\contentsline {subsection}{\tocsubsection {}{2.3}{Etale and syntomic regulators}}{7}
\contentsline {subsection}{\tocsubsection {}{2.4}{The rigid cohomology of modular curves}}{10}
\contentsline {subsection}{\tocsubsection {}{2.5}{Analytic description of the syntomic regulator}}{11}
\contentsline {subsection}{\tocsubsection {}{2.6}{Hida's $p$-adic $L$-function}}{15}
\contentsline {section}{\tocsection {}{3}{$\Lambda $-adic classes}}{17}
\contentsline {subsection}{\tocsubsection {}{3.1}{Norm-compatible systems of elements}}{17}
\contentsline {subsection}{\tocsubsection {}{3.2}{The $\Lambda $-adic Beilinson-Flach cohomology class}}{18}
\contentsline {subsection}{\tocsubsection {}{3.3}{The explicit reciprocity law}}{19}
\contentsline {section}{\tocsection {}{}{References}}{21}