\contentsline {section}{\tocsection {}{1}{Introduction}}{1} \contentsline {section}{\tocsection {}{2}{Complex \$L\$-series}}{2} \contentsline {subsection}{\tocsubsection {}{2.1}{Eisenstein series}}{3} \contentsline {subsection}{\tocsubsection {}{2.2}{Rankin's method}}{4} \contentsline {subsection}{\tocsubsection {}{2.3}{A factorisation of critical values}}{6} \contentsline {subsection}{\tocsubsection {}{2.4}{Beilinson's formula}}{7} \contentsline {subsection}{\tocsubsection {}{2.5}{Algebraicity}}{8} \contentsline {section}{\tocsection {}{3}{\$p\$-adic \$L\$-functions}}{9} \contentsline {subsection}{\tocsubsection {}{3.1}{Rankin's \$p\$-adic \$L\$-functions}}{9} \contentsline {subsection}{\tocsubsection {}{3.2}{Factorisation of \$p\$-adic \$L\$-functions}}{10} \contentsline {section}{\tocsection {}{4}{\$p\$-adic regulators}}{11} \contentsline {subsection}{\tocsubsection {}{4.1}{The regulator on \$K_2\$}}{12} \contentsline {subsection}{\tocsubsection {}{4.2}{The general case}}{14} \contentsline {section}{\tocsection {}{5}{The \$p\$-adic Beilinson formula}}{15} \contentsline {subsection}{\tocsubsection {}{5.1}{The main results}}{15} \contentsline {subsection}{\tocsubsection {}{5.2}{Relation with the work of Brunault and Gealy}}{15} \contentsline {section}{\tocsection {}{}{References}}{17}