\contentsline {section}{\numberline {1}A review of the classical setting}{3}
\contentsline {section}{\numberline {2}Elliptic units for real quadratic fields}{6}
\contentsline {subsection}{\numberline {2.1}$p$-adic measures}{8}
\contentsline {subsection}{\numberline {2.2}Double integrals}{10}
\contentsline {subsection}{\numberline {2.3}Splitting a two-cocycle}{11}
\contentsline {subsection}{\numberline {2.4}The main conjecture}{13}
\contentsline {subsection}{\numberline {2.5}Modular symbols and Dedekind sums}{14}
\contentsline {subsection}{\numberline {2.6}Measures and the Bruhat-Tits tree}{15}
\contentsline {subsection}{\numberline {2.7}Indefinite integrals}{17}
\contentsline {subsection}{\numberline {2.8}The action of complex conjugation and of $U_p$}{18}
\contentsline {section}{\numberline {3}Special values of zeta-functions}{19}
\contentsline {subsection}{\numberline {3.1}The zeta function}{19}
\contentsline {subsection}{\numberline {3.2}Values at negative integers}{21}
\contentsline {subsection}{\numberline {3.3}The $p$-adic valuation}{24}
\contentsline {subsection}{\numberline {3.4}The Brumer-Stark conjecture}{26}
\contentsline {subsection}{\numberline {3.5}Connection with the Gross-Stark conjecture}{27}
\contentsline {section}{\numberline {4}A Kronecker limit formula}{28}
\contentsline {subsection}{\numberline {4.1}Measures associated to Eisenstein series }{28}
\contentsline {subsection}{\numberline {4.2}Construction of the $p$-adic $L$-function}{30}
\contentsline {subsection}{\numberline {4.3}An explicit splitting of a two-cocycle}{31}
\contentsline {subsection}{\numberline {4.4}Generalized Dedekind Sums}{32}
\contentsline {subsection}{\numberline {4.5}Measures on $\@mathbb {Z}_p \times \@mathbb {Z}_p$}{34}
\contentsline {subsection}{\numberline {4.6}A partial modular symbol of measures on $\@mathbb {Z}_p \times \@mathbb {Z}_p$}{37}
\contentsline {subsection}{\numberline {4.7}{From} $\@mathbb {Z}_p \times \@mathbb {Z}_p$ to $\@mathbb {X}$}{38}
\contentsline {subsection}{\numberline {4.8}The measures $\mu $ and $\Gamma $-invariance}{41}