\contentsline {section}{\numberline {1}Duality and the $\@mathscr {L}$-invariant}{12}
\contentsline {subsection}{\numberline {1.1}Local cohomology groups}{14}
\contentsline {subsection}{\numberline {1.2}Global cohomology groups}{15}
\contentsline {subsection}{\numberline {1.3}A formula for the $\@mathscr {L}$-invariant}{17}
\contentsline {section}{\numberline {2}Hilbert modular forms}{18}
\contentsline {subsection}{\numberline {2.1}Definitions}{18}
\contentsline {subsection}{\numberline {2.2}Eisenstein Series}{20}
\contentsline {subsection}{\numberline {2.3}A product of Eisenstein series}{26}
\contentsline {subsection}{\numberline {2.4}The ordinary projection}{28}
\contentsline {subsection}{\numberline {2.5}Construction of a cusp form}{29}
\contentsline {section}{\numberline {3}$\Lambda $-adic forms}{31}
\contentsline {subsection}{\numberline {3.1}Definitions}{31}
\contentsline {subsection}{\numberline {3.2}$\Lambda $-adic Eisenstein series}{32}
\contentsline {subsection}{\numberline {3.3}A $\Lambda $-adic cusp form}{33}
\contentsline {subsection}{\numberline {3.4}The weight $1 + \varepsilon $ specialization}{34}
\contentsline {section}{\numberline {4}Galois representations}{37}
\contentsline {subsection}{\numberline {4.1}Representations attached to ordinary eigenforms}{37}
\contentsline {subsection}{\numberline {4.2}Construction of a cocycle}{40}