\contentsline {section}{\numberline {1}$p$-adic $L$-functions}{7}
\contentsline {subsection}{\numberline {1.1}Modular forms on quaternion algebras}{7}
\contentsline {subsection}{\numberline {1.2}$p$-adic Rankin $L$-functions}{12}
\contentsline {section}{\numberline {2}Selmer Groups}{16}
\contentsline {subsection}{\numberline {2.1}Galois representations and cohomology}{16}
\contentsline {subsection}{\numberline {2.2}Finite/singular structures}{20}
\contentsline {subsection}{\numberline {2.3}Definition of the Selmer group}{23}
\contentsline {section}{\numberline {3}Some Preliminaries}{25}
\contentsline {subsection}{\numberline {3.1}$\Lambda $-modules}{25}
\contentsline {subsection}{\numberline {3.2}Controlling the Selmer group}{26}
\contentsline {subsection}{\numberline {3.3}Rigid pairs}{28}
\contentsline {section}{\numberline {4}The Euler System Argument}{35}
\contentsline {subsection}{\numberline {4.1}The Euler system}{35}
\contentsline {subsection}{\numberline {4.2}The argument}{37}
\contentsline {section}{\numberline {5}Shimura curves}{43}
\contentsline {subsection}{\numberline {5.1}The moduli definition}{43}
\contentsline {subsection}{\numberline {5.2}The Cerednik-Drinfeld theorem}{45}
\contentsline {subsection}{\numberline {5.3}Character groups}{46}
\contentsline {subsection}{\numberline {5.4}Hecke operators and the Jacquet-Langlands correspondence}{48}
\contentsline {subsection}{\numberline {5.5}Connected components}{53}
\contentsline {subsection}{\numberline {5.6}Raising the level and groups of connected components}{57}
\contentsline {section}{\numberline {6}The theory of complex multiplication}{64}
\contentsline {section}{\numberline {7}Construction of the Euler System}{66}
\contentsline {section}{\numberline {8}The first explicit reciprocity law}{66}
\contentsline {section}{\numberline {9}The second explicit reciprocity law}{69}