\contentsline {section}{\numberline {1}$p$-adic $L$-functions}{6}
\contentsline {subsection}{\numberline {1.1}Modular forms on quaternion algebras}{6}
\contentsline {subsection}{\numberline {1.2}$p$-adic Rankin $L$-functions}{11}
\contentsline {section}{\numberline {2}Selmer Groups}{15}
\contentsline {subsection}{\numberline {2.1}Galois representations and cohomology}{15}
\contentsline {subsection}{\numberline {2.2}Definition of the Selmer group}{18}
\contentsline {section}{\numberline {3}Some Preliminaries}{22}
\contentsline {subsection}{\numberline {3.1}$\Lambda $-modules}{22}
\contentsline {subsection}{\numberline {3.2}Auxiliary primes}{23}
\contentsline {subsection}{\numberline {3.3}Admissible pairs}{27}
\contentsline {section}{\numberline {4}The Euler System Argument}{34}
\contentsline {subsection}{\numberline {4.1}The Euler system}{34}
\contentsline {subsection}{\numberline {4.2}The argument}{35}
\contentsline {section}{\numberline {5}Shimura curves}{40}
\contentsline {subsection}{\numberline {5.1}The moduli definition}{40}
\contentsline {subsection}{\numberline {5.2}The Cerednik-Drinfeld theorem}{42}
\contentsline {subsection}{\numberline {5.3}Character groups}{43}
\contentsline {subsection}{\numberline {5.4}Hecke operators and the Jacquet-Langlands correspondence}{45}
\contentsline {subsection}{\numberline {5.5}Connected components}{49}
\contentsline {subsection}{\numberline {5.6}Raising the level and groups of connected components}{53}
\contentsline {section}{\numberline {6}The theory of complex multiplication}{59}
\contentsline {section}{\numberline {7}Construction of the Euler System}{60}
\contentsline {section}{\numberline {8}The first explicit reciprocity law}{61}
\contentsline {section}{\numberline {9}The second explicit reciprocity law}{63}