\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}