\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}{24} \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}{29} \contentsline {section}{\numberline {4}The Euler System Argument}{36} \contentsline {subsection}{\numberline {4.1}The Euler system}{36} \contentsline {subsection}{\numberline {4.2}The argument}{37} \contentsline {section}{\numberline {5}Shimura curves}{44} \contentsline {subsection}{\numberline {5.1}The moduli definition}{44} \contentsline {subsection}{\numberline {5.2}The Cerednik-Drinfeld theorem}{46} \contentsline {subsection}{\numberline {5.3}Character groups}{47} \contentsline {subsection}{\numberline {5.4}Hecke operators and the Jacquet-Langlands correspondence}{49} \contentsline {subsection}{\numberline {5.5}Connected components}{54} \contentsline {subsection}{\numberline {5.6}Raising the level and groups of connected components}{58} \contentsline {section}{\numberline {6}The theory of complex multiplication}{65} \contentsline {section}{\numberline {7}Construction of the Euler System}{66} \contentsline {section}{\numberline {8}The first explicit reciprocity law}{67} \contentsline {section}{\numberline {9}The second explicit reciprocity law}{70}