\contentsline {section}{\numberline {1}Elliptic curves and modular forms}{5} \contentsline {subsection}{\numberline {1.1}Elliptic curves}{5} \contentsline {subsection}{\numberline {1.2}The Birch and Swinnerton-Dyer conjecture}{7} \contentsline {subsection}{\numberline {1.3}Modularity}{10} \contentsline {section}{\numberline {2}Cyclotomic $p$-adic $L$-functions}{12} \contentsline {subsection}{\numberline {2.1}The Mazur--Swinnerton-Dyer $p$-adic $L$-function}{13} \contentsline {subsection}{\numberline {2.2}The Mazur-Tate-Teitelbaum conjecture}{16} \contentsline {subsection}{\numberline {2.3}Results on the Mazur-Tate-Teitelbaum conjecture}{17} \contentsline {section}{\numberline {3}Schneider's approach}{18} \contentsline {subsection}{\numberline {3.1}Rigid analysis}{18} \contentsline {subsection}{\numberline {3.2}Shimura Curves}{20} \contentsline {subsection}{\numberline {3.3}Modular forms}{23} \contentsline {subsection}{\numberline {3.4}Schneider's distribution}{25} \contentsline {subsection}{\numberline {3.5}The Jacquet-Langlands correspondence}{27} \contentsline {subsection}{\numberline {3.6}Schneider's $p$-adic $L$-function}{30} \contentsline {section}{\numberline {4}Anticyclotomic $p$-adic $L$-functions}{31} \contentsline {subsection}{\numberline {4.1}The definite $p$-adic $L$-function}{31} \contentsline {subsection}{\numberline {4.2}The Iovita-Spiess construction}{39} \contentsline {subsection}{\numberline {4.3}Heegner points and the indefinite $p$-adic $L$-function}{40} \contentsline {subsection}{\numberline {4.4}The anticyclotomic $p$-adic Birch and Swinnerton-Dyer conjecture}{44} \contentsline {section}{\numberline {5}Theorems in the anticyclotomic setting}{47} \contentsline {subsection}{\numberline {5.1}Results on conjecture 4.8\hbox {}}{47} \contentsline {subsection}{\numberline {5.2}Heegner points and connected components}{49} \contentsline {subsection}{\numberline {5.3}Exceptional zero results via rigid analysis}{50} \contentsline {subsection}{\numberline {5.4}A $p$-adic Birch and Swinnerton-Dyer conjecture}{56} \contentsline {section}{\numberline {6}Heegner points for real quadratic fields}{59} \contentsline {subsection}{\numberline {6.1}Double integrals}{59} \contentsline {subsection}{\numberline {6.2}$p$-adic $L$-functions and theta-elements}{61} \contentsline {subsection}{\numberline {6.3}A conjecture of Mazur-Tate type}{66} \contentsline {subsection}{\numberline {6.4}Leading terms of theta-elements}{67} \contentsline {subsection}{\numberline {6.5}Heegner points attached to real quadratic fields}{70}