\contentsline {chapter}{\numberline {1}The class number formula}{10}
\contentsline {section}{\numberline {1.1}Dirichlet's analytic class number formula}{10}
\contentsline {section}{\numberline {1.2}Gross's refined class number formula}{12}
\contentsline {section}{\numberline {1.3}A refined class number formula for derivatives of Dirichlet $L$-series}{14}
\contentsline {chapter}{\numberline {2}Partial proof of the refined class number formula for circular units}{18}
\contentsline {section}{\numberline {2.1}The Euler system of circular units}{18}
\contentsline {section}{\numberline {2.2}Divisibility properties of the circular units}{19}
\contentsline {section}{\numberline {2.3}Formal properties of ${\theta ^{'}(\omega ,S)}$}{24}
\contentsline {section}{\numberline {2.4}The order of vanishing of ${\theta ^{'}(\omega ,S)}$}{25}
\contentsline {section}{\numberline {2.5}The leading coefficient}{26}
\contentsline {chapter}{\numberline {3}The Birch and Swinnerton-Dyer conjecture}{31}
\contentsline {section}{\numberline {3.1}The classical case}{31}
\contentsline {section}{\numberline {3.2}Modular elliptic curves}{33}
\contentsline {section}{\numberline {3.3}The Mazur-Tate conjectures}{35}
\contentsline {section}{\numberline {3.4}A conjecture of Mazur-Tate type for derivatives of $L$-functions}{39}
\contentsline {chapter}{\numberline {4}Partial proof of the refined Birch and Swinnerton-Dyer conjecture for Heegner points}{45}
\contentsline {section}{\numberline {4.1}The descent formalism}{45}
\contentsline {section}{\numberline {4.2}The Euler system of Heegner points}{49}
\contentsline {section}{\numberline {4.3}Derivatives in group rings}{51}
\contentsline {section}{\numberline {4.4}The Heegner cohomology classes}{53}
\contentsline {section}{\numberline {4.5}An application of the Chebotarev density theorem}{57}
\contentsline {section}{\numberline {4.6}A vanishing theorem for derivatives of Heegner points}{59}
\contentsline {section}{\numberline {4.7}The first non-vanishing derivatives of Heegner points}{64}
\contentsline {section}{\numberline {4.8}The order of vanishing of ${\theta ^{'}(E,S)}$}{68}
\contentsline {section}{\numberline {4.9}The leading coefficient}{70}