\contentsline {chapter}{\tocchapter {Chapter}{}{Preface}}{xi}
\contentsline {chapter}{\tocchapter {Chapter}{1}{Elliptic curves}}{1}
\contentsline {section}{\tocsection {}{1.1}{Elliptic curves}}{1}
\contentsline {section}{\tocsection {}{1.2}{The Mordell-Weil theorem}}{3}
\contentsline {section}{\tocsection {}{1.3}{The Birch and Swinnerton-Dyer conjecture}}{6}
\contentsline {section}{\tocsection {}{1.4}{$L$-functions}}{7}
\contentsline {section}{\tocsection {}{1.5}{Some known results}}{8}
\contentsline {section}{\tocsection {}{}{Further results and references}}{9}
\contentsline {section}{\tocsection {}{}{Exercises}}{10}
\contentsline {chapter}{\tocchapter {Chapter}{2}{Modular forms}}{13}
\contentsline {section}{\tocsection {}{2.1}{Modular forms}}{13}
\contentsline {section}{\tocsection {}{2.2}{Hecke operators}}{14}
\contentsline {section}{\tocsection {}{2.3}{Atkin-Lehner theory}}{16}
\contentsline {section}{\tocsection {}{2.4}{$L$-series}}{17}
\contentsline {section}{\tocsection {}{2.5}{Eichler-Shimura theory}}{18}
\contentsline {section}{\tocsection {}{2.6}{Wiles' theorem}}{20}
\contentsline {section}{\tocsection {}{2.7}{Modular symbols}}{21}
\contentsline {section}{\tocsection {}{}{Further results and references}}{25}
\contentsline {section}{\tocsection {}{}{Exercises}}{26}
\contentsline {chapter}{\tocchapter {Chapter}{3}{Heegner points on $X_0(N)$}}{29}
\contentsline {section}{\tocsection {}{3.1}{Complex multiplication}}{29}
\contentsline {section}{\tocsection {}{3.2}{Heegner points}}{33}
\contentsline {section}{\tocsection {}{3.3}{Numerical examples}}{34}
\contentsline {section}{\tocsection {}{3.4}{Properties of Heegner points}}{35}
\contentsline {section}{\tocsection {}{3.5}{Heegner systems}}{36}
\contentsline {section}{\tocsection {}{3.6}{Relation with the Birch and Swinnerton-Dyer conjecture}}{37}
\contentsline {section}{\tocsection {}{3.7}{The Gross-Zagier formula}}{39}
\contentsline {section}{\tocsection {}{3.8}{Kolyvagin's theorem}}{40}
\contentsline {section}{\tocsection {}{3.9}{Proof of the Gross-Zagier-Kolyvagin theorem}}{40}
\contentsline {section}{\tocsection {}{}{Further results}}{41}
\contentsline {section}{\tocsection {}{}{Exercises}}{42}
\contentsline {chapter}{\tocchapter {Chapter}{4}{Heegner points on Shimura curves}}{45}
\contentsline {section}{\tocsection {}{4.1}{Quaternion algebras}}{46}
\contentsline {section}{\tocsection {}{4.2}{Modular forms on quaternion algebras}}{47}
\contentsline {section}{\tocsection {}{4.3}{Shimura curves}}{49}
\contentsline {section}{\tocsection {}{4.4}{The Eichler-Shimura construction, revisited}}{50}
\contentsline {section}{\tocsection {}{4.5}{The Jacquet-Langlands correspondence}}{50}
\contentsline {section}{\tocsection {}{4.6}{The Shimura-Taniyama-Weil conjecture, revisited}}{51}
\contentsline {section}{\tocsection {}{4.7}{Complex multiplication for ${\@mathcal H}/\Gamma _{N^+,N^-}$}}{51}
\contentsline {section}{\tocsection {}{4.8}{Heegner systems}}{52}
\contentsline {section}{\tocsection {}{4.9}{The Gross-Zagier formula}}{53}
\contentsline {section}{\tocsection {}{}{References}}{54}
\contentsline {section}{\tocsection {}{}{Exercises}}{54}
\contentsline {chapter}{\tocchapter {Chapter}{5}{Rigid analytic modular forms}}{57}
\contentsline {section}{\tocsection {}{5.1}{$p$-adic uniformisation}}{57}
\contentsline {section}{\tocsection {}{5.2}{Rigid analytic modular forms}}{60}
\contentsline {section}{\tocsection {}{5.3}{$p$-adic line integrals}}{63}
\contentsline {section}{\tocsection {}{}{Further results}}{65}
\contentsline {section}{\tocsection {}{}{Exercises}}{65}
\contentsline {chapter}{\tocchapter {Chapter}{6}{Rigid analytic modular parametrisations}}{67}
\contentsline {section}{\tocsection {}{6.1}{Rigid analytic modular forms on quaternion algebras}}{67}
\contentsline {section}{\tocsection {}{6.2}{The \v Cerednik-Drinfeld theorem}}{68}
\contentsline {section}{\tocsection {}{6.3}{The $p$-adic Shimura-Taniyama-Weil conjecture}}{68}
\contentsline {section}{\tocsection {}{6.4}{Complex multiplication, revisited}}{69}
\contentsline {section}{\tocsection {}{6.5}{An example}}{70}
\contentsline {section}{\tocsection {}{6.6}{$p$-adic $L$-functions, d'apr\`es Schneider-Iovita-Spiess}}{73}
\contentsline {section}{\tocsection {}{6.7}{A Gross-Zagier formula}}{74}
\contentsline {section}{\tocsection {}{}{Further results}}{75}
\contentsline {section}{\tocsection {}{}{Exercises}}{75}
\contentsline {chapter}{\tocchapter {Chapter}{7}{Totally real fields}}{79}
\contentsline {section}{\tocsection {}{7.1}{Elliptic curves over number fields}}{79}
\contentsline {section}{\tocsection {}{7.2}{Hilbert modular forms}}{80}
\contentsline {section}{\tocsection {}{7.3}{The Shimura-Taniyama-Weil conjecture}}{82}
\contentsline {section}{\tocsection {}{7.4}{The Eichler-Shimura construction for totally real fields}}{83}
\contentsline {section}{\tocsection {}{7.5}{The Heegner construction}}{84}
\contentsline {section}{\tocsection {}{7.6}{A preview of Chapter 8}}{85}
\contentsline {section}{\tocsection {}{}{Further results}}{86}
\contentsline {chapter}{\tocchapter {Chapter}{8}{ATR points}}{87}
\contentsline {section}{\tocsection {}{8.1}{Period integrals}}{87}
\contentsline {section}{\tocsection {}{8.2}{Generalities on group cohomology}}{88}
\contentsline {section}{\tocsection {}{8.3}{The cohomology of Hilbert modular groups}}{89}
\contentsline {section}{\tocsection {}{8.4}{ATR points}}{93}
\contentsline {section}{\tocsection {}{}{References}}{95}
\contentsline {section}{\tocsection {}{}{Exercises}}{95}
\contentsline {chapter}{\tocchapter {Chapter}{9}{Integration on ${\@mathcal H}_p\times {\@mathcal H}$}}{97}
\contentsline {section}{\tocsection {}{9.1}{Discrete arithmetic subgroups of ${\rm {\bf SL}}_2({\@mathbb Q}_p)\times {\rm {\bf SL}}_2({\@mathbb R})$}}{98}
\contentsline {section}{\tocsection {}{9.2}{Forms on ${\@mathcal H}_p\times {\@mathcal H}$}}{99}
\contentsline {section}{\tocsection {}{9.3}{Periods}}{101}
\contentsline {section}{\tocsection {}{9.4}{Some $p$-adic cocycles}}{104}
\contentsline {section}{\tocsection {}{9.5}{Stark-Heegner points}}{105}
\contentsline {section}{\tocsection {}{9.6}{Computing Stark-Heegner points }}{106}
\contentsline {section}{\tocsection {}{}{Further results}}{109}
\contentsline {section}{\tocsection {}{}{Exercises}}{109}
\contentsline {chapter}{\tocchapter {Chapter}{10}{Kolyvagin's theorem}}{113}
\contentsline {section}{\tocsection {}{10.1}{Bounding Selmer groups}}{114}
\contentsline {section}{\tocsection {}{10.2}{Kolyvagin cohomology classes}}{117}
\contentsline {section}{\tocsection {}{10.3}{Proof of Kolyvagin's theorem}}{121}
\contentsline {section}{\tocsection {}{}{References}}{122}
\contentsline {section}{\tocsection {}{}{Exercises}}{122}
\contentsline {chapter}{\tocchapter {Chapter}{}{Bibliography}}{125}