\contentsline {section}{\numberline {1}Preliminaries}{9} \contentsline {subsection}{\numberline {1.1}Zero-dimensional varieties}{9} \contentsline {subsection}{\numberline {1.2}Etale morphisms and the Chevalley--Weil theorem}{10} \contentsline {section}{\numberline {2}Faltings' theorem}{15} \contentsline {subsection}{\numberline {2.1}Prelude: the Shafarevich problem}{15} \contentsline {subsection}{\numberline {2.2}First reduction: the Kodaira--Parshin trick}{17} \contentsline {subsection}{\numberline {2.3}Second reduction: passing to the jacobian}{19} \contentsline {subsection}{\numberline {2.4}Third reduction: passing to isogeny classes}{19} \contentsline {subsection}{\numberline {2.5}Fourth reduction: from isogeny classes to $\ell$-adic representations}{21} \contentsline {subsection}{\numberline {2.6}The isogeny conjecture}{24} \contentsline {subsection}{\numberline {2.7}The finiteness principle for rational $\ell$-adic representations}{26} \contentsline {subsection}{\numberline {2.8}A summary of Faltings' proof}{28} \contentsline {section}{\numberline {3}Modular curves and Mazur's theorem}{29} \contentsline {subsection}{\numberline {3.1}Modular curves}{29} \contentsline {subsection}{\numberline {3.2}Mazur's criterion}{30} \contentsline {subsection}{\numberline {3.3}The jacobian $J_0(p)$}{32} \contentsline {subsection}{\numberline {3.4}The Birch and Swinnerton-Dyer conjecture}{36} \contentsline {subsection}{\numberline {3.5}Hecke theory}{37} \contentsline {subsection}{\numberline {3.6}The winding quotient}{39} \contentsline {subsection}{\numberline {3.7}More results and questions}{40} \contentsline {section}{\numberline {4}Fermat curves}{42} \contentsline {subsection}{\numberline {4.1}Motivation for the strategy}{43} \contentsline {subsection}{\numberline {4.2}Galois representations associated to Frey curves}{45} \contentsline {subsection}{\numberline {4.3}Modular forms and Galois representations}{47} \contentsline {subsection}{\numberline {4.4}Serre's conjecture}{48} \contentsline {subsection}{\numberline {4.5}The Shimura--Taniyama conjecture}{49} \contentsline {subsection}{\numberline {4.6}A summary of Wiles' proof}{51} \contentsline {section}{\numberline {5}Elliptic curves}{53} \contentsline {subsection}{\numberline {5.1}Modular parametrisations}{54} \contentsline {subsection}{\numberline {5.2}Heegner points}{55} \contentsline {subsection}{\numberline {5.3}Proof of Theorem 5.3\hbox {} }{58} \contentsline {subsection}{\numberline {5.4}Modularity of elliptic curves over totally real fields}{59} \contentsline {subsection}{\numberline {5.5}Shimura curves}{60} \contentsline {subsection}{\numberline {5.6}Stark--Heegner points}{61}