\relax \newlabel{chap:intro}{{}{2}} \newlabel{eqn:typicaleqn}{{1}{2}} \newlabel{question:one}{{1}{3}} \newlabel{question:two}{{2}{4}} \newlabel{question:twobis}{{3}{4}} \newlabel{question:three}{{4}{4}} \newlabel{thm:genus0}{{5}{5}} \newlabel{thm:mordell-weil}{{7}{6}} \newlabel{thm:siegel-faltings}{{8}{6}} \citation{hugo} \citation{mazur_ihes} \citation{marusia} \citation{pierre} \citation{darmon_cbms} \citation{samit_john} \citation{john} \citation{matt1} \citation{john} \citation{matt1} \citation{matt2} \@writefile{toc}{\contentsline {section}{\numberline {1}Preliminaries}{9}} \newlabel{sec:preliminaries}{{1}{9}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Zero-dimensional varieties}{9}} \citation{szpiro_bourbaki} \citation{hugo} \citation{hugo} \newlabel{thm:hermite_minkowski}{{1.1}{10}} \newlabel{thm:minkowski_q}{{1.2}{10}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2}Etale morphisms and the Chevalley--Weil theorem}{10}} \newlabel{thm:chevalley_weil}{{1.5}{11}} \citation{hurwitz} \newlabel{klein}{{3}{12}} \newlabel{eqn:defpi}{{4}{12}} \citation{elkies_abc} \@writefile{toc}{\contentsline {section}{\numberline {2}Faltings' theorem}{15}} \newlabel{sec:faltings}{{2}{15}} \newlabel{thm:faltings_mordell}{{2.1}{15}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Prelude: the Shafarevich problem}{15}} \newlabel{sec:shafarevich}{{2.1}{15}} \citation{mazur_amsbull} \citation{shafarevich_stockholm} \citation{fontaine} \newlabel{conj:shafarevich}{{2.3}{16}} \newlabel{conj:shafcurve}{{2.3}{16}} \newlabel{conj:shafav}{{2.3}{16}} \citation{parshin1} \citation{mazur_amsbull} \citation{faltings_wustholz} \citation{parshin1} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}First reduction: the Kodaira--Parshin trick}{17}} \newlabel{sec:firstred}{{2.2}{17}} \newlabel{thm:kodaira-parshin}{{2.5}{17}} \citation{mazur_amsbull} \citation{cornell_silverman} \@writefile{toc}{\contentsline {subsection}{\numberline {2.3}Second reduction: passing to the jacobian}{19}} \newlabel{sec:secondred}{{2.3}{19}} \newlabel{prop:red2}{{2.8}{19}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.4}Third reduction: passing to isogeny classes}{19}} \newlabel{sec:thirdred}{{2.4}{19}} \citation{deligne_bourbaki} \citation{cornell_silverman} \citation{faltings_wustholz} \citation{szpiro_bourbaki} \citation{deligne_bourbaki} \citation{zarhin_parshin} \newlabel{thm:faltings_finiteisog}{{2.11}{20}} \newlabel{thm:faltings_fin1}{{2.12}{20}} \newlabel{thm:faltings_fin2}{{2.13}{20}} \newlabel{prop:conclusion}{{2.14}{21}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.5}Fourth reduction: from isogeny classes to $\ell $-adic representations}{21}} \newlabel{sec:fourthred}{{2.5}{21}} \citation{weil_book} \newlabel{thm:propsvla}{{2.15}{22}} \newlabel{lemma:projector}{{2.16}{23}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.6}The isogeny conjecture}{24}} \newlabel{sec:tate_conj}{{2.6}{24}} \newlabel{thm:isogeny}{{2.18}{24}} \newlabel{thm:tate}{{2.19}{24}} \newlabel{thm:tateend}{{2.20}{25}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.7}The finiteness principle for rational $\ell $-adic representations}{26}} \newlabel{sec:finitereps}{{2.7}{26}} \newlabel{thm:finrat}{{2.21}{26}} \newlabel{lemma:effcheb}{{2.22}{26}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.8}A summary of Faltings' proof}{28}} \newlabel{sec:faltings_summary}{{2.8}{28}} \@writefile{toc}{\contentsline {section}{\numberline {3}Modular curves and Mazur's theorem}{29}} \newlabel{sec:mazur}{{3}{29}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}Modular curves}{29}} \newlabel{sec:modular_curves}{{3.1}{29}} \newlabel{thm:mazur}{{3.1}{30}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Mazur's criterion}{30}} \newlabel{thm:mazur_criterion}{{3.4}{31}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.3}The jacobian $J_0(p)$}{32}} \newlabel{eqn:hecke_correspondences}{{6}{33}} \citation{darmon_cbms} \newlabel{eqn:hecke_q}{{7}{34}} \newlabel{eqn:hecke_tp}{{8}{34}} \newlabel{eqn:hecke_n}{{9}{34}} \newlabel{prop:hecke_semisimple}{{3.5}{34}} \citation{darmon_cbms} \newlabel{thm:eichler_shimura}{{3.6}{35}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.4}The Birch and Swinnerton-Dyer conjecture}{36}} \newlabel{conj:functional_equation}{{3.8}{37}} \newlabel{conj:bsd}{{3.9}{37}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.5}Hecke theory}{37}} \citation{darmon_cbms} \newlabel{eqn:mellin}{{10}{38}} \newlabel{eqn:signEQ1}{{12}{38}} \newlabel{eqn:fe}{{13}{38}} \newlabel{thm:hecke_hasse_weil}{{3.10}{38}} \newlabel{thm:koly_logachev}{{3.11}{39}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.6}The winding quotient}{39}} \citation{marusia} \citation{mazur_ihes} \citation{mazur_ihes} \newlabel{thm:finite_winding}{{3.12}{40}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.7}More results and questions}{40}} \citation{mazur_ihes} \citation{mazur_inventiones} \citation{serre_inv} \citation{mazur_ihes} \citation{kamienny} \citation{merel} \newlabel{thm:mazurihes}{{3.14}{41}} \newlabel{thm:mazur_ripd}{{3.15}{41}} \citation{marusia} \citation{cremona} \newlabel{conj:modpisog}{{3.18}{42}} \@writefile{toc}{\contentsline {section}{\numberline {4}Fermat curves}{42}} \newlabel{sec:wiles}{{4}{42}} \citation{hugo} \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}Motivation for the strategy}{43}} \newlabel{sec:motivationst}{{4.1}{43}} \newlabel{eqn:genfermat}{{14}{43}} \newlabel{eqn:frey0}{{15}{44}} \newlabel{eqn:invfrey}{{16}{44}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.2}Galois representations associated to Frey curves}{45}} \newlabel{thm:local_frey}{{4.1}{45}} \newlabel{eqn:tatid}{{17}{45}} \newlabel{eqn:ordlql}{{18}{46}} \newlabel{eqn:ordin}{{19}{46}} \newlabel{thm:irreducibility}{{4.2}{46}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.3}Modular forms and Galois representations}{47}} \newlabel{thm:propgalrep}{{4.4}{47}} \citation{darmon_cbms} \citation{serre} \@writefile{toc}{\contentsline {subsection}{\numberline {4.4}Serre's conjecture}{48}} \newlabel{eqn:rhomodp}{{20}{48}} \citation{serre} \citation{khare_wintenberger} \citation{kisin} \newlabel{eqn:Nrhoabc}{{21}{49}} \newlabel{eqn:krhoabc}{{22}{49}} \newlabel{conj:serre}{{4.5}{49}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.5}The Shimura--Taniyama conjecture}{49}} \citation{wiles} \citation{taylor_wiles} \citation{ddt} \citation{bcdt} \citation{taylor_artin} \citation{taylor} \citation{hst} \citation{cht} \newlabel{thm:wiles}{{4.6}{50}} \newlabel{thm:ribet}{{4.7}{50}} \citation{serre} \citation{kisin} \@writefile{toc}{\contentsline {subsection}{\numberline {4.6}A summary of Wiles' proof}{51}} \citation{pierre} \citation{bcdt} \@writefile{toc}{\contentsline {section}{\numberline {5}Elliptic curves}{53}} \newlabel{sec:elliptic}{{5}{53}} \newlabel{thm:gzk}{{5.3}{53}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.1}Modular parametrisations}{54}} \newlabel{thm:modpar}{{5.4}{54}} \newlabel{eqn:defmodpar}{{23}{54}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.2}Heegner points}{55}} \newlabel{sec:heegner_points}{{5.2}{55}} \newlabel{hyp:heegner}{{5.6}{55}} \citation{darmon_cbms} \citation{gross_zagier} \newlabel{eqn:defpk}{{24}{56}} \newlabel{eqn:conjpk}{{25}{56}} \newlabel{thm:gross_zagier}{{5.7}{56}} \newlabel{eqn:gz}{{26}{56}} \citation{zhang1} \citation{zhang2} \citation{howard1} \citation{howard2} \citation{murty_murty} \citation{gross_durham} \citation{darmon_cbms} \newlabel{eqn:defpchi}{{27}{57}} \newlabel{prop:murty_murty}{{5.9}{57}} \newlabel{thm:kolyvagin}{{5.10}{57}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Proof of Theorem 5.3\hbox {} }{58}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Modularity of elliptic curves over totally real fields}{59}} \newlabel{conj:st_tr}{{5.12}{59}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.5}Shimura curves}{60}} \citation{zhang1} \citation{john} \citation{matt1} \citation{john} \citation{matt1} \citation{longo} \@writefile{toc}{\contentsline {subsection}{\numberline {5.6}Stark--Heegner points}{61}} \citation{oda_book} \newlabel{conj:oda}{{5.15}{62}} \newlabel{eqn:def_phi}{{28}{62}} \citation{darmon_logan} \citation{charollois_darmon} \citation{darmon_logan} \citation{charollois_darmon} \newlabel{conj:stark_heegner_F}{{5.16}{63}} \citation{darmon_cbms} \citation{darmon_logan} \citation{darmon_logan} \citation{charollois_darmon} \citation{darmon_logan} \citation{charollois_darmon} \citation{darmon_logan} \citation{darmon_hpxh} \citation{darmon_hpxh} \citation{darmon_cbms} \citation{darmon_pollack} \citation{darmon_hpxh} \citation{bd_genus} \citation{trifkovic} \citation{trifkovic} \citation{matt2} \bibcite{bcdt}{BCDT} \bibcite{bd_genus}{BD} \bibcite{charollois_darmon}{CD} \bibcite{hugo}{Chap} \bibcite{pierre}{Char} \bibcite{cht}{CHT} \bibcite{Cassels-Frohlich}{CF} \bibcite{cornell_silverman}{CS} \bibcite{cremona}{Cr} \bibcite{darmon_hpxh}{Da1} \bibcite{darmon_cbms}{Da2} \bibcite{ddt}{DDT} \bibcite{darmon_granville}{DG} \bibcite{darmon_logan}{DL} \bibcite{darmon_pollack}{DP} \bibcite{samit_john}{DV} \bibcite{deligne_bourbaki}{De} \bibcite{elkies_abc}{El} \bibcite{faltings}{Fa} \bibcite{faltings_wustholz}{FW} \bibcite{fontaine}{Fo} \bibcite{frey1}{Fr1} \bibcite{Frey}{Fr2} \bibcite{matt1}{Gre1} \bibcite{matt2}{Gre2} \bibcite{gross_durham}{Gr} \bibcite{gross_zagier}{GZ} \bibcite{H1}{Hel} \bibcite{howard1}{Ho1} \bibcite{howard2}{Ho2} \bibcite{hst}{HST} \bibcite{hurwitz}{Hur} \bibcite{husemoller}{Hus} \bibcite{kamienny}{Ka} \bibcite{kisin}{Ki} \bibcite{koly1}{Ko1} \bibcite{koly2}{Ko2} \bibcite{khare_wintenberger}{KW} \bibcite{longo}{Lo} \bibcite{mazur_ihes}{Ma1} \bibcite{mazur_inventiones}{Ma2} \bibcite{mazur_amsbull}{Ma3} \bibcite{merel}{Me} \bibcite{murty_murty}{MM} \bibcite{oda_book}{Oda} \bibcite{oesterle}{Oe} \bibcite{parshin1}{Pa1} \bibcite{marusia}{Reb} \bibcite{ribet1}{Rib1} \bibcite{serre_inv}{Se1} \bibcite{serre}{Se2} \bibcite{shafarevich_stockholm}{Sha} \bibcite{Shimura-red-book}{Shi} \bibcite{Silverman}{Si} \bibcite{szpiro_bourbaki}{Sz} \bibcite{tate_abelian}{Ta} \bibcite{taylor}{Tay1} \bibcite{taylor_artin}{Tay2} \bibcite{trifkovic}{Tr} \bibcite{taylor_wiles}{TW} \bibcite{john}{Vo} \bibcite{weil_book}{We} \bibcite{wiles}{Wi} \bibcite{zhang1}{Zh1} \bibcite{zhang2}{Zh2} \bibcite{zarhin_parshin}{ZP}