\relax \citation{BD1} \newlabel{thm:main_conjecture}{{1}{2}} \citation{mtt} \citation{BD1} \citation{BD2} \citation{BD5} \citation{BD1} \citation{BD1} \citation{mtt} \citation{bd_derived_heights} \citation{bd_derived_regulators} \citation{bd_derived_heights} \newlabel{cor:main1}{{2}{3}} \newlabel{eqn:conjeq}{{{1}}{3}} \newlabel{cor:main2}{{3}{3}} \citation{zhang} \citation{gross_montreal} \citation{BD2} \citation{vatsal1} \citation{vatsal2} \newlabel{cor:mainzhang}{{4}{4}} \newlabel{cor:mainvatsal}{{5}{4}} \citation{BD1} \newlabel{assumptions}{{6}{5}} \@writefile{toc}{\contentsline {section}{\numberline {1}$p$-adic $L$-functions}{6}} \newlabel{sec:p-adic_L-function}{{1}{6}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.1}Modular forms on quaternion algebras}{6}} \newlabel{subsec:modularquaternion}{{1.1}{6}} \newlabel{eqn:formpairing}{{{2}}{7}} \citation{eichler} \citation{shimura_book} \newlabel{prop:jacquet_langlands}{{1.2}{9}} \newlabel{eqn:dblcst}{{{9}}{9}} \newlabel{eqn:dblcstn}{{{10}}{9}} \newlabel{prop:stw}{{1.3}{10}} \@writefile{toc}{\contentsline {subsection}{\numberline {1.2}$p$-adic Rankin $L$-functions}{11}} \newlabel{subsec:p-adic_L-functions}{{1.2}{11}} \newlabel{eqn:completion}{{{11}}{11}} \newlabel{eqn:defGinftytilde}{{{12}}{11}} \newlabel{eqn:palphae}{{{16}}{12}} \newlabel{eqn:tildenu}{{{19}}{12}} \citation{mtt} \citation{BD1} \newlabel{eqn:defginfty}{{{20}}{13}} \newlabel{lemma:sign_fe}{{1.4}{13}} \newlabel{def:rankin}{{1.5}{13}} \citation{zhang} \@writefile{toc}{\contentsline {section}{\numberline {2}Selmer Groups}{15}} \newlabel{sec:selmer}{{2}{15}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.1}Galois representations and cohomology}{15}} \newlabel{subsec:cohomology}{{2.1}{15}} \newlabel{ass:selmer}{{7}{16}} \newlabel{lemma:ribet_appl}{{2.1}{16}} \@writefile{toc}{\contentsline {subsection}{\numberline {2.2}Definition of the Selmer group}{18}} \newlabel{prop:isotropy}{{2.2}{19}} \newlabel{lemma:struct_l_split}{{2.3}{19}} \newlabel{lemma:struct_l_inert}{{2.4}{20}} \newlabel{eqn:resproj}{{26}{20}} \newlabel{eqn:globalrec}{{{27}}{22}} \newlabel{prop:global_duality}{{2.7}{22}} \@writefile{toc}{\contentsline {section}{\numberline {3}Some Preliminaries}{22}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.1}$\Lambda$-modules}{22}} \newlabel{prop:Lambda}{{3.1}{23}} \newlabel{eqn:pseudo}{{{28}}{23}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.2}Auxiliary primes}{23}} \citation{bd_derived_regulators} \newlabel{lemma:local_calc}{{3.2}{24}} \newlabel{lemma:local_free}{{3.3}{24}} \newlabel{thm:chebotarev1}{{3.4}{24}} \newlabel{eqn:rule_action}{{{30}}{25}} \newlabel{eqn:chebotarev_condition1}{{{31}}{25}} \citation{bd_derived_regulators} \citation{bd_derived_regulators} \citation{bd_derived_regulators} \newlabel{prop:h1free}{{3.5}{26}} \newlabel{thm:control}{{3.6}{26}} \newlabel{eqn:esGinv}{{32}{27}} \@writefile{toc}{\contentsline {subsection}{\numberline {3.3}Admissible pairs}{27}} \newlabel{sec:admissible_pairs}{{3.3}{27}} \citation{ramakrishna} \citation{ddt} \citation{wiles} \citation{wiles} \citation{ddt} \citation{ddt} \newlabel{prop:isolated_selmer}{{3.8}{29}} \citation{ddt} \citation{ddt} \citation{ddt} \citation{ddt} \newlabel{prop:selmeradjoint}{{3.9}{30}} \newlabel{prop:selmertrivadj}{{3.10}{30}} \newlabel{thm:chebotarev2}{{3.12}{31}} \newlabel{eqn:chebotarev_condition2}{{{33}}{32}} \newlabel{eqn:chebotarev_condition2case2}{{{34}}{32}} \newlabel{thm:chebotarev2prime}{{3.13}{33}} \newlabel{prop:congadmissible}{{3.14}{33}} \newlabel{eqn:congform}{{{35}}{33}} \@writefile{toc}{\contentsline {section}{\numberline {4}The Euler System Argument}{34}} \newlabel{sec:euler}{{4}{34}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.1}The Euler system}{34}} \newlabel{sec:euler_system}{{4.1}{34}} \newlabel{thm:residue}{{4.1}{34}} \newlabel{thm:value}{{4.2}{34}} \newlabel{cor:reciprocity}{{4.3}{35}} \@writefile{toc}{\contentsline {subsection}{\numberline {4.2}The argument}{35}} \newlabel{thm:mainf}{{4.4}{35}} \newlabel{eqn:mainphi}{{{36}}{35}} \newlabel{lemma:propertieskappaprime}{{4.5}{36}} \newlabel{lemma:relationell}{{4.6}{36}} \newlabel{eqn:sumkappatilde}{{{37}}{37}} \newlabel{lemma:drop}{{4.8}{38}} \newlabel{lemma:crucial}{{4.9}{38}} \newlabel{eqn:chain1}{{{38}}{38}} \newlabel{eqn:chain2}{{{39}}{38}} \newlabel{eqn:esselmerduals}{{{40}}{39}} \citation{roberts} \citation{BD1} \citation{boutot-carayol} \citation{buzzard} \citation{deligne-rapoport} \citation{buzzard} \@writefile{toc}{\contentsline {section}{\numberline {5}Shimura curves}{40}} \newlabel{sec:curves}{{5}{40}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.1}The moduli definition}{40}} \newlabel{subsec:moduli}{{5.1}{40}} \citation{bas.appendix} \citation{deligne-rapoport} \citation{katz-mazur} \citation{buzzard} \citation{bas.appendix} \citation{boutot-carayol} \citation{joli.mathann} \citation{boutot-carayol} \citation{BD3} \citation{drinfeld} \citation{joli.mathann} \citation{boutot-carayol} \@writefile{toc}{\contentsline {subsection}{\numberline {5.2}The Cerednik-Drinfeld theorem}{42}} \newlabel{subsec:cerednik}{{5.2}{42}} \newlabel{thm:cerednik}{{5.1}{42}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.3}Character groups}{43}} \newlabel{subsec:characters}{{5.3}{43}} \citation{bas.appendix} \citation{boutot-carayol} \newlabel{prop:characters}{{5.2}{44}} \newlabel{prop:characters+}{{5.3}{44}} \newlabel{prop:graph}{{5.4}{44}} \citation{joli.duke} \citation{ribet1} \citation{ribet} \citation{vigneras} \newlabel{prop:characters-}{{5.5}{45}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.4}Hecke operators and the Jacquet-Langlands correspondence}{45}} \newlabel{subsec:hecke}{{5.4}{45}} \newlabel{lemma:cosets}{{5.6}{45}} \citation{BD4} \newlabel{prop:identifications}{{5.7}{46}} \citation{BD1} \newlabel{prop:operators}{{5.8}{47}} \citation{BD2} \citation{BD3} \citation{ribet} \citation{ribet1} \citation{ribet1} \newlabel{prop:sequence}{{5.9}{48}} \citation{grothendieck} \citation{bas.appendix} \newlabel{cor:jacquet-langlands}{{5.10}{49}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.5}Connected components}{49}} \newlabel{subsec:components}{{5.5}{49}} \newlabel{prop:grothendieck}{{5.11}{49}} \newlabel{cor:components}{{5.12}{50}} \newlabel{prop:structure}{{5.13}{50}} \citation{ribet} \citation{grothendieck} \citation{bas.appendix} \citation{ribet} \citation{joli.compositio} \citation{bas.appendix} \citation{raynaud.picard} \newlabel{prop:specialization}{{5.14}{52}} \@writefile{toc}{\contentsline {subsection}{\numberline {5.6}Raising the level and groups of connected components}{53}} \newlabel{sec:raising}{{5.6}{53}} \newlabel{thm:raising}{{5.15}{53}} \citation{ribet} \newlabel{lemma:multiplicity}{{5.16}{55}} \newlabel{eqn:uniformization}{{{42}}{55}} \newlabel{eqn:[]versus<>}{{{43}}{55}} \newlabel{thm:multiplicity}{{5.17}{55}} \citation{BLR} \newlabel{eqn:tate1}{{{44}}{56}} \newlabel{eqn:tate2}{{{45}}{56}} \newlabel{eqn:tate3}{{46}{56}} \newlabel{eqn:tate4}{{{48}}{56}} \newlabel{eqn:tate5}{{{49}}{56}} \citation{BD1} \newlabel{cor:multiplicity}{{5.18}{58}} \newlabel{eqn:tate6}{{{51}}{58}} \@writefile{toc}{\contentsline {section}{\numberline {6}The theory of complex multiplication}{59}} \newlabel{sec:heegner}{{6}{59}} \@writefile{toc}{\contentsline {section}{\numberline {7}Construction of the Euler System}{60}} \newlabel{sec:construction}{{7}{60}} \citation{BD3} \@writefile{toc}{\contentsline {section}{\numberline {8}The first explicit reciprocity law}{61}} \newlabel{sec:explicit1}{{8}{61}} \newlabel{lemma:residue_reformulation}{{8.1}{61}} \citation{BD3} \@writefile{toc}{\contentsline {section}{\numberline {9}The second explicit reciprocity law}{63}} \newlabel{sec:explicit2}{{9}{63}} \newlabel{eqn:kummer}{{{52}}{63}} \newlabel{eqn:kummer2}{{{53}}{63}} \newlabel{eqn:isomodell2}{{{54}}{63}} \newlabel{eqn:kummer3}{{{55}}{63}} \citation{ihara2} \citation{ihara2} \citation{ihara1} \citation{cornut} \newlabel{eqn:jacobi}{{{56}}{64}} \newlabel{lemma:ihara}{{9.1}{64}} \newlabel{eqn:equivariance}{{{57}}{64}} \newlabel{prop:ihara}{{9.2}{64}} \newlabel{eqn:jss}{{{58}}{64}} \citation{waterhouse} \citation{ribet} \newlabel{lemma:gamma=g}{{9.3}{65}} \citation{BD5} \citation{vatsal2} \citation{BD5} \bibcite{boutot-carayol}{BC} \bibcite{bd_derived_regulators}{BD0} \bibcite{bd_derived_heights}{BD$\frac {1}{2}$} \bibcite{BD1}{BD1} \bibcite{BD2}{BD2} \bibcite{BD3}{BD3} \bibcite{BD4}{BD4} \bibcite{BD5}{BD5} \bibcite{bosch}{BLR} \bibcite{BLR}{BoLeRi} \bibcite{buzzard}{Bu} \bibcite{cornut}{Co} \bibcite{daghigh}{Dag} \bibcite{darmon_thesis}{Da} \bibcite{ddt}{DDT} \bibcite{deligne-rapoport}{DR} \bibcite{drinfeld}{Dr} \bibcite{bas.appendix}{Ed} \bibcite{eichler}{Ei} \bibcite{gross_montreal}{Gr1} \bibcite{gross_zagier}{Gr2} \bibcite{grothendieck}{Groth} \bibcite{ihara1}{I1} \bibcite{ihara2}{I2} \bibcite{joli.mathann}{JoLi1} \bibcite{joli.compositio}{JoLi2} \bibcite{joli.duke}{JoLi3} \bibcite{katz-mazur}{KM} \bibcite{mtt}{MTT} \bibcite{mazur_special}{Ma1} \bibcite{ramakrishna}{Ram} \bibcite{raynaud.picard}{Ray} \bibcite{ribet1}{Ri1} \bibcite{ribet}{Ri2} \bibcite{roberts}{Ro} \bibcite{shimura_book}{Sh} \bibcite{vatsal1}{Va1} \bibcite{vatsal2}{Va2} \bibcite{vigneras}{Vi} \bibcite{waterhouse}{Wa} \bibcite{wiles}{W} \bibcite{zhang}{Zh}