\relax \citation{rubin} \citation{perrin-riou} \citation{pr-book} \citation{bdp1} \citation{rubin} \citation{rubin} \@writefile{toc}{\contentsline {section}{\tocsection {}{1}{Introduction}}{1}} \newlabel{mthm:rubin-main}{{1}{1}} \newlabel{eqn:rubin}{{1.1}{1}} \citation{rubin} \citation{perrin-riou-gz} \citation{bdp1} \citation{rubin} \newlabel{eqn:rubin-eval1}{{1.2}{2}} \newlabel{eqn:rubin-eval2}{{1.3}{2}} \newlabel{eqn:ratio-rubin}{{1.4}{2}} \newlabel{eqn:perrin-riou}{{1.5}{2}} \newlabel{eqn:resnuAQ}{{1.6}{2}} \newlabel{eqn:ass-sign}{{1.7}{2}} \citation{bdp1} \citation{bdp3} \citation{bdp1} \citation{bdp3} \citation{bdp5} \citation{schappacher} \newlabel{eqn:discK-is-odd}{{1.8}{3}} \newlabel{eqn:cond-nu-D}{{1.9}{3}} \newlabel{eqn:defOmegaBnunu}{{1.10}{3}} \newlabel{eqn:defBnunu}{{1.11}{3}} \newlabel{mthm:rubin-general}{{2}{3}} \@writefile{toc}{\contentsline {section}{\tocsection {}{2}{Hecke characters and periods}}{3}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.1}{Algebraic Hecke characters}}{4}} \newlabel{eqn:AHC-defn1}{{2.1}{4}} \newlabel{eqn:AHC-defn2}{{2.2}{4}} \newlabel{eqn:heckechartranslation}{{2.3}{4}} \newlabel{def:alghcinE}{{2.3}{5}} \newlabel{sec:abvargross10}{{2.2}{5}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.2}{Abelian varieties associated to characters of type $(1,0)$}}{5}} \newlabel{def:cm-ab-var}{{2.4}{5}} \citation{shimura-zeta} \newlabel{thm:casselman}{{2.5}{6}} \newlabel{eqn:defni}{{2.4}{6}} \newlabel{lemma:defomegapsichi}{{2.6}{7}} \newlabel{eqn:iastOmega}{{2.5}{7}} \newlabel{eqn:defomegapsichi0}{{2.6}{7}} \newlabel{eqn:defB-asatwist}{{2.7}{7}} \newlabel{eqn:desc-Bpsichi}{{2.9}{7}} \newlabel{eqn:GkactOmega}{{2.10}{7}} \newlabel{eqn:echi-prod}{{2.11}{7}} \newlabel{def:gauss-sum}{{2.7}{8}} \newlabel{eqn:echi-prod-1}{{2.12}{8}} \newlabel{lemma:compare-differentials}{{2.8}{8}} \newlabel{eqn:intro-lambda}{{2.13}{8}} \newlabel{eqn:intro-lambda-bis}{{2.14}{8}} \newlabel{lemma:kartiks-favorite-lemma}{{2.9}{8}} \newlabel{sec:periods-hecke}{{2.3}{8}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.3}{Complex periods and special values of $L$-functions}}{8}} \newlabel{eqn:complex-periodA}{{2.15}{8}} \citation{schappacher} \citation{gs} \citation{blasius} \citation{deshalit} \newlabel{lemma:periodrelation1}{{2.10}{9}} \newlabel{eqn:periodrelation1}{{2.16}{9}} \newlabel{prop:alg-ratio}{{2.11}{9}} \newlabel{thm:blasius}{{2.12}{9}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{2.4}{$p$-adic periods}}{9}} \newlabel{eqn:padic-periodA}{{2.17}{9}} \newlabel{def:rubin-general}{{2.13}{9}} \citation{bdp1} \citation{deshalit} \newlabel{lemma:periodrelation1p}{{2.14}{10}} \newlabel{eqn:periodrelation1p}{{2.18}{10}} \newlabel{prop:alg-ratio-p}{{2.15}{10}} \newlabel{eqn:periodrelations2p}{{2.19}{10}} \@writefile{toc}{\contentsline {section}{\tocsection {}{3}{$p$-adic $L$-functions and Rubin's formula}}{10}} \newlabel{sec:rubinsformula}{{3}{10}} \newlabel{sec:katz-L}{{3.1}{10}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.1}{The Katz $p$-adic $L$-function}}{10}} \newlabel{thm:interpolation}{{3.1}{10}} \newlabel{eqn:interpkatz}{{3.1}{10}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces Critical infinity types for the Katz $p$-adic $L$-function}}{11}} \newlabel{pic:pic1}{{1}{11}} \newlabel{cor:blasius-p}{{3.3}{11}} \newlabel{eqn:key}{{3.2}{11}} \citation{bdp1} \newlabel{def:self-dual}{{3.4}{12}} \newlabel{prop:interp-sd}{{3.5}{12}} \newlabel{eqn:interp-sd}{{3.3}{12}} \newlabel{rem:selfdual}{{3.7}{12}} \newlabel{eqn:sigmapm}{{3.4}{12}} \newlabel{sec:p-adic-Rankin-L}{{3.2}{12}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.2}{$p$-adic Rankin $L$-series}}{12}} \newlabel{def:central-critical-char}{{3.9}{12}} \citation{bdp1} \citation{bdp1} \citation{bdp1} \newlabel{def:SigmaccN}{{3.10}{13}} \newlabel{eqn:alg-part-rankin}{{3.5}{13}} \newlabel{eqn:explicit-C}{{3.6}{13}} \newlabel{eqn:defLpRankin}{{3.7}{13}} \newlabel{sec:p-adic-gz}{{3.3}{13}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.3}{A $p$-adic Gross-Zagier formula}}{13}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces Critical infinity types for the $p$-adic Rankin $L$-function}}{14}} \newlabel{pic:pic2}{{2}{14}} \newlabel{eqn:pullbackfwt2}{{3.8}{14}} \newlabel{eqn:heegner-divisor-1}{{3.9}{14}} \citation{bdp1} \newlabel{eqn:heegner-point}{{3.10}{15}} \newlabel{thm:gross-zagier}{{3.12}{15}} \newlabel{eqn:from-bdp}{{3.11}{15}} \newlabel{eqn:gz-bdp}{{3.12}{15}} \newlabel{eqn:gz-rubin}{{3.13}{15}} \citation{ogg} \citation{zagier} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.4}{A factorisation of the $p$-adic Rankin $L$-series}}{16}} \newlabel{prop:theta_psi}{{3.13}{16}} \newlabel{lemma:strong-heegner-hypothesis}{{3.14}{16}} \newlabel{eqn:choice-of-fN}{{3.14}{16}} \newlabel{lemma:Sigma12}{{3.15}{16}} \newlabel{thm:factoring}{{3.16}{16}} \newlabel{eqn:factoring}{{3.15}{16}} \newlabel{eqn:factorLp}{{3.16}{17}} \newlabel{eqn:factorLalg}{{3.17}{17}} \newlabel{cor:factoring}{{3.17}{17}} \newlabel{sec:rubin}{{3.5}{17}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.5}{Proof of Rubin's Theorem}}{17}} \newlabel{def:good-pair}{{3.18}{17}} \newlabel{rem:goodpair}{{3.19}{17}} \newlabel{rem:goodpair1}{{3.20}{18}} \newlabel{eqn:mod-par-psi}{{3.19}{18}} \newlabel{eqn:heegner-point-psi}{{3.20}{18}} \newlabel{prop:good-pair-provisional}{{3.21}{18}} \newlabel{eqn:good-pair-provisional}{{3.21}{18}} \newlabel{eqn:gzpsichi}{{3.22}{18}} \newlabel{eqn:fact-psi-chi}{{3.23}{18}} \newlabel{eqn:period-simp}{{3.24}{18}} \newlabel{eqn:period-simp-2}{{3.25}{18}} \newlabel{eqn:heegner-point-bis}{{3.26}{18}} \newlabel{lemma:simpaa}{{3.22}{19}} \newlabel{lemma:simpb}{{3.24}{19}} \newlabel{prop:good-pair}{{3.25}{19}} \newlabel{eqn:good-pair}{{3.27}{19}} \citation{rohrlich} \citation{greenberg} \citation{greenberg} \citation{greenberg} \newlabel{prop:choose-psi-chi}{{3.26}{20}} \newlabel{eqn:fields-of-coeffs}{{3.28}{20}} \newlabel{lemma:ralph-greenberg}{{3.27}{20}} \newlabel{eqn:newLvalue}{{3.29}{20}} \newlabel{eqn:fields-of-coeffs-easier}{{3.30}{21}} \newlabel{eqn:step3}{{3.31}{21}} \newlabel{eqn:fields-of-coeffs-even-easier}{{3.32}{21}} \newlabel{eqn:the-clincher}{{3.33}{21}} \citation{serre-book} \citation{ddt} \citation{ddt} \newlabel{eqn:reschi0}{{3.34}{22}} \newlabel{eqn:condHprime}{{3.35}{22}} \newlabel{eqn:condition-on-S}{{3.36}{22}} \citation{gs} \citation{shimura-zeta} \newlabel{sec:rubin-motivic}{{3.6}{23}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.6}{Elliptic curves with complex multiplication}}{23}} \newlabel{eqn:bringinF}{{3.37}{23}} \newlabel{eqn:galactB}{{3.6}{23}} \newlabel{eqn:endoverK}{{3.38}{23}} \newlabel{eqn:prod-form}{{3.39}{23}} \newlabel{eqn:TactB}{{3.40}{23}} \newlabel{eqn:cmgaloisaction1}{{3.41}{24}} \newlabel{eqn:cmgaloisaction2}{{3.42}{24}} \newlabel{eqn:cmgaloisaction3}{{3.43}{24}} \newlabel{eqn:varphicocyle}{{3.44}{24}} \newlabel{prop:minor-variant-indeed}{{3.28}{24}} \newlabel{eqn:sub-kartik}{{3.45}{24}} \newlabel{eqn:restscalars-rationalpoints}{{3.46}{24}} \newlabel{prop:mucomponent}{{3.29}{24}} \newlabel{eqn:galois-equi}{{3.47}{24}} \newlabel{thm:rubin-ec}{{3.30}{25}} \newlabel{eqn:rubin-conseq-gen}{{3.49}{25}} \newlabel{eqn:rewrite-Omega-nu-gen}{{3.50}{25}} \newlabel{eqn:rewrite-point}{{3.51}{25}} \newlabel{sec:simple-setting}{{3.7}{25}} \@writefile{toc}{\contentsline {subsection}{\tocsubsection {}{3.7}{A special case}}{25}} \newlabel{eqn:canonical-hecke-character}{{3.52}{25}} \newlabel{lemma:classify-nus}{{3.32}{25}} \newlabel{eqn:def-chi-part}{{3.53}{25}} \newlabel{thm:rubin-specialised}{{3.33}{25}} \bibcite{bdp1}{BDP-gz} \bibcite{bdp3}{BDP-ch} \bibcite{bdp5}{BDP-co} \bibcite{blasius}{Bl} \bibcite{ddt}{DDT} \bibcite{deshalit}{deS} \bibcite{gs}{GS} \bibcite{greenberg}{Gre} \bibcite{ogg}{Ogg} \bibcite{perrin-riou-gz}{PR1} \bibcite{perrin-riou}{PR2} \bibcite{pr-book}{PR3} \bibcite{rohrlich}{Ro} \bibcite{rubin}{Ru} \bibcite{schappacher}{Scha} \bibcite{serre-book}{Se} \bibcite{shimura-zeta}{Shi} \bibcite{zagier}{Za} \newlabel{eqn:rubin-conseq}{{3.54}{26}} \newlabel{eqn:rewrite-Omega-nu}{{3.55}{26}} \newlabel{eqn:rewrite-log-omega-nu}{{3.56}{26}} \newlabel{eqn:rubin-formula-sp}{{3.57}{26}} \@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{26}} \newlabel{tocindent-1}{0pt} \newlabel{tocindent0}{12.7778pt} \newlabel{tocindent1}{17.77782pt} \newlabel{tocindent2}{29.38873pt} \newlabel{tocindent3}{0pt}