\relax 
\@writefile{toc}{\contentsline {section}{\tocsection {}{1}{The Group Law on an Elliptic Curve}}{2}}
\newlabel{sec:grouplaw}{{1}{2}}
\newlabel{defn:elliptic}{{1.1}{2}}
\newlabel{defn:grouplaw}{{1.2}{2}}
\newlabel{lem:pregplaw}{{1.3}{2}}
\newlabel{thm:gplaw}{{1.4}{3}}
\newlabel{comm:birathom}{{1.5}{4}}
\newlabel{comm:affineell}{{1.6}{4}}
\newlabel{thm:wform}{{1.7}{4}}
\newlabel{comm:quartic}{{1.8}{5}}
\newlabel{comm:wform}{{1.9}{5}}
\newlabel{ex:gplaw}{{1.10}{6}}
\newlabel{lem:expl}{{1.11}{6}}
\newlabel{ell:degen}{{1.12}{7}}
\newlabel{defn:tors}{{1.13}{7}}
\newlabel{exs:tors}{{1.14}{7}}
\newlabel{thm:hasse}{{1.15}{8}}
\newlabel{ex:hasse}{{1.16}{8}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{2}{The $p$-adic Numbers ${\mathbb  Q}_p$}}{9}}
\newlabel{sec:padics}{{2}{9}}
\newlabel{defn:valuation}{{2.1}{9}}
\newlabel{defn:padicval}{{2.2}{9}}
\newlabel{ex:padicval}{{2.3}{10}}
\newlabel{lem:padicval}{{2.4}{10}}
\newlabel{comm:maxcomm}{{2.5}{10}}
\newlabel{defn:converge}{{2.6}{10}}
\newlabel{exs:convpadic}{{2.7}{11}}
\newlabel{defn:padics}{{2.8}{12}}
\newlabel{exs:padicexp}{{2.9}{13}}
\newlabel{comm:localglobal}{{2.10}{13}}
\newlabel{defn:residuefield}{{2.11}{13}}
\newlabel{thm:series}{{2.12}{14}}
\newlabel{comm:maxseries}{{2.13}{14}}
\newlabel{thm:hensel}{{2.14}{15}}
\newlabel{ex:hensel}{{2.15}{16}}
\newlabel{cor:squares}{{2.16}{16}}
\newlabel{ex:squares}{{2.17}{16}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{3}{The Reduction Map on an Elliptic Curve}}{17}}
\newlabel{sec:red}{{3}{17}}
\newlabel{defn:reduction}{{3.1}{17}}
\newlabel{ex:reduction}{{3.2}{17}}
\newlabel{defn:curvereduc}{{3.3}{18}}
\newlabel{defn:lift}{{3.4}{18}}
\newlabel{ex:lift}{{3.5}{18}}
\newlabel{thm:lifting}{{3.6}{18}}
\newlabel{thm:redhom}{{3.7}{19}}
\newlabel{cor:redhom}{{3.8}{20}}
\newlabel{defn:kerred}{{3.9}{20}}
\newlabel{lem:wz}{{3.10}{22}}
\newlabel{comm:laurent}{{3.11}{23}}
\newlabel{lem:ellformal}{{3.12}{24}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{4}{Formal Groups}}{25}}
\newlabel{sec:formalgroups}{{4}{25}}
\newlabel{defn:formalgroup}{{4.1}{25}}
\newlabel{ex:formalgroup}{{4.2}{25}}
\newlabel{lem:formalinv}{{4.3}{25}}
\newlabel{defn:formalhom}{{4.4}{26}}
\newlabel{ex:homgmga}{{4.5}{26}}
\newlabel{defn:formalmultm}{{4.6}{26}}
\newlabel{lem:psinv}{{4.7}{26}}
\newlabel{lem:multmiso}{{4.8}{27}}
\newlabel{defn:invdiff}{{4.9}{27}}
\newlabel{thm:invdiffunique}{{4.11}{28}}
\newlabel{cor:invdiffs}{{4.12}{29}}
\newlabel{cor:multp}{{4.13}{29}}
\newlabel{defn:formallog}{{4.14}{30}}
\newlabel{thm:formallog}{{4.15}{30}}
\newlabel{comm:exp}{{4.16}{30}}
\newlabel{defn:fm}{{4.17}{30}}
\newlabel{lem:torfm}{{4.18}{31}}
\newlabel{thm:ptors}{{4.19}{31}}
\newlabel{cor:elltor}{{4.20}{32}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{5}{Global Torsion}}{33}}
\newlabel{sec:torsion}{{5}{33}}
\newlabel{lem:inj}{{5.1}{33}}
\newlabel{ex:torsff}{{5.2}{33}}
\newlabel{lem:torsint}{{5.3}{34}}
\newlabel{thm:nagelllutz}{{5.4}{35}}
\newlabel{ex:nagellutz}{{5.5}{35}}
\newlabel{comm:torsgenform}{{5.6}{36}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{6}{A $2$-isogeny on an Elliptic Curve}}{37}}
\newlabel{sec:2isog}{{6}{37}}
\newlabel{lem:phi}{{6.1}{39}}
\newlabel{lem:preimages}{{6.2}{40}}
\newlabel{lem:phiG}{{6.3}{41}}
\newlabel{defn:qmap}{{6.4}{41}}
\newlabel{lem:qhom}{{6.5}{42}}
\newlabel{lem:imqfinite}{{6.6}{42}}
\newlabel{comm:imhqfinite}{{6.7}{44}}
\newlabel{thm:HphiGfinite}{{6.8}{45}}
\newlabel{cor:weakmw}{{6.9}{45}}
\newlabel{thm:weakmw}{{6.10}{46}}
\newlabel{comm:bool}{{6.11}{46}}
\newlabel{ex:isogdesc2}{{6.12}{47}}
\newlabel{comm:sha}{{6.13}{49}}
\newlabel{comm:galcoh}{{6.14}{51}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{7}{The Mordell-Weil Theorem}}{53}}
\newlabel{sec:mw}{{7}{53}}
\newlabel{defn:heightfunction}{{7.1}{53}}
\newlabel{thm:fgA}{{7.2}{53}}
\newlabel{lem:ECht}{{7.3}{54}}
\newlabel{thm:mw}{{7.4}{55}}
\newlabel{comm:rank}{{7.5}{56}}
\newlabel{ex:rank2}{{7.6}{56}}
\@writefile{toc}{\contentsline {section}{\tocsection {}{8}{Cryptography}}{57}}
\newlabel{sec:crypto}{{8}{57}}
\newlabel{ex:pollard}{{8.1}{58}}
\newlabel{ex:ecmethod}{{8.2}{61}}
\bibcite{cas:ell}{1}
\bibcite{sil:AEC1}{2}
\newlabel{tocindent-1}{0pt}
\newlabel{tocindent0}{15.01373pt}
\newlabel{tocindent1}{20.88867pt}
\newlabel{tocindent2}{0pt}
\newlabel{tocindent3}{0pt}
\@writefile{toc}{\contentsline {section}{\tocsection {}{}{References}}{62}}