\usepackage{amsmath,amssymb,amsthm,amscd}
\include{V3,88}

\pagestyle{myheadings}
\markboth{\sc \quad Franz Lemmermeyer \hfill Elliptic Curves}
         {{\sc Elliptic Curves}} 

\newcommand{\lblequ}[1]{\begin{equation}\label{#1}}
\newcommand{\ReF}[1]{{\rm(\ref{#1})}}
\newcommand{\qrm}[1]{\qquad\mbox{{\rm#1}}}
\newcommand{\mrm}[1]{{\rm #1}}
\renewcommand{\footnoterule}{\hrule}
\newcommand{\footn}[1]{\hspace{-.2em}
             \footnote{\addtolength{\baselineskip}{-1.9pt})#1})}
\newcommand{\doppsum}[2]{\sum_{\overset{\scriptstyle#1}{\scriptstyle#2}}}
\newcommand{\dopprod}[2]{\prod_{\overset{\scriptstyle#1}{\scriptstyle#2}}}
\newcommand{\Alt}[4]{\left\{\begin{array}{ll}#1&#2\\#3&#4\end{array}\right.}
\newcommand{\Altwenn}[4]{\Alt{#1}{\mbox{\ wenn\ }#2}{#3}{\mbox{\ wenn\ }#4}}


\newcommand{\srt}[2]{\sqrt[\raisebox{0.2ex}{$\scriptstyle{#1\,\,}$}]{#2}\,}
\newcommand{\rsp}{\raisebox{0em}[2.7ex][1.3ex]{\rule{0em}{2ex} }}
\newcommand{\0}{{\mathcal O}}
\newcommand{\A}{{\mathbb A}}
\newcommand{\G}{{\mathbb G}}
\newcommand{\aA}{{\mathbb A}}
\newcommand{\bP}{{\mathbb P}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\Qt}{{\mathbb Q}^\times}
\newcommand{\Qts}{\Q^{\times\,2}}
\newcommand{\Kt}{{K^\times}}
\newcommand{\Kts}{K^{\times\,^2}}
\newcommand{\bQ}{{\overline{\mathbb Q}}}
\newcommand{\bD}{\overline{\Delta}}
\newcommand{\cC}{\mathcal C}
\newcommand{\cH}{\mathcal H}
\newcommand{\cO}{\mathcal O}
\newcommand{\cP}{\mathcal P}
\newcommand{\cT}{\mathcal T}
\newcommand{\End}{{\rm End}}
\newcommand{\bE}{\overline{E}}
\newcommand{\bF}{\overline{F}}
\newcommand{\bG}{\overline{G}}
\newcommand{\bH}{\overline{H}}
\newcommand{\bT}{\overline{T}}
\newcommand{\bK}{\overline{K}}
\newcommand{\balpha}{\overline{\alpha}}
\newcommand{\bbeta}{\overline{\beta}}
\newcommand{\bphi}{\overline{\phi}}
\newcommand{\ba}{\overline{a}}
\newcommand{\bb}{\overline{b}}
\newcommand{\bc}{\overline{c}}
\newcommand{\bg}{\overline{g}}
\newcommand{\bl}{\overline{l}}
\newcommand{\bm}{\overline{m}}
\newcommand{\bp}{\overline{P}}
\newcommand{\br}{\overline{r}}
\newcommand{\bs}{\overline{s}}
\newcommand{\bt}{\overline{t}}
\newcommand{\bx}{\overline{x}}
\newcommand{\by}{\overline{y}}
\newcommand{\bz}{\overline{z}}
\newcommand{\olf}{{\overline{f}}}
\newcommand{\olg}{{\overline{g}}}
\newcommand{\olh}{{\overline{h}}}
\newcommand{\oli}{{\overline{i}}}
\newcommand{\sid}{{\mbox{\rm {\scriptsize id}}}}
\newcommand{\sab}{{\mbox{\rm {\scriptsize ab}}}}
\newcommand{\shom}{{\mbox{\rm {\scriptsize h}}}}
\newcommand{\limp}{{\lim \limits_{\longleftarrow}}\,}
\newcommand{\limi}{{\lim \limits_{\longrightarrow}}\,}
\newcommand{\R}{{\mathbb R}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\N}{{\mathbb N}}
\newcommand{\Z}{{\mathbb Z}}
\newcommand{\OO}{\mathcal O}
\newcommand{\cV}{\mathcal V}
\newcommand{\cv}{\text{\rm{\em v}}}
\newcommand{\cE}{\mathcal E}
\newcommand{\car}{\operatorname{char}\,}
\newcommand{\disc}{\operatorname{disc}\,}
\newcommand{\Aut}{\operatorname{Aut}\,}
\newcommand{\Cl}{\operatorname{Cl}\,}
\newcommand{\res}{\operatorname{res}}
\newcommand{\Gal}{\operatorname{Gal}\,}
\newcommand{\Spec}{\operatorname{Spec}\,}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\Res}{\operatorname{Res}\,}
\newcommand{\Tr}{\operatorname{Tr}\,}
\newcommand{\ns}{{\operatorname{ns}}}
\newfont{\cyr}{wncyb10}
\newcommand{\TS}{\mbox{\cyr Sh}}
\newcommand{\im}{\operatorname{im}\,}
\newcommand{\id}{\operatorname{id}\,}
\newcommand{\cok}{\operatorname{coker}\,}
\newcommand{\la}{\langle}
\newcommand{\ra}{\rangle}
\newcommand{\eps}{\varepsilon}
\newcommand{\hra}{\hookrightarrow}
\newcommand{\lra}{\longrightarrow}
\newcommand{\Lra}{\Longrightarrow}
\newcommand{\impl}{\Longrightarrow}
\newcommand{\plim}{\Longleftarrow}
\newcommand{\too}{\longmapsto}
\newcommand{\ds}{\displaystyle}
\newcommand{\ts}{\textstyle}
\newcommand{\dx}{\overset{.}{x}}
\newcommand{\dy}{\overset{.}{y}}
\newcommand{\trs}{{\scriptstyle{\rm tors}}}
\newcommand{\hN}{\widehat{h}}
\newcommand{\tx}{\widetilde{x}}
\newcommand{\ty}{\widetilde{y}}
\newcommand{\tP}{\widetilde{P}}
\newcommand{\tQ}{\widetilde{Q}}

\newtheorem{lemma}{Lemma}
\newtheorem{prop}[lemma]{Proposition}
\newtheorem{thm}[lemma]{Theorem}
\newtheorem{cor}[lemma]{Corollary}