\def\bold{\bf}
\def\F{\bold F}    % use for finite fields
\def\Q{\bold Q}
\def\C{\bold C}
\def\R{\bold R}
\def\Z{\bold Z}
\def\divides{\,\vert\,}       
\def\notdivides{\,\not\vert\,}
\def\SwD{Swinnerton-Dyer}
\def\BSwD{Birch and \SwD}
\def\heightpair#1#2{\langle #1, #2 \rangle}
\def\Ls{L(E,s)}
\def\frac#1#2{{#1\over #2}}
\def\Sha{\hbox{$\amalg\kern-.39em\amalg$}}
%
\magnification 1200
\nopagenumbers
\vskip1in
\centerline{\bf BASIC DEFINITIONS}
\bigskip
\noindent
Elliptic curve $E$: 
$$
y^2+a_1xy+a_3y=x^3+a_2x^2+a_4x+a_6
$$
or
$$
Y^2=X^3-{c_4\over 48}X - {c_6\over 864}
$$
with
$$
c_4^3-c_6^2=1728\Delta
$$
We have
$$
E({\bf Q}) = F\oplus {\bf Z}^r
$$
where $F$ is finite, and $r$ is the {\bf rank\/} of $E$.

\noindent
Define the {\bf Hasse-Weil} $L$-series:
$$
\Ls = \prod_{p\divides \Delta}\left(1-a_pp^{-s}\right)^{-1}
\prod_{p\notdivides \Delta}\left(1-a_pp^{-s}+p^{1-2s}\right)^{-1}
$$
where  $a_p=p+1-|E(\F_p)|$.
%\bigskip
\vskip1in
\centerline{\bf ALL OUR CALCULATIONS ASSUME}
\smallskip
\centerline{\bf THE STANDARD CONJECTURES}
\medskip
\item {1.} Taniyama-Weil: $L(E,s)$ comes from a cusp form $\sum a_n q^n$ of
weight 2 and conductor $N$, and satisfies a functional equation with sign $w$.
\item {2.} Birch-Swinnerton-Dyer: leading term at $s=1$:
$$
\lim_{s\to1}\frac{\Ls}{(s-1)^r} =  \Omega
\frac{|\Sha|\det\left(\heightpair{P_i}{P_j}\right)} {|F|^2}
\prod_{p\divides\Delta}c_p. 
$$
\item {3.} $|\Delta|=p$, $p$ prime, $\Rightarrow N=\Delta$, and
$w=\left(\frac{-c_6}{N}\right)$.

\bye