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{\LARGE Frank's Mock Qual.}

\par\noindent {\Large \sc ELLIPTIC CURVES:}
Our curve is
$$E:\quad y^2 = x^3+4\cdot x.$$
\begin{enumerate}
\item Find $j$: {\em Ans. 1728}\\
\item Find $\omega=\omega_E$: {\em Ans. $\frac{dx}{2y}$}\\
\begin{enumerate}
\item  Compute explicitely the order of the $0$ of $\omega$
  at the point $P=(0,2)$. 
\item Show that $t_P^*\omega=\omega$ where $t_P$ is
translation by $P$.
\end{enumerate}
\item Compute the ring $\End(E/\C)$. {\em Ans. \Z[i].}\\

\item Let $p$ be an odd prime and $K=\Q(E[p])$ the field
obtained by adjoining the coordinates of $p$-torsion
points on $E$ to $\Q$.  Use the Weil pairing to show
that $K/\Q$ is ramified at $p$. 



\par\noindent {\Large \sc ALGEBRAIC GEOMETRY:}


\par\noindent {\Large \sc COMPLEX ANALYSIS:}

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