\documentclass{amsart}
\input ellc/emac
%\input emac

\title{Take Home Exam 2}
\author{Topics in Algebraic Geometry: Elliptic Curves}
\begin{document}
\maketitle

\begin{enumerate}

\item Consider the cubic $E_t: (x+y+z)(xy+yz+zx)=txyz$.
      For which values of $t$ is this an elliptic curve?
      Show that if $E_t$ is an elliptic curve, then 
      $E_t(\Q)_\trs$ has a subgroup of order $6$.

\item Let $\Gamma$ be a subgroup of finite index in $\SL_2(\Z)$.
      Let $\SL_2(\Z) = \bigcup \gamma_i\Gamma$ be a decomposition
      of $\SL_2(\Z)$ into left cosets modulo $\Gamma$, where 
      $i = 1, 2, \ldots, m$. Show that every cusp for $\Gamma$ has
      the form $\gamma_i^{-1}(\infty) \in \bP^1\Q$. 

\item Compute the cusps for $\Gamma_0(2)$. Where are the cusps
      located in the fundamental domain $\cF$ (cf. lecture notes)?
      Which parts of the boundary of $\cF$ are equivalent under
      the action of $\Gamma_0(2)$? Explain why, after gluing these
      parts together, you get a surface of genus $0$ (with the
      cusps missing). 

\item Compute the rank of $E: y^2 = x^3 - 25x$ using $2$-descent.
      Find rational points on each of the curves $C_{a,b,c}$ in
      $W(E)$. Find the torsion subgroup as well as a rational point 
      in $E(\Q) \setminus E(\Q)_\trs$.

\end{enumerate}

\end{document} 
