\documentclass[12pt]{amsart}

\newif\ifpdfstoll
\ifx\pdfoutput\undefined
  \pdfstollfalse
\else
  \pdfoutput=1
  \pdfstolltrue
\fi

\ifpdfstoll
  \usepackage[pdftex]{epsfig}
  \usepackage[pdftex,colorlinks=true,%
              pdftitle={Torsion points on elliptic curves},%
              pdfauthor={William Stein, Michael Stoll}]{hyperref}
  \newcommand{\Gr}[2]{\psfig{file=#1.pdf,width=#2}}
\else
  \usepackage{epsfig}
  \newcommand{\Gr}[2]{\psfig{file=#1.eps,width=#2}}
\fi

\usepackage{amssymb}  % defines \nmid, for example
\usepackage{latexsym} % for \Box
\usepackage{comment}

\usepackage[all]{xy}

\DeclareFontEncoding{OT2}{}{} % to enable usage of cyrillic fonts
\newcommand{\textcyr}[1]{%
 {\fontencoding{OT2}\fontfamily{cmr}\fontseries{m}\fontshape{n}\selectfont #1}}

\newcommand{\Sha}{{\mbox{\textcyr{Sh}}}}

\newcommand{\Z}{{\mathbb Z}}
\newcommand{\Q}{{\mathbb Q}}
\newcommand{\R}{{\mathbb R}}
\newcommand{\F}{{\mathbb F}}
\newcommand{\C}{{\mathbb C}}
\newcommand{\BP}{{\mathbb P}}
\newcommand{\BA}{{\mathbb A}}
\newcommand{\CO}{{\mathcal O}}
\newcommand{\CV}{{\mathcal V}}
\newcommand{\CZ}{{\mathcal Z}}
\newcommand{\CF}{{\mathcal F}}
\newcommand{\HH}{{\mathcal H}}
\newcommand{\To}{\longrightarrow}
\newcommand{\Too}{\;\longrightarrow\;}
\newcommand{\TO}{\quad\To\quad}
\newcommand{\tensor}{\otimes}
\renewcommand{\Bar}{\overline}
\newcommand{\disc}{\operatorname{disc}}
\newcommand{\ks}{\bar{k}}
\newcommand{\GL}{\operatorname{GL}}
\newcommand{\SL}{\operatorname{SL}}
\newcommand{\PGL}{\operatorname{PGL}}
\newcommand{\PSL}{\operatorname{PSL}}
\newcommand{\U}{\operatorname{U}}
\newcommand{\Gal}{\operatorname{Gal}}
\newcommand{\Aut}{\operatorname{Aut}}
\newcommand{\Hom}{\operatorname{Hom}}
\newcommand{\Spec}{\operatorname{Spec}}
\newcommand{\Sel}{\operatorname{Sel}}
\newcommand{\eps}{\varepsilon}
\newcommand{\Sym}{\operatorname{Sym}}
\newcommand{\Cl}{\operatorname{Cl}}
\renewcommand{\Re}{\operatorname{Re}}
\renewcommand{\Im}{\operatorname{Im}}
\newcommand{\Gm}{\mathbb{G}_m}
\newcommand{\red}{{\text{\rm red}}}
\newcommand{\half}{\tfrac{1}{2}}
\newcommand{\lcm}{\operatorname{lcm}}
\newcommand{\Tr}{\operatorname{Tr}}
\newcommand{\isom}{\cong}
\newcommand{\inj}{\hookrightarrow}
\newcommand{\dual}{\vee}
\newcommand{\Res}{\operatorname{Res}}
\newcommand{\dist}{\operatorname{dist}}
\newcommand{\splt}{{\operatorname{split}}}
\newcommand{\rank}{{\text{rank}}}
\newcommand{\tors}{{\text{tors}}}
\newcommand{\floor}[1]{{\left\lfloor #1 \right\rfloor}}
\newcommand{\ceil}[1]{{\left\lceil #1 \right\rceil}}
\newcommand{\xx}{{\mathbf x}}
\newcommand{\emp}[1]{{#1}}

\newenvironment{steplist}[1]%
   {\begin{list}{}{\settowidth{\labelwidth}{#1}%
                   \setlength{\leftmargin}{\labelwidth}%
                   \addtolength{\leftmargin}{\labelsep}}}%
   {\end{list}}

\newtheorem{Theorem}{Theorem}%[section]
\newtheorem{Lemma}[Theorem]{Lemma}
\newtheorem{Proposition}[Theorem]{Proposition}
\newtheorem{Corollary}[Theorem]{Corollary}
\newtheorem{Conjecture}[Theorem]{Conjecture}

\theoremstyle{definition}
\newtheorem{Definition}[Theorem]{Definition}
\newtheorem{Example}[Theorem]{Example}
\newtheorem{Examples}[Theorem]{Examples}
\newtheorem{Remark}[Theorem]{Remark}
\newtheorem{Remarks}[Theorem]{Remarks}
\newtheorem{Question}[Theorem]{Question}
\newtheorem{Problem}[Theorem]{Problem}

%\numberwithin{equation}{section}

\addtolength{\hoffset}{-1cm}
\addtolength{\textwidth}{2cm}

\setlength{\parskip}{0.8ex plus 0.1ex minus 0.1ex}
\setlength{\parindent}{0mm}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

\title[Torsion points on elliptic curves]%
      {Torsion points on elliptic curves \\ over quartic number fields}

\author{Sheldon Kamienny}
\address{University of Southern California,
         3620 South Vermont Ave., KAP 108,
         Los Angeles, California 90089-2532, USA}
\email{kamienny@usc.edu}

\author{William Stein}
\address{University of Washington,
         Department of Mathematics,
         Box 354350,
         Seattle, WA 98195-4350, USA.}
\email{wstein@gmail.com}

\author{Michael Stoll}
\address{Mathematisches Institut,
         Universit\"at Bayreuth,
         95440 Bayreuth, Germany.}
\email{Michael.Stoll@uni-bayreuth.de}

\date{May 11, 2010}

%\keywords{}
%\subjclass[2000]{Primary 11D41, Secondary 11G30, 14G05, 14G25}

\begin{abstract}
  We complete the proof that there are no elliptic curves over a number field~$K$
  of degree $\le 4$ that have a $K$-rational point of prime order~$> 17$.
\end{abstract}

\maketitle

%\renewcommand{\baselinestretch}{1.1}
%\renewcommand{\arraystretch}{1.3}

%=============================================================================

\section{Introduction}

For an integer $d \ge 1$, we let $S(d)$ be the set of primes~$p$ such there
exists an elliptic curve~$E$ over a number field~$K$ of degree~$\le d$
with a $K$-rational point of order~$p$ in~$E(K)$. Mazur has famously proved
that
\[ S(1) = \{2, 3, 5, 7\} . \]
Kamienny and Mazur showed that
\[ S(2) = \{2, 3, 5, 7, 11, 13\}, \]
and Parent, building on earlier work by Kamienny, proved that
\[ S(3) = \{2, 3, 5, 7, 11, 13\}. \]

Our main result is the following.

\begin{Theorem} \label{ThmMain}
  \[ S(4) = \{2, 3, 5, 7, 11, 13, 17\} . \]
\end{Theorem}

\begin{proof}
  Kamienny and Stein~\cite{KS} show that $S(4)$ does not contain any $p > 31$.
  Theorem~\ref{Thm19and23} below shows that $19, 23 \notin S(4)$.
  Theorem~\ref{Thm31} below shows that $31 \notin S(4)$.
  Finally, Theorem~\ref{Thm29} below shows that $29 \notin S(4)$.
\end{proof}

{\sf Should be extended and proper references added.}

%=============================================================================

\section{The result}

We first state a general results on points of degree~$\le d$ on a curve over~$\Q$.
All curves will be assumed to be smooth, projective and geometrically integral.
We will use $C^{(d)}$ to denote the $d$th symmetric power of~$C$.

\begin{Proposition} \label{Prop1}
  Let $C/\Q$ be a curve with Jacobian~$J$, let $d \ge 1$ be an integer,
  and let $\ell$ be a prime of good reduction for~$C$.
  Let $P_0 \in C(\Q)$ be chosen as base-point
  for an embedding $\iota : C \to J$. We make the following assumptions.
  \begin{enumerate}
    \item \label{ass1} $J(\Q)$ is finite.
    \item \label{ass2} $\ell > 2$ or $J(\Q)[2]$ injects into $J(\F_\ell)$
          (for example, $\#J(\Q)$ is odd).
    \item \label{ass3} There is no rational function in~$\Q(C)^\times$ of degree~$\le d$.
    \item \label{ass4} The reduction map $C(\Q) \to C(\F_\ell)$ is surjective.
    \item \label{ass5} If $P \in C(\bar{\F}_\ell) \setminus C(\F_\ell)$ is a point
          of degree~$d' \le d$, then $\Tr_{\F_{\ell^{d'}}/\F_\ell} \iota(P) \in J(\F_\ell)$
          is not in the image of~$J(\Q)$.
  \end{enumerate}
  Then the only points of degree~$\le d$ on~$C$ are the rational points on~$C$.
\end{Proposition}

\begin{proof}
  We will use $\rho_X$ to denote the reduction map $X(\Q) \to X(\F_\ell)$, where
  $X$ is a smooth projective variety over~$\Q$ with good reduction at~$\ell$.
  
  From assumptions \ref{ass1} and~\ref{ass2} (and the fact that $\ell$ is a good prime)
  we can deduce that $\rho_J : J(\Q) \to J(\F_\ell)$ is injective. Assumption~\ref{ass3}
  implies that the map $C^{(d)}(\Q) \to J(\Q)$ induced by~$\iota$ is also injective.
  Considering the commutative diagram
  \[ \xymatrix{ C^{(d)}(\Q) \ar@{^(->}[r]^{\iota} \ar[d]_{\rho_{C^{(d)}}}
                 & J(\Q) \ar@{^(->}[d]^{\rho_J} \\
                C^{(d)}(\F_\ell) \ar[r]^{\iota}
                 & J(\F_\ell),
              }
  \]
  we conclude that $\rho_{C^{(d)}}$ is injective as well.
  
  Assumption~\ref{ass5} tells us that the image of~$\rho_{C^{(d)}}$ is contained
  in the image~$C^{(d)}(\F_\ell)_\splt$ of $C(\F_\ell)^d$ in~$C^{(d)}(\F_\ell)$.
  We now consider the diagram
  \[ \xymatrix{ C(\Q)^d \ar[r] \ar@{->>}[d]_{\rho_{C^d}}
                 & C^{(d)}(\Q) \ar@{^(->}[d]^{\rho_{C^{(d)}}} \\
                {C(\F_\ell)^d} \ar@{->>}[r]
                 & {C^{(d)}(\F_\ell)_{\splt} .}
              }
  \]
  The left hand vertical map is surjective by assumption~\ref{ass4}, the right hand
  vertical map is injective by the argument above. We conclude that $C(\Q)^d$
  surjects onto~$C^{(d)}(\Q)$. This implies the claim, since a non-rational point
  of degree~$\le d$ on~$C$ would induce a point in~$C^{(d)}(\Q)$ that is not
  in the image of~$C(\Q)^d$.
\end{proof}

We can apply this to torsion points on elliptic curves as follows.
We use the cusp at infinity as a base-point on~$X_1(p)$.
Note that $X_1(p)$ has $p-1$ cusps, half of which are rational, the other half
being conjugate and defined over the maximal real subfield of~$\Q(\mu_p)$.
{\sf William: Can you confirm this?}

\begin{Theorem} \label{Thm2}
  Let $p$ and $\ell$ be primes with $\ell \neq p$, and let $d \ge 1$.
  Assume that
  \begin{enumerate}
    \item \label{assT1} $J_1(p)(\Q)$ is finite.
    \item \label{assT2} $\ell > 2$ or $\#J_1(p)(\Q)$ is odd.
    \item \label{assT3} The order of~$\ell$ in~$\F_p^\times/\{\pm1\}$ is $> d$.
    \item \label{assT4} $p > (\sqrt{\ell} + 1)^2$.
    \item \label{assT5} If $E/\F_{\ell^e}$ is an elliptic curve that has an
          $\F_{\ell^e}$-rational point of order~$p$ and $e \le d$, then the
          trace of its image in~$J_1(p)$ is not in the image of~$J_1(p)(\Q)$.
  \end{enumerate}
  Then there are no elliptic curves $E/K$ with $[K : \Q] \le d$ that have
  a $K$-rational point of order~$p$, or else there are infinitely many.
\end{Theorem}

\begin{proof}
  If $p \le 7$, then there are even infinitely many elliptic curves over~$\Q$
  with a rational point of order~$p$. So in the remainder of the proof, we
  can assume that $p \ge 11$.
  
  We use Prop.~\ref{Prop1}. The first two assumptions in the theorem imply the
  first two assumptions of the proposition. We will deal with Assumption~\ref{ass3}
  in the proposition later.
  
  Assumption~\ref{assT3} of the theorem implies that the only cusps of degree~$\le d$
  on~$X_1(p)/\F_\ell$ are the images of the rational cusps (the order of~$\ell$
  in~$\F_p^\times/\{\pm1\}$ gives the length of the Frobenius-orbits on the
  images of the non-rational cusps). Assumption~\ref{assT4} of the theorem
  shows that there all points in~$X_1(p)(\F_\ell)$ are cusps. Together with
  the fact that $X_1(p)(\Q)$ consists of cusps (since $p \ge 11$), this implies
  assumption~\ref{ass4} of the proposition. Assumption~\ref{assT5} of the theorem
  then implies assumption~\ref{ass5} of the proposition, since there are no other
  $\F_{\ell^e}$-rational points except for the $\F_\ell$-rational cusps.
  
  Now we consider assumption~\ref{ass3} of the proposition. If it holds, we can
  apply the proposition, and we find that there are no non-rational points of
  degree~$\le d$ on~$X_1(p)$. This implies that there are no elliptic curves
  over a number field~$K$ of degree~$\le d$ with a $K$-rational point of order~$p$,
  since the rational points on~$X_1(p)$ are all cusps (by our assumption that
  $p \ge 11$).
  
  If assumption~\ref{ass3} of the proposition does not hold, then 
  $X_1(p)^{(d)} \to J_1(p)$ is not an embedding. Let $Y$ be a fiber of positive
  dimension. If $Y$ contains a rational point, then $Y \cong \BP^n$ for some $n \ge 1$,
  and so $Y$ contains infinitely many rational points. Since the image
  of~$X_1(p)(\Q)^d$ in~$X_1(p)^{(d)}(\Q)$ is finite ($X_1(p)(\Q)$ is finite),
  the infinitely many rational points on~$Y$ involve infinitely many non-cuspidal
  points. Hence there are infinitely many elliptic curves with the required
  property. If, on the other hand, none of the positive-dimensional fibers
  contains a rational point, then the proof of Prop.~\ref{Prop1} applies,
  and there are no elliptic curves with the required properties. 
\end{proof}

\begin{Corollary} \label{Cor3}
  Let $p$ be a prime and $d \ge 1$ such that
  \begin{enumerate}
    \item \label{assC1} $J_1(p)(\Q)$ is finite.
    \item \label{assC2} $\#J_1(p)(\Q)[2]$ injects into~$J_1(p)(\F_2)$ and $p > (2^{d/2}+1)^2$,
          or $p > (3^{d/2}+1)^2$.
  \end{enumerate}
  The either there are infinitely many elliptic curves $E/K$ with
  $[K : \Q] \le d$ such that $E(K)$ contains a point of order~$p$,
  or else there are none.
\end{Corollary}

\begin{proof}
  We take $\ell = 2$ or $\ell = 3$ in Thm~\ref{Thm2}. The first two assumptions in
  the theorem are satisfied, and so is assumption~\ref{assT4}. Assumption~\ref{assT3}
  follows from $\ell^d < p + 1$, which is a consequence of assumption~\ref{assC2}
  in the corollary. Finally, assumption~\ref{assT5} again follows from the
  second assumption in the corollary, since by the Hasse bound, any elliptic
  curve $E/\F_{\ell^e}$ with $e \le d$ satisfies $\#E(\F_{\ell^e}) < p$.
  Therefore the theorem applies.
\end{proof}

We quote Prop.~6.2.1.\ of~\cite{CES} (in an equivalent formulation),
adding some more information from Section~6.2 in~loc.cit.

\begin{Proposition} \label{PropRank0}
  The primes $p$ such that $J_1(p)$ has rank zero are the primes
  $p \le 31$ and $41$, $47$, $59$, and~$71$.
  
  For all of these except possibly $p = 29$, the Mordell-Weil group is
  generated by differences of rational cusps, and for all except
  $p = 17, 29, 31$ and~$41$, the order of $J_1(p)(\Q)$ is odd.
\end{Proposition}

We also quote the main result from~\cite{JKP}.

\begin{Theorem} \label{ThmInfMany}
  Exactly the following torsion structures occur infinitely often for
  elliptic curves over number fields~$K$ of degree~$\le 4$.
  \begin{align*}
    \Z/m\Z           &  \quad\text{for $m \le 18$, or $m \in \{20, 21, 22, 24\}$}, \\
    \Z/2\Z \times \Z/2v\Z &  \quad\text{for $v \le 9$}, \\
    \Z/3\Z \times \Z/3v\Z &  \quad\text{for $v \le 3$}, \\
    \Z/4\Z \times \Z/4v\Z &  \quad\text{for $v \le 2$}, \\
    \Z/5\Z \times \Z/5\Z  &  \quad\text{and} \\
    \Z/6\Z \times \Z/6\Z  & .
  \end{align*}
  In particular, there are infinitely many such elliptic curves with a $K$-rational
  point of prime order~$p$ if and only if $p \le 17$.
\end{Theorem}

\begin{Corollary} \label{Cor29and31}
  Let $p = 29$ or~$p = 31$.
  If $J_1(p)(\Q)[2] \to J_1(p)(\F_2)$ is injective, then $p \notin S(4)$.
\end{Corollary}

\begin{proof}
  By Prop.~\ref{PropRank0}, $J_1(29)(\Q)$ and~$J_1(31)(\Q)$ are both finite.
  Since $(2^{4/2} + 1)^2 < 29$, Cor.~\ref{Cor3} applies.
  By Thm.~\ref{ThmInfMany}, there are at most finitely many elliptic
  curves over a number field~$K$ of degree~$\le 4$ that have a $K$-rational
  point of order~$> 17$. Corollary~\ref{Cor3} therefore shows that there are
  in fact no such curves with a $K$-rational point of order~$p$.
\end{proof}

We can use this to deal with~$p = 31$.

\begin{Theorem} \label{Thm31}
  \[ 31 \notin S(4) . \]
\end{Theorem}

\begin{proof}
  From~\cite{CES}, we know that the 2-part of~$J_1(31)(\Q)$ has order~4.
  We construct $X_1(31)$ and the intermediate curve $X_3(31)$ explicitly
  over~$\F_2$ (they are cut out by quadratic relations between the corresponding
  cusp forms). We find the $\F_2$-rational cusps on~$X_1(31)$ and their
  image on~$X_3(31)$. We then check that their differences generate a subgroup
  of~$J_3(31)(\F_2)$ of order a multiple of~4. This shows that the composition
  \[ J_1(31)(\Q)[2] \To J_1(31)(\F_2) \To J_3(31)(\F_2) \]
  is injective, whence the first map is also injective. The claim then
  follows from Cor.~\ref{Cor29and31}. The computations were done
  using {\sf MAGMA}~\cite{Magma}.
\end{proof}

We have not been able to show that $J_1(29)(\Q)[2]$ injects into~$J_1(29)(\F_2)$.
However, with a bit more computational work, we can still obtain the desired
result.

\begin{Theorem} \label{Thm29}
  \[ 29 \notin S(4) . \]
\end{Theorem}

\begin{proof}
  We apply Thm.~\ref{Thm2} with $\ell = 11$. Let $X = X_1(29)$, and let $Y$ be its
  genus~8 quotient by the action of the diamond operator $\langle 12 \rangle$
  (note that $12^2 \equiv -1 \bmod 29$, so we have a double cover). Since
  $(\sqrt{11} + 1)^2 < 29$, the only $\F_{11}$-points on~$X$ are the images of the
  rational cusps ($11$ generates $\F_{29}^\times$, so the other cusps don't split).
  We find all points on~$X/\F_{11}$ of degree 2, 3 and 4. We map them down to~$Y$.
  Taking the image of the infinite cusp as base-point, we embed $Y$ ito its
  Jacobian~$J$. It is known~\cite{CES} that the subgroup generated by the differences
  of rational cusps on~$X/\Q$ has index at most~8 in the Mordell-Weil group
  of~$J_1(29)$. We find that
  \[ J(\F_{11}) \cong (\Z/2\Z)^5 \times \Z/2 \cdot 7 \cdot 17 \cdot 43 \cdot 1933\,\Z ,\]
  so that the image of Mordell-Weil is included in the subgroup~$G$ obtained as the
  cuspidal subgroup plus
  the 2-torsion subgroup. This subgroup~$G$ has index $17 \cdot 1933 = 32861$
  in~$J(\F_{11})$, so there is a good chance that our points will miss it.
  Indeed, taking the image of each prime divisor on~$X_1(29)/\F_{11}$ of degree
  $2 \le d \le 4$ and subtracting $d$~times the base-point, we can check that
  the class of the resulting divisor in~$J(\F_{11})$ is not contained in~$G$.
  This implies that the corresponding
  points on~$X_1(29)/\F_{11}$ have the property that the trace of the image
  in~$J_1(29)$ is not in the image of the Mordell-Weil group. Therefore,
  assumption~\ref{assT5} in Thm.~\ref{Thm2} is satisfied. The first asumption
  holds by Prop.~\ref{PropRank0}, and the remaining ones are clearly satisfied.
  Using Thm.~\ref{ThmInfMany}, we obtain the result.
  
  The computations were done with {\sf MAGMA} again.
\end{proof}

% \begin{Proposition} \label{Prop29}
%   If differences of the rational cusps on~$X_1(29)$ generate~$J_1(29)(\Q)$, then
%   $29 \notin S(4)$.
% \end{Proposition}
% 
% \begin{proof}
%   Again using {\sf MAGMA}, we construct $X_1(29)$ and its quotient $X_7(29)$
%   over~$\F_3$. We check that the images of the points of degrees 3 and~4
%   on~$X_1(29)/\F_3$ (for degree~$\le 2$, there are only the cusps) on~$X_7(29)$
%   do not map into the subgroup generated by the images of the cusps
%   in~$J_7(29)(\F_3)$. This establishes assumption~\ref{assT5} of Thm.~\ref{Thm2},
%   if the condition in the statement of the proposition is satisfied. The
%   other assumptions also hold, so Thm.~\ref{Thm2} together with Thm.~\ref{ThmInfMany}
%   implies the claim.
% \end{proof}

\begin{Theorem} \label{Thm19and23}
  $S(4)$ does not contain $19$ or~$23$.
\end{Theorem}

\begin{proof}
  By Prop.~\ref{PropRank0}, $J_1(19)(\Q)$ and $J_1(23)(\Q)$
  are both finite and of odd order. Taking $p = 19$ or~$23$, $d = 4$, $\ell = 2$
  in Thm.~\ref{Thm2},
  we see that the first four assumptions are satisfied. We need to check
  the last assumption. In fact, an exhaustive enumeration shows that there
  are no elliptic curves over~$\F_{2^4}$ with a point of order~19 or~23, so the
  last assumption is trivially satisfied. (For smaller~$e$, it follows from
  the Hasse bound, since $(2^{3/2} + 1)^2 < 19$.)
  Invoking Thm.~\ref{ThmInfMany} again, we see that Thm.~\ref{Thm2} shows that
  19 and~23 are both not contained in~$S(4)$.
\end{proof}

\begin{Remark}
  If $E/\F_{2^e}$ is an elliptic curve with $e \le 4$ and a point of order
  $p = 19$ or~$23$, then the Hasse bound forces $e = 4$ and $\#E(\F_{2^4}) = p$.
  So the number of points must be odd. This implies that~$E$ is supersingular,
  so $E$ is a twist of an elliptic curve defined over~$\F_{2^2}$. All those
  curves (they are the curves with $j = 0$) have $\#E(\F_{2^4}) \equiv 1 \bmod 4$.
  Does this follow from some theoretical result? If so, we can eliminate the
  computational part from the proof (other than that going into Prop.~\ref{PropRank0}
  and Thm.~\ref{ThmInfMany}, of course.)
\end{Remark}

%=============================================================================

\begin{thebibliography}{99}
  \frenchspacing

  \bibitem{Magma} {\sf MAGMA} is described in
    W. Bosma, J. Cannon and C. Playoust,
    {\it The Magma algebra system I: The user language},
    J. Symb. Comp. {\bf 24} (1997), 235--265. \\
    (Also see the Magma home page at 
    {\tt http://www.maths.usyd.edu.au:8000/u/magma/}\,.)

  \bibitem{CES}
    B. Conrad, B. Edixhoven, and W.A. Stein,
    {\it $J_1(p)$ has connected fibers},
    Documenta Math. {\bf 8} (2003), 331--408.

  \bibitem{JKP}
    D. Jeon, C.H. Kim and E. Park.
    {\it On the torsion of elliptic curves over quartic number fields},
    J. London Math. Soc. {\bf 74} (2006), 1--12.
    doi:10.1112/S0024610706022940 

  \bibitem{KS}
    S. Kamienny and W.A. Stein,
    {\it Torsion points on elliptic curves over quartic number fields},
    Draft manuscript (2010).

\end{thebibliography}

\end{document}

