Explicit higher descents on elliptic curves

(Un exposé de John Cremona au séminaire de théorie des nombres de Chevaleret le 15 décembre 2003)

Résumé :

Let $E/K$ be an elliptic curve over a number field. Descent on~$E$
attempts to get information on both the Mordell-Weil group $E(K)$ and
the Shafarevich-Tate group $\Sha(E/K)$. For each $n\ge2$, there is an
exact sequence
\[ 0 \To E(K)/n E(K) \To \Sel^{(n)}(E/K) \To \Sha(E/K)[n] \To 0 \,,\]
where $\Sel^{(n)}(E/K)$ is the $n$-Selmer group. Our goal is to
compute the $n$-Selmer group, and represent its elements explicitly as
curves $C\subset\PP^{n-1}$ (when $n\ge3$). Having this representation
allows searching for points on $C$ (which in turn give points
in~$E(K)$, since $C$ may be seen as an $n$-covering of~$E$), and also
doing higher descents.

Traditionally, only $2$-descent (over $\Q$) has been fully
implemented: in this case $C\rightarrow\PP^{1}$ is a double cover
rather than an embedding, and elements of $\Sel^{(2)}(E/K)$ are
represented by curves of the form $Y^2=g(X)$ where $g$ is a quartic.
Our goal has been to be equally explicit for $n>2$. Our algorithm is
fully worked out for all odd prime~$n$, and has been implemented in
Magma for $K=\Q$ and $n=3$.

The talk will be illustrated by numerical examples.

(Joint work with T.A. Fisher (Cambridge), C. O'Neil(MIT), D. Simon (Caen) and M.Stoll (Bremen)