### Explicit Descent over X(3) and X(5), by Catherine H. O'Neil

We split the program of explicit descent of elliptic curves into two parts.
For $n=3$ and $n=5,$ we first display a model for the universal elliptic curve
$E$ with full level $n$ structure and describe the map of rational points of
$E$ to the cohomology group $H^1(G, E[n]).$ Second, we find models in
$\PP^{n-1}$ of principal homogeneous spaces of $E$ corresponding to all possible
elements of $H^1(G, E[n]),$ i.e. for those elements with trivial period-index
obstruction. For this we use the relationship established in \cite{me2} between
the period-index obstruction and the norm symbol, a generalization of the
Hilbert symbol.

Catherine H. O'Neil <coneil@math.mit.edu>