### 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>