Given a curve
of the form
over a number field, one can use descents to obtain explicit bounds on the
Mordell-Weil rank of the Jacobian or to prove that the curve has no rational
points. We show how, having performed such a descent, one can easily obtain
additional information which may rule out the existence of rational divisors on
of degree
prime to
.
This can yield sharper bounds on the Mordell-Weil rank by demonstrating
the existence of nontrivial elements in the Shafarevich-Tate group. As an
example we compute the Mordell-Weil rank of the Jacobian of a genus
curve over
by determining
that the
-primary
part of the Shafarevich-Tate group is isomorphic to
.