  Using Fourier-Mukai transformations, we prove some results about the ring of
unipotent vector bundles on elliptic curves in positive characteristics. This
ring was determined by Atiyah in characteristic zero, who showed that it is a
polynomial ring in one variable. It turns out that the situation in
characteristic p>0 is completely different and rather bizarre: the ring is
nonnoetherian and contains a subring whose spectrum contains infinitely many
copies of Spec(Z), which are glued with successively higher and higher
infinitesimal identification at the point corresponding to the prime p.
