  Let E/Q be an elliptic curve and let p be an odd supersingular prime for E.
In this article, we study the simplest case of Iwasawa theory for elliptic
curves, namely when E(Q) is finite, III(E/Q) has no p-torsion and the Tamagawa
factors for E are all prime to p. Under these hypotheses, we prove that E(Q_n)
is finite and make precise statemens about the size and structure of the
p-power part of III(E/Q_n). Here Q_n is the n-th step in the cyclotomic
Z_p-extension of Q.
