Title:
The Euler system of cyclotomic units and the converse to Herbrand's theorem

Abstract:
This is the easiest nontrivial example of Kolyvagin's Euler system
method used to bound Selmer and Tate-Shafarevich groups (here, the group
in question is the \chi-eigenspace of the p-part of the class group of
the p-th cyclotomic field where \chi is a nontrivial even character of
the Galois group of this field over its prime field). As an application,
we get an easy numerical criterion for verifying FLT(p) in terms of the
p-divisibility of Bernoulli numbers. The main conjecture of Iwasawa
theory may be proved by an extension of this method.