We explain how to use results from Iwasawa theory to obtain
information about pparts of TateShafarevich groups of specific
elliptic curves over Q. Our method provides a practical way to
compute Sha(E/Q)(p) in many cases when traditional
pdescent methods are completely impractical and also in
situations where results of Kolyvagin do not apply, e.g., when the
rank of the MordellWeil group is greater than 1. We apply our
results along with a computer calculation to show that
Sha(E/Q)[p]=0 for the 1,534,422 pairs (E,p) consisting of a nonCM elliptic
curve E over Q
with conductor <= 30,000, rank >=2, and good
ordinary primes p with 5 <= p < 1000 and surjective mod p
representation.
