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We explain how to use results from Iwasawa theory to obtain
information about p-parts of Tate-Shafarevich groups of specific
elliptic curves over Q. Our method provides a practical way to
compute Sha(E/Q)(p) in many cases when traditional
p-descent methods are completely impractical and also in
situations where results of Kolyvagin do not apply, e.g., when the
rank of the Mordell-Weil 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 non-CM 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.
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