Algorithms for the arithmetic of elliptic curves using Iwasawa theory

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 #X(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 X(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.