Algorithms for the Arithmetic of Elliptic Curves using Iwasawa Theory

William Stein
Christian Wuthrich

To appear in Mathematics of Computation

Latex Source

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.