Algorithms for the arithmetic of elliptic curves using Iwasawa theory

Abstract

We explain how to use results from Iwasawa theory to obtain information about $p$-parts of Tate-Shafarevich groups of specific elliptic curves over $\mathbb {Q}$. Our method provides a practical way to compute $\#\Sha (E/\mathbb {Q})(p)$ 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 $\Sha (E/\mathbb {Q})[p]=0$ for the 1,534,422 pairs $(E,p)$ consisting of a non-CM elliptic curve $E$ over $\mathbb {Q}$ with conductor $\leq 30{,}000$, rank $\geq 2$, and good ordinary primes $p$ with $5 \leq p < 1000$ and surjective mod-$p$ representation.