Algorithms for the arithmetic of elliptic curves using Iwasawa theory

Abstract

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 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.

Proving the Birch and Swinnerton-Dyer conjecture for specific elliptic curves of analytic rank zero and one

“…There is also a relevant algorithm of Stein and Wuthrich based on the work of Kato, PerrinRiou and Schneider (a preprint is available at [54] and the algorithm is implemented in Sage [55]). Suppose that the elliptic curve E and the prime p = 2 are such that E does not have additive reduction at p and ρ E,p is either surjective or reducible.…”

“…These 31 remaining optimal curves are shown in Table 2. If E is in the set {681b1, 1913b1, 2006e1, 2429b1, 2534e1, 2534f 1, 2541d1, 2674b1, 2710c1, 2768c1, 2849a1, 2955b1, 3054a1, 3306b1, 3536h1, 3712j1, 3954c1, 4229a1, 4592f 1, 4606b1}, then the algorithm of Stein and Wuthrich [54] proves the desired upper bound. For the rest of the curves except for 2366d1 and 4914n1, the mod-3 representations are surjective.…”

“…For 2900d1, Corollary 4.8 together with a point search shows that the Heegner index is at most 23 for discriminant −71; hence, Kolyvagin's inequality provides the upper bound of 2 in this case. For 3185c1, the algorithm of Stein and Wuthrich [54] provides the upper bound of 2. In all twelve cases [17] (and the appendix of [2]) found visible non-trivial parts of X(Q, E) [5].…”

“…For such pairs (E, p), we can compute the Heegner index or an upper bound for it, which gives an upper bound on ord p (X (Q, E)). When the results of Kolyvagin and Jetchev were not strong enough to prove BSD(E, p) using the first available Heegner discriminant, the algorithm of Stein and Wuthrich [54] was (although to be fair the former may be strong enough using other Heegner discriminants in these cases). This algorithm always provides a bound in this situation, since p > 3 is prime of non-additive reduction such that ρ E,p is surjective and E is rank zero.…”

“…Kolyvagin's Theorem 4.4 then rules out many pairs (E, p) right away. Then some combination of Theorems 5.2, 5.3 and 5.4 and the algorithm of Stein and Wuthrich [54] will rule out many more pairs. If no combination of these techniques works for the first Heegner index one computes, then another Heegner discriminant must be used.…”