On Elliptic Curves with Large Tate–Shafarevich Groups

Abstract

The Tate-Shafarevich group X(E/K) of an elliptic curve E over a number field K is defined as X(E/K) := Ker H 1 (K, E(K)) −→ v H 1 (K v , E(K v) , where v runs over all (archimedean and non-archimedean) primes of K. This X(E/K) describes the failure of "Hasse principle" for the torsors of E/K. It is conjectured that X(E/K) is finite (still unknown in general). The following question is very natural, but we don't know the answer. Question 1. For each prime p, does there exist an elliptic curve E over Q such that the p-torsion subgroup X(E/Q)[p] of X(E/Q) is nonzero? We can give examples of such elliptic curves for small primes. • E : y 2 = x 3 + 17x ⇒ X(E/Q)[2] = 0. (Lind, Reichardt, '40s) • E : y 2 = x 3 − 24300 ⇒ X(E/Q)[3] = 0. (Selmer, 1951) • E : y 2 + xy = x 3 − 3301465x − 2309192023 ⇒ X(E/Q)[5] = 0. • E : y 2 + xy = x 3 − 3674496x − 2711401518 ⇒ X(E/Q)[7] = 0. • E : y 2 = x 3 +x 2 −21477749985x−1211529110734587 ⇒ X(E/Q)[13] = 0. Remark. One can verify the above assertions for p = 5 and 7 by a p-descent argument (cf. [Be], [Fi]) or by a result of Cassels [Ca2] together with the fact that rankE(Q) = 0. The second argument also works for p = 13 (cf. [Ma]). • E : y 2 = x 3 − 20675209x ⇒ X(E/Q)[11] = 0. • E : y 2 = x 3 − 239228089x ⇒ X(E/Q)[17] = 0. • E : y 2 = x 3 − 258904415517049x ⇒ X(E/Q)[211] = 0. Remark. These curves have complex multiplication by Z[ √ −1]. For such CM curves, Rubin [Ru] proved the full Birch and Swinnerton-Dyer conjecture (mod-ulo 2-parts). The above assertions is verified by using this fact and a result of Tunnell [Tu]. Another computation can be found in [Ro]. Moreover the following question has already been solved affirmatively for p ≤ 5.

On the arithmetic of twists of superelliptic curves

“…However, for some elliptic curves, more is known. For instance, it is proved in [1] as a generalization of Lemmermeyer's work [8] that the 2-part of the TateShafarevich groups of quadratic twists of the elliptic curves considered in this paper can be arbitrarily large.…”

On Positive Proportion of Rank-Zero Twists of Elliptic Curves Over

“…The 2-descent method is explained in the last chapter of Silverman's book [17] (see also [1,4]). For clarity we specify the 2-descent method for elliptic curves E A,B given by (1.1) below.…”

On the Tate–Shafarevich group of semistable elliptic curves with a rational 3-torsion

“…This problem has been studied for n = 2 by Bölling [3], Kramer [8], Lemmermeyer [9] and Atake [1], for n = 3 by Cassels [6], and for n = 5 by Fisher [7]. The families of elliptic curves considered in those works may 2000 Mathematics Subject Classification: 11G05, 11G07, 14H52.…”

“…be divided into two types: one is the family of (quadratic ( [3], [9], [1]) or cubic ( [6])) twists of a fixed elliptic curve, and the other is a one-parameter family of semistable elliptic curves with non-constant j-invariant ( [8], [7]). …”