Die Ordnung der Schafarewitsch‐Tate‐Gruppe kann beliebig groß werden

Abstract

Sei k ein endlicher algebraischer Zahlkorper, d eine uber k definierte elliptische Kurve. Besitzt A wenigstens einen uber k definierten Punkt 0, so kann auf A eine abeldche Gruppenstruktur definiert werden, wobei o die Rolle des Nullelements uberiiimmt. d wird zu einer iiber k definierten eindimensionalen abelschen Mannigfaltigkeit, und man erhalt umgekehrt alle auf diese Weise. Zur Abkurzung wollen wir d als abelsclze Kurve bezeichnen. Fur jeden Oberkorper K von k sei .-Z(K) die Gruppe der uber K definierten (kurz: K-ratiotzaletz) Punkte von CR sowie K eine algebraische AbschlieBung von K . Das Ziel dieser Arbeit besteht in dem Nachweis, dalj die Ordnungen der

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]). …”

“…The discriminant of E is given by ∆ E = (a 3 − 27b)b 3 , and E has split multiplicative reduction at every prime in Σ(E/Q) = Σ Q ((a 3 − 27b)b). Let k be a number field containing a cubic root of unity ζ.…”

Average size of 2-Selmer groups of elliptic curves, II

“…While it is known that the order of Sel 2 (E/Q) can be arbitrarily large (cf. [1], [11]), the study of its average value has attracted the attention of some authors. For instance, with purely analytic tools, Heath-Brown ( [7], [8]) studied the congruent number curves and his results provide very good understanding of the distribution of the orders of 2-Selmer groups of such curves.…”

Some families of non-congruent numbers

“…Birch and Swinnerton-Dyer [2], Razar [32], Lagrange [19,20], Wada [39] and Nemenzo [27]), employ the Cassels pairing (see e.g. Aoki [1], Bölling [3], Cassels [5], and McGuinness [26]), compare the Selmer groups Sel (ψ) ( E/Q) and Sel (2) (E/Q) as done by Kramer [18] (essentially, the methods mentioned so far are all equivalent to the classical second 2-descent), or use the method usually attributed to Lind [25] but actually going back (in a slightly different context) to Rédei [33] and Dirichlet [8] (I learned this technique from Stroeker & Top [38] and used it in [22] and [21]). In this paper, we continue to use this last method; as we shall see, it will allow us to obtain results that are stronger than those provided by simple second 2-descents.…”