The $p$-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large

Kloosterman, Remke

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 3, pp. 787-800.

Nous montrons dans ce papier que pour chaque nombre premier $p\ge 5$, la dimension de la partie de $p$-torsion du groupe de Tate et Shafarevich, $Ш(E/K)$, peut être arbitrairement grande, où $E$ est une courbe elliptique définie sur un corps de nombres $K$ de degré borné par une constante dépendant seulement de $p$. En utilisant ce résultat, nous obtenons aussi que la partie de $p$-torsion du $Ш(A/\mathbb{Q})$ peut être arbitrairement grande, pour des variétées abéliennes $A$ de dimension bornée par une constante dépendant seulement de $p$.

In this paper we show that for every prime $p\ge 5$ the dimension of the $p$-torsion in the Tate-Shafarevich group of $E/K$ can be arbitrarily large, where $E$ is an elliptic curve defined over a number field $K$, with $[K:\mathbb{Q}]$ bounded by a constant depending only on $p$. From this we deduce that the dimension of the $p$-torsion in the Tate-Shafarevich group of $A/\mathbb{Q}$ can be arbitrarily large, where $A$ is an abelian variety, with $dimA$ bounded by a constant depending only on $p$.

Mots clés : Tate-Shafarevich group, elliptic curve, abelian variety

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