The p-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large
Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 787-800.

In this paper we show that for every prime p5 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:] 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/ can be arbitrarily large, where A is an abelian variety, with dimA bounded by a constant depending only on p.

Nous montrons dans ce papier que pour chaque nombre premier p5, 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/) peut être arbitrairement grande, pour des variétées abéliennes A de dimension bornée par une constante dépendant seulement de p.

Published online:
DOI: 10.5802/jtnb.521
Keywords: Tate-Shafarevich group, elliptic curve, abelian variety
Remke Kloosterman 1

1 Institute for Mathematics and Computer Science (IWI) University of Groningen P.O. Box 800 NL-9700 AV Groningen, The Netherlands Current address: Institut für Geometrie Universität Hannover Welfengarten 1 D-30167 Hannover, Germany
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Remke Kloosterman. The $p$-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large. Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 3, pp. 787-800. doi : 10.5802/jtnb.521. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.521/

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