In this paper we show that for every prime the dimension of the -torsion in the Tate-Shafarevich group of can be arbitrarily large, where is an elliptic curve defined over a number field , with bounded by a constant depending only on . From this we deduce that the dimension of the -torsion in the Tate-Shafarevich group of can be arbitrarily large, where is an abelian variety, with bounded by a constant depending only on .
Nous montrons dans ce papier que pour chaque nombre premier , la dimension de la partie de -torsion du groupe de Tate et Shafarevich, , peut être arbitrairement grande, où est une courbe elliptique définie sur un corps de nombres de degré borné par une constante dépendant seulement de . En utilisant ce résultat, nous obtenons aussi que la partie de -torsion du peut être arbitrairement grande, pour des variétées abéliennes de dimension bornée par une constante dépendant seulement de .
@article{JTNB_2005__17_3_787_0, author = {Kloosterman, Remke}, title = {The $p$-part of {Tate-Shafarevich} groups of elliptic curves can be arbitrarily large}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {787--800}, publisher = {Universit\'e Bordeaux 1}, volume = {17}, number = {3}, year = {2005}, doi = {10.5802/jtnb.521}, mrnumber = {2212126}, zbl = {1153.11313}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.521/} }
TY - JOUR AU - Kloosterman, Remke TI - The $p$-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large JO - Journal de théorie des nombres de Bordeaux PY - 2005 SP - 787 EP - 800 VL - 17 IS - 3 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.521/ DO - 10.5802/jtnb.521 LA - en ID - JTNB_2005__17_3_787_0 ER -
%0 Journal Article %A Kloosterman, Remke %T The $p$-part of Tate-Shafarevich groups of elliptic curves can be arbitrarily large %J Journal de théorie des nombres de Bordeaux %D 2005 %P 787-800 %V 17 %N 3 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.521/ %R 10.5802/jtnb.521 %G en %F JTNB_2005__17_3_787_0
Kloosterman, Remke. 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. http://archive.numdam.org/articles/10.5802/jtnb.521/
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