Algebraic Geometry/Group Theory
An analogue for elliptic curves of the Grunwald–Wang example
Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 47-50.

We give examples of elliptic curves / and rational points P() such that P is divisible by 4 in ( v ) for each rational place v but P is not divisible by 4 in (). This is an analogue of a well-known example, with 𝔾 m in place of : namely, P=16 is a rational 8-th power locally almost everywhere, but not globally in * =𝔾 m ().

Nous donnons des exemples de courbes elliptiques / et de points rationnels P() tels que P soit divisible par 4 dans ( v ) pour tout place rationnelle v, sans que P soit divisible par 4 dans (). Il s'agit d'un analogue d'un exemple bien connu, avec 𝔾 m à la place de : on sait que, dans * =𝔾 m (), P=16 est localement une puissance 8-ième presque partout, mais P n'est pas une puissance 8-ième globalement.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2003.10.034
Dvornicich, Roberto 1; Zannier, Umberto 2

1 Dipartimento di Matematica, Università di Pisa, via Buonarroti, 2, 56127 Pisa, Italy
2 Istituto Universitario di Architettura D.C.A. S.Croce, 191 (Tolentini), 30135 Venezia, Italy
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Dvornicich, Roberto; Zannier, Umberto. An analogue for elliptic curves of the Grunwald–Wang example. Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 47-50. doi : 10.1016/j.crma.2003.10.034. http://www.numdam.org/articles/10.1016/j.crma.2003.10.034/

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