An analogue for elliptic curves of the Grunwald–Wang example

Roberto Dvornicich; Umberto Zannier

doi:10.1016/j.crma.2003.10.034

Algebraic Geometry/Group Theory

An analogue for elliptic curves of the Grunwald–Wang example

Presented by: Jean-Pierre Serre

Roberto Dvornicich ¹; Umberto Zannier 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

Comptes Rendus. Mathématique, Volume 338 (2004) no. 1, pp. 47-50.

Abstract (Original language)
Résumé

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

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

Received: 2003-04-30
Accepted: 2003-10-21
Published online: 2003-12-09

DOI: 10.1016/j.crma.2003.10.034

Author's affiliations:

Roberto Dvornicich ¹; Umberto Zannier 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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     title = {An analogue for elliptic curves of the {Grunwald{\textendash}Wang} example},
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Roberto Dvornicich; Umberto Zannier. 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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2003.10.034/

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[4] J.-P. Serre Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math., Volume 15 (1972), pp. 259-331

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