Local-global divisibility of rational points in some commutative algebraic groups

Roberto Dvornicich; Umberto Zannier

Bulletin de la Société Mathématique de France (2001)

  • Volume: 129, Issue: 3, page 317-338
  • ISSN: 0037-9484

Abstract

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Let 𝒜 be a commutative algebraic group defined over a number field  k . We consider the following question:Let r be a positive integer and let P ∈ 𝒜 ( k ) . Suppose that for all but a finite number of primes v of k , we have P = r D v for some D v ∈ 𝒜 ( k v ) . Can one conclude that there exists D ∈ 𝒜 ( k ) such that P = r D ?A complete answer for the case of the multiplicative group 𝔾 m is classical. We study other instances and in particular obtain an affirmative answer when r is a prime and  𝒜 is either an elliptic curve or a torus of small dimension with respect to r . Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when r is a prime.

How to cite

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Dvornicich, Roberto, and Zannier, Umberto. "Local-global divisibility of rational points in some commutative algebraic groups." Bulletin de la Société Mathématique de France 129.3 (2001): 317-338. <http://eudml.org/doc/272481>.

@article{Dvornicich2001,
abstract = {Let $\mathcal \{A\}$ be a commutative algebraic group defined over a number field $k$. We consider the following question:Let $r$ be a positive integer and let $P\in \mathcal \{A\}(k)$. Suppose that for all but a finite number of primes $v$ of $k$, we have $P=rD_v$ for some $D_v\in \mathcal \{A\}(k_v)$. Can one conclude that there exists $D\in \mathcal \{A\}(k)$ such that $P=rD$?A complete answer for the case of the multiplicative group $\{\mathbb \{G\}\}_m$ is classical. We study other instances and in particular obtain an affirmative answer when $r$ is a prime and $\mathcal \{A\}$ is either an elliptic curve or a torus of small dimension with respect to $r$. Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when $r$ is a prime.},
author = {Dvornicich, Roberto, Zannier, Umberto},
journal = {Bulletin de la Société Mathématique de France},
keywords = {rationality questions; rational points},
language = {eng},
number = {3},
pages = {317-338},
publisher = {Société mathématique de France},
title = {Local-global divisibility of rational points in some commutative algebraic groups},
url = {http://eudml.org/doc/272481},
volume = {129},
year = {2001},
}

TY - JOUR
AU - Dvornicich, Roberto
AU - Zannier, Umberto
TI - Local-global divisibility of rational points in some commutative algebraic groups
JO - Bulletin de la Société Mathématique de France
PY - 2001
PB - Société mathématique de France
VL - 129
IS - 3
SP - 317
EP - 338
AB - Let $\mathcal {A}$ be a commutative algebraic group defined over a number field $k$. We consider the following question:Let $r$ be a positive integer and let $P\in \mathcal {A}(k)$. Suppose that for all but a finite number of primes $v$ of $k$, we have $P=rD_v$ for some $D_v\in \mathcal {A}(k_v)$. Can one conclude that there exists $D\in \mathcal {A}(k)$ such that $P=rD$?A complete answer for the case of the multiplicative group ${\mathbb {G}}_m$ is classical. We study other instances and in particular obtain an affirmative answer when $r$ is a prime and $\mathcal {A}$ is either an elliptic curve or a torus of small dimension with respect to $r$. Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when $r$ is a prime.
LA - eng
KW - rationality questions; rational points
UR - http://eudml.org/doc/272481
ER -

References

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  1. [1] N. Bourbaki – « Groupes et algèbres de Lie », ch. 2 et 3, Hermann, Paris, 1972. Zbl0483.22001MR573068
  2. [2] J.-L. Colliot-Thélène & J.-J. Sansuc – « La r -équivalence sur les tores », Ann. Sci. École Norm. Sup.10 (1977), p. 175–229. Zbl0356.14007MR450280
  3. [3] J.-J. Sansuc – « Groupe de Brauer et arithmétique des groupes linéaires sur un corps de nombres », J. reine angew. Math. 327 (1981), p. 12–80. Zbl0468.14007MR631309
  4. [4] J.-P. Serre – Algebraic Groups and Class Fields, Springer Verlag, 1988. Zbl0703.14001MR918564

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