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 -
