[Trivialité de pour certaines classes de congruence de N]
Soit N un nombre premier. On donne un critère permettant de vérifier si les points rationnels de la courbe modulaire Xsplit(N) sont triviaux (c'est-à-dire des pointes ou des points fournis par la multiplication complexe). On montre ensuite que ce critère est satisfait si N est assez grand et vérifie certaines congruences explicites.
We give a criterion to check if, given a prime number N, the only rational points of the modular curve Xsplit(N) are trivial (i.e., cusps or points furnished by complex multiplication). We then prove that this criterion is verified for large enough N satisfying some explicit congruences.
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@article{CRMATH_2003__336_5_377_0,
author = {Parent, Pierre},
title = {Triviality of $ X_{\mathrm{split}}\mathrm{(N)(}\mathbb{Q})$ for certain congruence classes {of~\protect\emph{N}}},
journal = {Comptes Rendus. Math\'ematique},
pages = {377--380},
publisher = {Elsevier},
volume = {336},
number = {5},
year = {2003},
doi = {10.1016/S1631-073X(03)00078-5},
language = {en},
url = {http://www.numdam.org/articles/10.1016/S1631-073X(03)00078-5/}
}
TY - JOUR
AU - Parent, Pierre
TI - Triviality of $ X_{\mathrm{split}}\mathrm{(N)(}\mathbb{Q})$ for certain congruence classes of N
JO - Comptes Rendus. Mathématique
PY - 2003
SP - 377
EP - 380
VL - 336
IS - 5
PB - Elsevier
UR - http://www.numdam.org/articles/10.1016/S1631-073X(03)00078-5/
DO - 10.1016/S1631-073X(03)00078-5
LA - en
ID - CRMATH_2003__336_5_377_0
ER -
%0 Journal Article
%A Parent, Pierre
%T Triviality of $ X_{\mathrm{split}}\mathrm{(N)(}\mathbb{Q})$ for certain congruence classes of N
%J Comptes Rendus. Mathématique
%D 2003
%P 377-380
%V 336
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/S1631-073X(03)00078-5/
%R 10.1016/S1631-073X(03)00078-5
%G en
%F CRMATH_2003__336_5_377_0
Parent, Pierre. Triviality of $ X_{\mathrm{split}}\mathrm{(N)(}\mathbb{Q})$ for certain congruence classes of N. Comptes Rendus. Mathématique, Tome 336 (2003) no. 5, pp. 377-380. doi : 10.1016/S1631-073X(03)00078-5. http://www.numdam.org/articles/10.1016/S1631-073X(03)00078-5/
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