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.
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.
Accepted:
Published online:
Pierre Parent 1
@article{CRMATH_2003__336_5_377_0, author = {Pierre Parent}, 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}, }
Pierre Parent. Triviality of $ X_{\mathrm{split}}\mathrm{(N)(}\mathbb{Q})$ for certain congruence classes of N. Comptes Rendus. Mathématique, Volume 336 (2003) no. 5, pp. 377-380. doi : 10.1016/S1631-073X(03)00078-5. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(03)00078-5/
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