Triviality of Xsplit(N)(Q) for certain congruence classes of N

We give a criterion to check if, given a prime number N, the only rational points of the modular curve X_split(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. To cite this article: P. Parent, C. R. Acad. Sci. Paris, Ser. I 336 (2003).