Towards the triviality of $X_0^+ (p^r) (mathbb{Q})$ for r > 1

[摘要] We give a criterion to check if, given a prime power pr with r > 1, the only rational points of the modular curve X0+ (pr) are trivial (i.e. cusps or points furnished by complex multiplication). We then prove that this criterion is verified if p satisfies explicit congruences. This applies in particular to the modular curves Xsplit (p), which intervene in the problem of Serre concerning uniform surjectivity of Galois representations associated to division points of elliptic curves.

[学科分类] 数学（综合）