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

Abstract

We give a criterion to check if, given a prime power p^r with r > 1, the only rational points of the modular curve X₀⁺ (p^r) 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 X_split (p), which intervene in the problem of Serre concerning uniform surjectivity of Galois representations associated to division points of elliptic curves.