Quadratic Chabauty for Atkin–Lehner quotients of modular curves of prime level and genus 4, 5, 6

We use the method of quadratic Chabauty on the quotients X 0 (N) of modular curves X0(N) by their Fricke involutions to provably compute all the rational points of these curves for prime levels N of genus four, five, and six. We find that the only such curves with exceptional rational points are of levels 137 and 311. In particular there are no exceptional rational points on those curves of genus five and six. More precisely, we determine the rational points on the curves X 0 (N) for N = 137, 173, 199, 251, 311, 157, 181, 227, 263, 163, 197, 211, 223, 269, 271, 359. 1 Introduction The curve X 0 (N) is a quotient of the modular curve X0(N) by the Atkin-Lehner involution wN (also called the Fricke involution). The non-cuspidal points of X 0 (N) classify unordered pairs of elliptic curves together with a cyclic isogeny of degree N between them, where the Atkin-Lehner involution wN sends an isogeny to its dual. The set X + 0 (N)(Q) consists of cusps, points corresponding to CM elliptic curves (CM points), and points corresponding to quadratic Q-curves without complex multiplication. The points of the third kind are referred to as exceptional points. There have been several works related to the study of Q-rational points on Atkin-Lehner quotients of modular curves (see [BDMTV21; Cas09; DL20; EL19; Gal96; Gal99; Gal02; Mer18; Mom87]) and on Atkin-Lehner quotients of Shimura curves (see [Cla03; PY07]). Especially relevant for this work are the articles [Gal96; Gal99; Gal02] in which Galbraith constructs models of all such curves of genus ≤ 5 except for X 0 (263), and conjectures that he has found all exceptional points on these curves. Building on work of Galbraith, Mercuri [Mer18] constructs models for such curves of genus 6 and 7 of prime level and shows that up to a (very) large naive height, there are no exceptional points on six of these curves (those with N = 163, 197, 229, 269 and 359). Towards verifying Galbraith’s conjecture, Balakrishnan, Dogra, Müller, Tuitman, and Vonk [BDMTV21] show that of the curves X 0 (N) of prime level N and genus 2 and 3, the only curves with exceptional rational points are those with level N = 73, 103, 191. The rational points on X 0 (N) for N = {67, 73, 103} were computed by Balakrishnan, Best, Bianchi, Lawrence, Müller, Triantafillou, and Vonk [BBBLMTV19] using quadratic Chabauty and the Mordell-Weil *nadzaga@grad.hr „varul.math@gmail.com …lea.beneish@mail.mcgill.ca §mic181@ucsd.edu ¶shivac@uchicago.edu †Timo.Keller@uni-bayreuth.de **boyaw@math.princeton.edu 02020 Mathematics subject classification: 14G05 (primary); 11G30; 11G18