Exceptional points on Atkin--Lehner quotients
Abstract
We study the rational points on the star curve X0*(N) := X0(N)/W(N), the quotient of the classical modular curve X0(N) by the full group of Atkin--Lehner involutions, for squarefree levels N. Rational points on X0*(N) parameterize ℚ-curves, i.e.\ elliptic curves E/ℚ̅ that are isogenous to all of their Galois conjugates. Elkies conjectures that X0*(N) has only CM or cuspidal rational points for all large enough N. We call any other rational points "exceptional". In this article, we provide new examples of exceptional points in genus 3 and 4, and we give evidence that no exceptional points exist in genus g ≥ 5. Moreover, we investigate the underlying geometric reasons that might "explain" why these exceptional points arise in the first place, in the vein of Ogg and Mazur. In particular, we propose geometric explanations for Galbraith's exceptional points on X0*(137) and X0*(311).
Related
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BibTeX
@misc{assaf2026exceptional,
title = {{Exceptional points on Atkin--Lehner quotients}},
author = {Eran Assaf and Sachi Hashimoto and Ari Shnidman},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.00516},
eprint = {2609.00516},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.00516},
}
RIS
TY - UNPB
TI - Exceptional points on Atkin--Lehner quotients
AU - Eran Assaf
AU - Sachi Hashimoto
AU - Ari Shnidman
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.00516
UR - https://arxiv.org/abs/2609.00516
AB - We study the rational points on the star curve X_0^*(N) := X_0(N)/W(N), the quotient of the classical modular curve X_0(N) by the full group of Atkin--Lehner involutions, for squarefree levels N. Rational points on X_0^*(N) parameterize ℚ-curves, i.e.\ elliptic curves E/ℚ̅ that are isogenous to all of their Galois conjugates. Elkies conjectures that X_0^*(N) has only CM or cuspidal rational points for all large enough N. We call any other rational points "exceptional". In this article, we provide new examples of exceptional points in genus 3 and 4, and we give evidence that no exceptional points exist in genus g ≥ 5. Moreover, we investigate the underlying geometric reasons that might "explain" why these exceptional points arise in the first place, in the vein of Ogg and Mazur. In particular, we propose geometric explanations for Galbraith's exceptional points on X_0^*(137) and X_0^*(311).
ER -