Vol. 15, No. 3, 2021

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Computing integral points on $X_{\mathrm{ns}}^+(p)$

Aurélien Bajolet, Yuri Bilu and Benjamin Matschke

Vol. 15 (2021), No. 3, 569–608
Abstract

We develop a general method for computing integral points on modular curves, based on Baker’s inequality. As an illustration, we show that for $11\le p<101$, the only integral points on the curve ${X}_{ns}^{+}\left(p\right)$ are the CM points.

 To the memory of Alan Baker

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Keywords
modular curves, normalizers, nonsplit Cartan subgroups, integral points, Serre's uniformity problem, economical modular units, Baker–Davenport method, lattice point enumeration
Mathematical Subject Classification 2010
Primary: 11-04
Secondary: 11G16, 11Y40, 14G05
Milestones
Revised: 27 July 2020
Accepted: 10 October 2020
Published: 20 May 2021
Authors
 Aurélien Bajolet Lycée Jean Dautet La Rochelle France Yuri Bilu Institut de Mathématiques de Bordeaux Université de Bordeaux & CNRS Talence France Benjamin Matschke Department of Mathematics and Statistics Boston University Boston, MA United States