Computing integral points on Xns+(p)

Bajolet, Aurélien; Bilu, Yuri; Matschke, Benjamin

doi:10.2140/ant.2021.15.569

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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

DOI: 10.2140/ant.2021.15.569

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 $11 \leq p < 101$ , the only integral points on the curve $X_{ns}^{+} (p)$ are the CM points.

To the memory of Alan Baker

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

Received: 23 October 2018

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

Vol. 15, No. 3, 2021