Hostname: page-component-7f64f4797f-9t7q9 Total loading time: 0 Render date: 2025-11-07T16:20:56.978Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  21 April 2005

Pierre J. R. Parent
Pierre J. R. Parent
Affiliation:
A2X, UFR mathématiques-informatique, Université de Bordeaux 1, 351 cours de la libération, 33405 Talence cedex, FrancePierre.Parent@math.u-bordeaux1.fr
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

Keywords

elliptic curvesmodular curvesrational pointsGross–Heegner points11G1811G0514G10

Information

Type
Research Article
Information
Compositio Mathematica , Volume 141 , Issue 3 , May 2005 , pp. 561 - 572
Copyright
Foundation Compositio Mathematica 2005