Heights on squares of modular curves

Parent, Pierre

doi:10.2140/ant.2018.12.2065

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Heights on squares of modular curves

Pierre Parent

Appendix: Pascal Autissier

Vol. 12 (2018), No. 9, 2065–2122

DOI: 10.2140/ant.2018.12.2065

Abstract

We develop a strategy for bounding from above the height of rational points of modular curves with values in number fields, by functions which are polynomial in the curve’s level. Our main technical tools come from effective Arakelov descriptions of modular curves and jacobians. We then fulfill this program in the following particular case:

If $p$ is a not-too-small prime number, let $X_{0} (p)$ be the classical modular curve of level $p$ over $ℚ$ . Assume Brumer’s conjecture on the dimension of winding quotients of $J_{0} (p)$ . We prove that there is a function $b (p) = O (p^{5} log p)$ (depending only on $p$ ) such that, for any quadratic number field $K$ , the $j$ -height of points in $X_{0} (p) (K)$ which are not lifts of elements of $X_{0}^{+} (p) (ℚ)$ is less or equal to $b (p)$ .

Keywords

modular curves, Arakelov geometry

Mathematical Subject Classification 2010

Primary: 11G18

Secondary: 14G05, 14G40

Milestones

Received: 15 July 2017

Revised: 29 May 2018

Accepted: 15 July 2018

Published: 21 December 2018

Authors

Pierre Parent
Institut de Mathématiques de Bordeaux Université Bordeaux Talence France

Pascal Autissier
Institut de Mathématiques de Bordeaux Université Bordeaux Talence France

Vol. 12, No. 9, 2018