Vol. 12, No. 9, 2018

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Heights on squares of modular curves

Pierre Parent

Appendix: Pascal Autissier

Vol. 12 (2018), No. 9, 2065–2122

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 X0(p) be the classical modular curve of level p over . Assume Brumer’s conjecture on the dimension of winding quotients of J0(p). We prove that there is a function b(p) = O(p5 logp) (depending only on p) such that, for any quadratic number field K, the j-height of points in X0(p)(K) which are not lifts of elements of X0+(p)() is less or equal to b(p).

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modular curves, Arakelov geometry
Mathematical Subject Classification 2010
Primary: 11G18
Secondary: 14G05, 14G40
Received: 15 July 2017
Revised: 29 May 2018
Accepted: 15 July 2018
Published: 21 December 2018
Pierre Parent
Institut de Mathématiques de Bordeaux
Université Bordeaux
Pascal Autissier
Institut de Mathématiques de Bordeaux
Université Bordeaux