# Heights on square of modular curves

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 level. Our main technical tools come from effective arakelovian 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 $b(p)=O(p^{13} )$ (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 lift of elements of $X_0 (p)/w_p (\Q )$, is less or equal to~$b(p)$.
Keywords :
Type de document :
Pré-publication, Document de travail
2016
Liste complète des métadonnées

https://hal.archives-ouvertes.fr/hal-01448316
Contributeur : Pierre Parent <>
Soumis le : vendredi 27 janvier 2017 - 17:22:10
Dernière modification le : samedi 28 janvier 2017 - 01:03:49

### Citation

Pierre Parent. Heights on square of modular curves . 2016. 〈hal-01448316〉

### Métriques

Consultations de la notice