Heights on square of modular curves
Résumé
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~$b(p)$.