Heights on squares 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 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 ‫.ޑ‬ 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)(‫)ޑ‬ is less or equal to b( p). MSC2010: primary 11G18; secondary 14G05, 14G40.

Néron-Tate heights of cycles on jacobians

“…Upper bounds for h ′ L (F ) and h ′ L (Z 2,α ) play an important role in a recent work [40] by P. Parent. For example, a combination of Theorem 1.3 and Corollary 1.4 yields the estimate…”