Heights on squares of modular curves
Language
en
Article de revue
This item was published in
Algebra & Number Theory. 2018-12 n° 12:9, p. 2065--2122
Mathematical Sciences Publishers
Date
2018-12English 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 ...Read more >
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 $\bf Q$. 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 $X_0^+ (p) ({\bf Q})$, is less or equal to $b(p)$.Read less <
English Keywords
Arakelov geometry and Diophantine applications
Modular curves
Origin
Hal imported