Heights on square of modular curves

TL;DR

This paper develops a method to bound the heights of rational points on modular curves using Arakelov geometry, and applies it to classical modular curves to establish explicit height bounds under certain conjectural assumptions.

Contribution

It introduces a polynomial-level bound strategy for heights on modular curves and applies it to $X_0(p)$, assuming Brumer's conjecture, providing explicit height bounds for rational points.

Findings

Bounded the height of non-lift rational points on $X_0(p)$ by $O(p^{5} \log p)$

Established a conditional height bound depending only on the prime level $p$

Applied Arakelov techniques to modular curves for effective height estimates.

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 $\bf Q$. 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 $b(p)$.