D\'ecompte dans une conjecture de Lang sur les corps de fonctions : cas des courbes

TL;DR

This paper proves a finiteness result and provides explicit bounds for the intersection of a non-isotrivial algebraic curve with a subgroup of its Jacobian over a global function field, extending previous work by Buium and Voloch.

Contribution

It generalizes a conjecture of Lang for function fields by establishing finiteness and explicit bounds for curve-Jacobian subgroup intersections in positive characteristic.

Findings

Proves finiteness of the intersection $X \,\cap\, \Gamma$.

Provides explicit upper bounds for the size of the intersection.

Extends previous results by Buium and Voloch to new cases.

Abstract

Let $X$ be a genus $d$ curve with $d\geq 2$ defined over a global function field $K$ of characteristic $p>0$ with $p>2d+1$. Suppose $X$ non-isotrivial. Let $\Gamma$ be a sub-group of $J(K_s)$, where $J$ is the jacobian of $X$ and $K_s$ is a separable closure of $K$, verifying $\Gamma/p \Gamma$ finite. Then one shows that $X\cap \Gamma$ has finite cardinal and one provides an explicit upper bound. This generalizes a result of Buium and Voloch.