On the $p$-adic distance between a point of finite order and a curve of genus higher or equal to two

TL;DR

This paper provides an explicit formula for the lower bound of the $p$-adic distance between a torsion point and a curve of genus at least two embedded in its Jacobian, extending previous results to a specific case.

Contribution

It derives an explicit formula for the $p$-adic distance bound involving analytic and Arakelov invariants for curves over number fields.

Findings

Explicit formula for the $p$-adic distance lower bound.

Involves analytic and Arakelov invariants of the curve.

Extends previous bounds to curves with models over number fields.

Abstract

Let $A$ be an abelian variety over ${\bf C}_p$ ($p$ a prime number) and $V\hookrightarrow A$ a closed subvariety. The conjecture of Tate-Voloch predicts that the $p$-adic distance from a torsion point $T\not\in V({\bf C}_p)$ to the variety $V$ is bounded below by a strictly positive constant. This conjecture is proven by Hrushovski and Scanlon, when $A$ has a model over $\bar{{\bf C}}_p$. We give an explicit formula for this constant, in the case where $V$ is a curve embedded into its Jacobian and $V$ has a model over a number field. This explicit formula involves analytic and arakelovian invariants of the curve.