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

Abstract

Let $A$ be an abelian variety over ${\mathbf{C}}_p$ ($p$ 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 $T\notin V({\mathbf{C}}_p)$ 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 ${\bar{\mathbf{C}}}_p$. 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.