Lower bound for the canonical height on abelian varieties over totally p-adic extensions

Abstract

Let $A/\mathbb{Q}$ be an abelian variety and let $\hat{h}$ be the canonical height on $A(\overline{\Q})$ associated to a symmetric ample line bundle $\mathcal{L}$ on $A$. We prove that $\hat h$ is bounded away from zero on non-torsion points of $A$ defined over the maximal totally $p$-adic extension of $\mathbb{Q}$, for all but finitely many primes $p$. More generally, for abelian varieties over a number field $K$, we obtain a similar height gap over certain infinite extensions of $K$, including Galois extensions with finite local degree at non-archimedean places.