On points of small height in infinite extensions

Abstract

In this paper, we introduce the notion of asymptotically positive infinite extensions of $\mathbb{Q}$, in the spirit of the Tsfasman-Vlăduţ theory of asymptotically exact families of number fields. For asymptotically positive extensions, we obtain lower bounds on the logarithmic Weil height, establishing the Bogomolov property for a wide range of infinite non-Galois extensions. Our result encompasses the famous theorem of E. Bombieri and U. Zannier on Bogomolov property for totally $p$-adic extensions of type $(e,f)$. Additionally, our theorem can be interpreted as a $p$-adic equidistribution result on conjugates of $α$, resonating with the archimedean equidistribution theorem à la F. Amoroso-M. Mignotte and Y. Bilu. In the parallel setting of elliptic curves, we derive lower bounds on the canonical height for points on an elliptic curve over asymptotically positive extensions, without any restriction on its reduction type. In particular, this extends a result of M. Baker in the context of totally $ν$-adic extensions, where the elliptic curve is assumed to have semistable reduction at $ν$.