A p-adic criterion for Lehmer’s conjecture

For a non-zero algebraic number [Formula: see text] of degree [Formula: see text], let [Formula: see text] denote its logarithmic Weil height. It is known that when [Formula: see text] is small, and [Formula: see text] is large, the conjugates of [Formula: see text] are clustered near the unit circle and have angular equidistribution in the complex plane about the origin. In this paper, we establish a [Formula: see text]-adic analog of this result by obtaining lower bounds for [Formula: see text] in terms of the number of its conjugates that lie in a finite extension of [Formula: see text], for some prime [Formula: see text]. As a consequence, we prove Lehmer’s conjecture for all [Formula: see text] such that [Formula: see text] many of its conjugates lie in a finite extension of [Formula: see text].