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 analogue 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.