A p-adic criterion for Lehmer's conjecture

2025/07/27 by Dixit, Anup B., Kala, Sushant
#11G50 #11R04 #11R06 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.20141

Abstract

For a non-zero algebraic number α of degree d, let h(α) denote its logarithmic Weil height. It is known that when h(α) is small, and d is large, the conjugates of α are clustered near the unit circle and have angular equidistribution in the complex plane about the origin. In this paper, we establish a p-adic analogue of this result by obtaining lower bounds for h(α) in terms of the number of its conjugates that lie in a finite extension of ℚ_p, for some prime p. As a consequence, we prove Lehmer's conjecture for all α such that ≫ √(dlog d) many of its conjugates lie in a finite extension of ℚ_p.

Citations

• Lower bound on height of algebraic numbers and low lying zeros of the Dedekind zeta-function
• Equidistribution quantitative des points de petite hauteur sur la droite projective
• Equidistribution of small points, rational dynamics, and potential theory
• An inequality in the theory of arithmetic on algebraic varieties

Related

• Lehmer's Problem, McKay's Correspondence, and 2,3,7
• A note on logarithmic equidistribution
• A Proof of the Conjecture of Lehmer and of the Conjecture of Schinzel-Zassenhaus
• Experimental investigations on Lehmer's conjecture for elliptic curves
• Analogues of Lehmer's conjecture in positive characteristic