On the non-Archimedean metric Mahler measure
Paul Fili, Charles L. Samuels
TL;DR
This work introduces a non-Archimedean analogue of the metric Mahler measure, $M_\n∞$, on the multiplicative group of algebraic numbers and proves that $M_\n∞(\alpha)=1$ if and only if $\alpha$ is a root of unity. It shows that $M_\n∞$ induces a projective height on the quotient group $\bar\mathbb Q^\times/\bar\mathbb Q^\times_{\mathrm{tors}}$ viewed as a $\mathbb Q$-vector space, and situates this height relative to existing notions like $M_1$, $M$, and $H$, including a method to compute $M_\n∞(\alpha)$ for surds. The paper then develops a general theory of heights on abelian groups, introducing metric and strong metric height constructions $\rho_1$ and $\rho_\infty$ from a base height $\rho$, and proves foundational properties about these constructions. Finally, it provides detailed proofs of the main results, leveraging Galois norms, Dobrowolski-type bounds, and $p$-adic unit considerations to establish the root-of-unity criterion and to derive explicit values and bounds for $M_\n∞$, including surd cases.
Abstract
Recently, Dubickas and Smyth constructed and examined the metric Mahler measure and the metric naïve height on the multiplicative group of algebraic numbers. We give a non-Archimedean version of the metric Mahler measure, denoted $M_\infty$, and prove that $M_\infty(α) = 1$ if and only if $α$ is a root of unity. We further show that $M_\infty$ defines a projective height on $\bar{\mathbb Q}^\times/ \bar{\mathbb Q}^\times_\mathrm{tors}$ as a vector space over $\mathbb Q$. Finally, we demonstrate how to compute $M_\infty(α)$ when $α$ is a surd.