Heights of algebraic numbers modulo multiplicative group actions

Abstract

Given a number field K and a subgroup G ⊂ K ∗ of the multiplicative group of K, Silverman defined the G-height H(θ;G) of an algebraic number θ as H(θ;G) := inf g∈G, n∈N

H

g1/nθ

, where H on the right is the usual absolute height. When G = EK is the units of K, such a height was introduced by Bergé and Martinet who found a formula for H(θ;EK) involving a curious product over the archimedean places of K(θ). We take the analogous product over all places of K(θ) and find that it corresponds to H(θ;K1), where K1 is the kernel of the norm map from K ∗ to Q ∗. We also find that a natural modification of this same product leads to H(θ;K ∗ ). This is a height function on algebraic numbers which is unchanged under multiplication by K ∗. For G = K1, or G = K ∗, we show that H(θ;G) = 1 if and only if θn ∈ G for some positive integer n. For these same G we also show that G-heights have the expected finiteness property: for any real number X and any integer N there are, up to multiplication by elements of G, only finitely many algebraic numbers θ such that H(θ;G) < X and [K(θ) : K]<N. For G = EK, all of these statements were proved by Bergé and Martinet.