Heights of algebraic numbers modulo multiplicative group actions Original Research Article

Given a number field K and a subgroup Gsubset ofK* of the multiplicative group of K, Silverman defined the G-height image of an algebraic number θ asimage 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 image 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 image, where K1 is the kernel of the norm map from K* to image. We also find that a natural modification of this same product leads to image. This is a height function on algebraic numbers which is unchanged under multiplication by K*. For G=K1, or G=K*, we show that image if and only if θnset membership, variantG 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 image and [K(θ):K]