Heights of algebraic numbers modulo multiplicative group actions. (English) Zbl 1235.11066
Let \(K\) be a number field, and let \(G\) be a subgroup of the multiplicative group of \(K\). Silverman defined the \(G\)-height of an algebraic number \(\theta\) by the formula \(\mathcal{H}(\theta,G):=\inf H(g^{1/n} \theta)\), where \(H\) is the usual absolute height and the infimum is taken over every \(g \in G\) and over every positive integer \(n\). Let \(K^1\) be the kernel of the norm map from \(K^*\) to \({\mathbb Q}^*\), and let \(E_K\) be the group of units of \(K\). Then \(\{1\} \subset E_K \subset K^1 \subset K^*\), and so
\[
H(\theta) \geq \mathcal{H}(\theta,E_K) \geq \mathcal{H}(\theta, K^1) \geq \mathcal{H}(\theta,K^*).
\]
The formula involving a product over the archimedean places of \(K(\theta)\) for \(\mathcal{H}(\theta,E_K)\) was established by A.-M. Bergé and J. Martinet [Sémin. Théor. Nombres, Univ. Bordeaux I 1987/1988, Exp. No. 11, 28 p. (1988; Zbl 0699.12013)]. The authors find similar formulas for \(\mathcal{H}(\theta, K^1)\) and \(\mathcal{H}(\theta,K^*)\) and show that \(K^1\)-height (resp. \(K^*\)-height) is equal to \(1\) if and only if \(\theta^n \in K^1\) (resp. \(\theta^n \in K^*\)) for some positive integer \(n\).
Reviewer: Artūras Dubickas (Vilnius)
Citations:
Zbl 0699.12013References:
| [1] | A.M. Bergé, J. Martinet, Minorations de hauteurs et petits régulateurs relatifs, Sém. Théorie des Nombres de Bordeaux, 1987-1988, exposé 11, 1988; A.M. Bergé, J. Martinet, Minorations de hauteurs et petits régulateurs relatifs, Sém. Théorie des Nombres de Bordeaux, 1987-1988, exposé 11, 1988 · Zbl 0699.12013 |
| [2] | Bergé, A. M.; Martinet, J., Notions relatives de régulateurs et de hauteurs, Acta Arith., 54, 156-170 (1989) · Zbl 0642.12011 |
| [3] | de la Maza, A. C., Counting points of bounded relative height, Mathematika, 50, 125-152 (2003) · Zbl 1080.11077 |
| [4] | Ghate, E.; Hironaka, E., The arithmetic and geometry of Salem numbers, Bull. Amer. Math. Soc. (N.S.), 38, 293-314 (2001) · Zbl 0999.11064 |
| [5] | Hindry, M.; Silverman, J., Diophantine Geometry: An Introduction (2000), Springer: Springer Berlin · Zbl 0948.11023 |
| [6] | Lang, S., Algebraic Number Theory (1970), Addison-Wesley: Addison-Wesley Reading, MA · Zbl 0211.38404 |
| [7] | Lang, S., Fundamentals of Diophantine Geometry (1983), Springer: Springer Berlin · Zbl 0528.14013 |
| [8] | Schanuel, S., Heights in number fields, Bull. Soc. Math. France, 107, 433-449 (1979) · Zbl 0428.12009 |
| [9] | Smyth, C. J., On the product of the conjugates outside the unit circle of an algebraic integer, Bull. London Math. Soc., 3, 169-175 (1971) · Zbl 0235.12003 |
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