Lower bounds for regulators of number fields in terms of their discriminants
Shabnam Akhtari1; Jeffrey D. Vaaler2
1 Department of Mathematics, University of Oregon, Eugene, Oregon 97403 USA
2 Department of Mathematics, University of Texas, Austin, Texas 78712 USA
Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 259-282.

We prove inequalities that compare the regulator of a number field with the absolute value of its discriminant. We refine the ideas in Silverman’s work [15] where such general inequalities are first proven. In order to prove our main theorems, we combine these refinements with the authors’ previous results on bounding the product of heights of relative units in a number field extension.

Nous prouvons une inégalité qui compare le régulateur d’un corps de nombres et la valeur absolue de son discriminant. Nous affinons les idées de Silverman [15] où de telles inégalités ont été prouvées pour la première fois. Pour démontrer nos théorèmes principaux, nous combinons ces méthodes avec les bornes pour le produit des hauteurs des unités relatives d’une extension de corps de nombres démontrées dans notre article antérieur.

Received:
Accepted:
Published online:
DOI: 10.5802/jtnb.1245
Classification: 11G50, 11R04, 11R27
Keywords: Regulator and discriminant of a number field, Weil height, Arakelov height

Shabnam Akhtari 1; Jeffrey D. Vaaler 2

1 Department of Mathematics, University of Oregon, Eugene, Oregon 97403 USA
2 Department of Mathematics, University of Texas, Austin, Texas 78712 USA
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights 
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Shabnam Akhtari; Jeffrey D. Vaaler. Lower bounds for regulators of number fields in terms of their discriminants. Journal de théorie des nombres de Bordeaux, Volume 35 (2023) no. 1, pp. 259-282. doi : 10.5802/jtnb.1245. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1245/

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