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, Tome 35 (2023) no. 1, pp. 259-282.

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.

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.

Reçu le :
Accepté le :
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DOI : 10.5802/jtnb.1245
Classification : 11G50, 11R04, 11R27
Mots clés : 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
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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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, Tome 35 (2023) no. 1, pp. 259-282. doi : 10.5802/jtnb.1245. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1245/

[1] Shabnam Akhtari; Jeffrey D. Vaaler Heights, regulators and Schinzel’s determinant inequality, Acta Arith., Volume 172 (2016) no. 3, pp. 285-298 | MR | Zbl

[2] Shabnam Akhtari; Jeffrey D. Vaaler Independent relative units of low height, Acta Arith., Volume 2002 (2022) no. 4, pp. 389-401 | DOI | MR | Zbl

[3] Francesco Amoroso; Sinnou David Le théorème de Dobrowolski en dimension supérieure, C. R. Acad. Sci. Paris, Volume 326 (1998) no. 10, pp. 1163-1166 | DOI | Zbl

[4] Francesco Amoroso; Evelina Viada Small points on subvarieties of tori, Comment. Math. Helv., Volume 87 (2012) no. 2, pp. 355-383 | DOI | MR | Zbl

[5] Enrico Bombieri; Walter Gubler Heights in Diophantine Geometry, New Mathematical Monographs, 4, Cambridge University Press, 2006

[6] Antone Costa; Eduardo Friedman Ratios of regulators in totally real extensions of number fields, J. Number Theory, Volume 37 (1991) no. 3, pp. 288-297 | DOI | MR | Zbl

[7] Antone Costa; Eduardo Friedman Ratios of regulators in extensions of number fields, Proc. Am. Math. Soc., Volume 119 (1993) no. 2, pp. 381-390 | DOI | MR | Zbl

[8] Thomas W. Cusick Lower bounds for regulators, Number theory, Noordwijkerhout 1983 (Lecture Notes in Mathematics), Volume 1068, Springer, 1984, pp. 63-73 | DOI | MR | Zbl

[9] Edward Dobrowolski On a question of Lehmer and the number of irreducible factors of a polynomial, Acta Arith., Volume 34 (1979), pp. 391-401 | DOI | MR | Zbl

[10] Eduardo Friedman Analytic formulas for the regulator of a number field, Invent. Math., Volume 98 (1989) no. 3, pp. 599-622 | DOI | MR | Zbl

[11] Władysław Narkiewicz Elementary and Analytic Theory of Algebraic Numbers, Springer Monographs in Mathematics, Springer, 2010

[12] Michael E. Pohst Regulatorabschätzungen für total reelle algebraische Zahlkörper, J. Number Theory, Volume 9 (1977), pp. 459-492 | DOI | Zbl

[13] Robert Remak Über Grössenbeziehungen zwischen Diskriminante und Regulator eines algebraischen Zahlkörpers, Compos. Math., Volume 10 (1952), pp. 245-285 | Numdam | MR | Zbl

[14] Joseph H. Silverman The Thue equation and height functions, Approximations diophantiennes et nombres transcendants (Progress in Mathematics), Volume 31, Birkhäuser, 1983, pp. 259-270 | MR | Zbl

[15] Joseph H. Silverman An inequality relating the regulator and the discriminant of a number field, J. Number Theory, Volume 19 (1984), pp. 437-442 | DOI | MR | Zbl

[16] Paul M. Voutier An effective lower bound for the height of algebraic numbers, Acta Arith., Volume 74 (1996) no. 1, pp. 81-95 | DOI | MR | Zbl

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