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.
Accepted:
Published online:
Keywords: Regulator and discriminant of a number field, Weil height, Arakelov height
Shabnam Akhtari 1; Jeffrey D. Vaaler 2

@article{JTNB_2023__35_1_259_0, author = {Shabnam Akhtari and Jeffrey D. Vaaler}, title = {Lower bounds for regulators of number fields in terms of their discriminants}, journal = {Journal de th\'eorie des nombres de Bordeaux}, pages = {259--282}, publisher = {Soci\'et\'e Arithm\'etique de Bordeaux}, volume = {35}, number = {1}, year = {2023}, doi = {10.5802/jtnb.1245}, language = {en}, url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1245/} }
TY - JOUR AU - Shabnam Akhtari AU - Jeffrey D. Vaaler TI - Lower bounds for regulators of number fields in terms of their discriminants JO - Journal de théorie des nombres de Bordeaux PY - 2023 SP - 259 EP - 282 VL - 35 IS - 1 PB - Société Arithmétique de Bordeaux UR - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1245/ DO - 10.5802/jtnb.1245 LA - en ID - JTNB_2023__35_1_259_0 ER -
%0 Journal Article %A Shabnam Akhtari %A Jeffrey D. Vaaler %T Lower bounds for regulators of number fields in terms of their discriminants %J Journal de théorie des nombres de Bordeaux %D 2023 %P 259-282 %V 35 %N 1 %I Société Arithmétique de Bordeaux %U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1245/ %R 10.5802/jtnb.1245 %G en %F JTNB_2023__35_1_259_0
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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