## Sur l’indépendance $$\ell$$-adique de nombres algébriques. (On $$\ell$$- adic independence of algebraic numbers).(French)Zbl 0571.12007

Let $$\ell$$ be a prime number, K a finite Galois extension of the field of rational numbers, and M a finitely generated G-submodule of $$K^{\times}$$. The author gives a conjectural description of the $$\ell$$- adic rank of M in terms of the character of its Galois representation. The main condition for the rank to be as large as one can expect should be that the $${\mathbb{Q}}[G]$$-module $${\mathbb{Q}}\otimes_{{\mathbb{Z}}}M$$ is monogenic.
This description provides 1) a generalization of Leopoldt’s conjecture (the author shows that his conjecture would follow from the algebraic independence of $${\mathbb{Q}}$$-linearly independent p-adic logarithms of algebraic numbers); 2) a generalization of a conjecture of Gross. The author proves his conjectures in some cases, e.g. when G is abelian.
Reviewer: M.Waldschmidt

### MSC:

 11S23 Integral representations 11J85 Algebraic independence; Gel’fond’s method 11R27 Units and factorization 11R52 Quaternion and other division algebras: arithmetic, zeta functions
Full Text:

### References:

 [1] Ax, J., On the units of an algebraic number field, Illinois J. math., 9, 584-589, (1965) · Zbl 0132.28303 [2] Brumer, A., On the units of algebraic number fields, Mathematica, 14, 121-124, (1967) · Zbl 0171.01105 [3] {\scM. Emsalem, H. H. Kisilevsky, and D. B. Wales}, Indépendance linéaire sur $$Q$$ de logarithmes p-adiques de nombres algébriques et rang p-adique du groupe des unités d’un corps de nombres. · Zbl 0547.12003 [4] {\scB. H. Gross}, On the behavior of p-adic L-series at a = 0. [5] Herbrand, J., Sur LES unités d’un corps algébrique, C. R. acad. sci. Paris, 192, 24-27, (1931) · Zbl 0001.00802 [6] Jaulent, J.-F., $$S$$-classes infinitésimales d’un corps de nombres algébriques, Ann. sci. inst. Fourier, 34, 2, (1984) · Zbl 0522.12014 [7] Miyake, M., On the units of an algebraic field, J. math. soc. Japan, 34, 515-525, (1982) · Zbl 0476.12004 [8] Waldschmidt, M., A lower bound for the p-adic rank of the units of an algebraic number field, () · Zbl 0541.12003
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.