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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
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References:

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