Heuristics in direction of a p-adic Brauer--Siegel theorem
Résumé
Let p be a fixed prime number. Let K be a totally real number field of discriminant D_K
and let T_K be the torsion group of the Galois group of the maximal abelian p-ramified
pro-p-extension of K (under Leopoldt's conjecture).
We conjecture the existence of a constant C_p>0 such that log(#T_K)≤C_p.log(√D_K)
when K varies in some specified families (e.g., fields of fixed degree).
In some sense, we try to suggest a p-adic analogue, of the classical Brauer--Siegel Theorem,
wearing on the valuation of the residue, at s=1, of the p-adic zeta-function of K.
We approach this question via the arithmetical study of T_K since p-adic analysis fails
contrary to the classical framework.
We give extensive numerical verifications for quadratic and cubic fields (cyclic or not)
and publish the PARI programs. Such a conjecture (if exact) reinforces our conjecture
that any number field is p-rational (i.e., T_K=1), for all p>>0.
Origine : Fichiers produits par l'(les) auteur(s)
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