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 shall use a different definition that of Washington, given in the 1980's,
to approach this question via the arithmetical study of T_K
since p-adic analysis seems to fail contrary to the classical framework.
We give extensive numerical verifications for quadratic and cubic fields
(cyclic or not) and publish the PARI/GP 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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