Heuristics and conjectures in direction of a p-adic Brauer--Siegel theorem

Georges Gras (1)

1 LMB - Laboratoire de Mathématiques de Besançon (UMR 6623)

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(\sqrt(D_K)) when K varies in some specified families (e.g., fields of fixed degree). In some sense, we suggest the existence of a p-adic analogue, of the classical Brauer--Siegel Theorem, wearing here on the valuation of the residue at s=1 (essentially equal to #T_K) of the p-adic zeta-function zeta_p(s) of K. We shall use a different definition that of Washington, given in the 1980's, and approach this question via the arithmetical study of T_K since p-adic analysis seems to fail because of possible abundant ``Siegel zeros'' of zeta_p(s), contrary to the classical framework. We give extensive numerical verifications for quadratic and cubic fields (cyclic or not) and publish the PARI/GP programs directly usable by the reader for numerical improvements. Such a conjecture (if exact) reinforces our conjecture that any fixed number field K is p-rational (i.e., T_K=1) for all p >> 0 .

Mots clés: Brauer--Siegel theorem; class field theory; p-adic regulators; Leopoldt's conjecture; zeta-functions

Domaines: Théorie des nombres [math.NT]

Georges Gras. Heuristics and conjectures in direction of a p-adic Brauer--Siegel theorem. 2018. ⟨hal-01682891v3⟩