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

Abstract

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) $\le$ 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 .