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 Cp > 0 such that log( # T K ) ≤ Cp • log( √ 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 ζ-function ζp(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 ζp(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. Abelian p-ramification -Main definitions and notationsLet K be a totally real number field of degree d, and let p ≥ 2 be a prime number fulfilling the Leopoldt conjecture in K. We denote by Cℓ K the p-class group of K (ordinary sense) and by E K the group of p-principal global units ε ≡ 1 (mod p|p p) of K.Let's recall from [9,12] the diagram of the so called abelian p-ramification theory, in which K c = KQ c is the cyclotomic Z p -extension of K (as compositum with that of Q), H K the p-Hilbert class field and H pr K the maximal abelian p-ramified (i.e., unramified outside p) pro-p-extension of K.is the group of p-principal units of the completion K p of K at p | p, where p is the maximal ideal of the ring of integers of K p .For any field k, let µ k be the group of roots of unity of k of p-power order. Then put W K := tor Zp U K = p|p µ Kp and W K := W K /µ K , where µ K = {1} or {±1}.