• DOI:10.1090/mcom/3395
• Corpus ID: 85531967

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

@article{Gras2019HeuristicsAC,
  title={Heuristics and conjectures in the direction of a p-adic Brauer-Siegel Theorem},
  author={Georges Gras},
  journal={Math. Comput.},
  year={2019},
  volume={88},
  pages={1929-1965}
}

• Georges Gras
• Published 12 January 2018
• Mathematics
• Math. Comput.

Let p be a fixed prime number. Let K be a totally real number field of discriminant DK and let TK 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(#TK) ≤ Cp · log( √ DK) 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 #TK) 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 TK 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., TK = 1) for all p ≫ 0. 1. Abelian p-ramification – Main definitions and notations Let 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 ClK the p-class group of K (ordinary sense) and by EK 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 = KQ is the cyclotomic Zp-extension of K (as compositum with that of Q), HK the p-Hilbert class field and H pr K the maximal abelian p-ramified (i.e., unramified outside p) pro-p-extension of K.