Heuristics and conjectures in the direction of a -adic Brauer-Siegel Theorem
Author:
Georges Gras
Journal:
Math. Comp. 88 (2019), 1929-1965
MSC (2010):
Primary 11S40, 11R37, 11R29, 11R42
DOI:
https://doi.org/10.1090/mcom/3395
Published electronically:
November 20, 2018
MathSciNet review:
3925492
Full-text PDF
View in AMS MathViewer
Abstract | References | Similar Articles | Additional Information
Abstract: Let be a fixed prime number. Let
be a totally real number field of discriminant
, and let
be the torsion group of the Galois group of the maximal abelian
-ramified pro-
-extension of
. We conjecture the existence of a constant
such that
when
varies in some specified families (e.g., fields of fixed degree). In some sense, we suggest the existence of a
-adic analogue, of the classical Brauer-Siegel Theorem, depending here on the valuation of the residue at
(essentially equal to
) of the
-adic zeta-function
of
. We shall use different definitions from that of Washington, given in the 1980s, and approach this question via the arithmetic study of
since
-adic analysis seems to fail because of possible abundant ``Siegel zeros'' of
, 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. We give some examples of families of number fields where
exists. Such a conjecture (if exact) reinforces our conjecture that any fixed number field
is
-rational (i.e.,
) for all
.
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Additional Information
Georges Gras
Affiliation:
Villa la Gardette, Chemin Château Gagnière F–38520 Le Bourg d’Oisans, France
Email:
g.mn.gras@wanadoo.fr
DOI:
https://doi.org/10.1090/mcom/3395
Keywords:
$p$-adic $\zeta$-functions,
class field theory,
$p$-ramification,
Brauer--Siegel Theorem,
discriminants,
$p$-adic regulators
Received by editor(s):
January 28, 2018
Received by editor(s) in revised form:
February 9, 2018, June 12, 2018, July 9, 2018, and August 2, 2018
Published electronically:
November 20, 2018
Article copyright:
© Copyright 2018
American Mathematical Society