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Mathematics of Computation

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

Author: Georges Gras
Journal: Math. Comp. 88 (2019), 1929-1965
MSC (2010): Primary 11S40, 11R37, 11R29, 11R42
Published electronically: November 20, 2018
MathSciNet review: 3925492
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Abstract: Let $ p$ be a fixed prime number. Let $ K$ be a totally real number field of discriminant $ D_K$, and let $ {\mathcal T}_K$ be the torsion group of the Galois group of the maximal abelian $ p$-ramified pro-$ p$-extension of $ K$. We conjecture the existence of a constant $ {\mathcal C}_p$ such that $ {\rm log}({\raise 1.5pt \hbox {${\scriptscriptstyle \char93 }$}} {\mathcal T}_K) \leq {\mathcal C}_p \cdot {\rm 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, depending here on the valuation of the residue at $ s=1$ (essentially equal to $ {\scriptscriptstyle \char93 } {\mathcal T}_K$) of the $ p$-adic zeta-function $ \zeta _p(s)$ of $ K$. We shall use different definitions from that of Washington, given in the 1980s, and approach this question via the arithmetic study of $ {\mathcal 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. We give some examples of families of number fields where $ {\mathcal C}_p$ exists. Such a conjecture (if exact) reinforces our conjecture that any fixed number field $ K$ is $ p$-rational (i.e., $ {\mathcal T}_K=1$) for all $ p \gg 0$.

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Additional Information

Georges Gras
Affiliation: Villa la Gardette, Chemin Château Gagnière F–38520 Le Bourg d’Oisans, France

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