The local \(\theta\)-regulators of an algebraic number: \(p\)-adic conjectures
type: Item
instance of: scholarly article
zbMATH Open document ID: 1351.11033
DOI: 10.4153/CJM-2015-026-3
author: Georges Gras
published in: Canadian Journal of Mathematics
publication date: 2016-06-03T00:00:00Z
full work available at URL: https://arxiv.org/abs/1511.01755
Mathematics Subject Classification ID: 11F85
Mathematics Subject Classification ID: 11R04
Mathematics Subject Classification ID: 20C15
Mathematics Subject Classification ID: 11C20
Mathematics Subject Classification ID: 11R37
Mathematics Subject Classification ID: 11R27
Mathematics Subject Classification ID: 11Y40
zbMATH DE Number: 6589334
zbMATH Keywords: \(p\)-adic regulators
zbMATH Keywords: Fermat quotients
zbMATH Keywords: Borel-Cantelli heuristics
zbMATH Keywords: Leopoldt-Jaulent conjecture
zbMATH Keywords: Frobenius group determinants
MaRDI profile type: Publication
arXiv ID: 1511.01755
title: The local \(\theta\)-regulators of an algebraic number: \(p\)-adic conjectures
review text: Let \(K/\mathbb Q\) be a normal extension of degree \(n\), with Galois group \(G\). Let \(\eta\in K^\ast\) such that the multiplicative \(\mathbb Z[G]\)-module \(F_\eta\) generated by \(\eta\) has \(\mathbb Z\)-rank \(n\). Given a rational prime \(p\), consider the \(p\)-adic regulator \(\mathcal R_p(\eta):=\mathrm{Frob}^G(\log_p(\eta))\), where the so called Frobenius group determinant is defined as \(\mathrm{Frob}^G(\log_p(\eta))=\det(\log_p(\eta^{\tau\sigma^{-1}}))_{\sigma,\tau\in G}\). Example: if \(K\) is totally real, the usual \(p\)-adic regulator \(\mathcal R_p(K)\) is given by a minor of order \((n-1)\) of \(\mathrm{Frob}^G(\log_p(\varepsilon))\), where \(\varepsilon\) is an adequate Minkowski unit, and Leopoldt's conjecture predicts the non nullity of \(\mathcal R_p(K)\). Getting back to \(\eta\), a natural extension of Leopoldt's conjecture , due to \textit{J.-F. Jaulent} [J. Number Theory 20, 149--158 (1985; Zbl 0571.12007)], asserts that the \(p\)-adic rank \(r_p(F_\eta):=(\dim_{\mathbb Q_p}(\mathbb Q_p\mathrm{Log}_p(F_\eta)))\) (with an obvious notation for \(\mathrm{Log}_p\)) and the \(\mathbb Z\)-rank \(\operatorname{rg}(F_\eta):=\dim_{\mathbb Q}(F_\eta):=\dim_{\mathbb Q}(F_\eta\otimes \mathbb Q)\) must coincide. It is not difficult to show that \(r_p(F_\eta)\) is equal to the rank (in the usual sense of linear algebra) of \(\mathcal R_p(\eta)\). From now on, assume that \(p\) is odd, unramified, coprime to \(\eta\). The main probabilistic/heuristic result of the present paper is that, ultimately, the Borel-Cantelli heuristics imply conjecturally that the normalized \(p\)-adic regulator \(\mathcal R^G_p(\eta):=\mathrm{Frob}^G\left(-\frac{1}{p}\log_p(\eta)\right)\) is not congruent to \(0\mod p\) outside a finite set of primes. More precisely, for a given \(\eta\):{\parindent=0.7cm\begin{itemize}\item[(1)] Assuming the existence of a precise binomial probability law (which we do not record here), the probability that \(\mathcal R^G_p(\eta)\equiv 0\mod…
reviewed by: Thong Nguyen Quang Do
description: scientific article; zbMATH DE number 6589334