The local \(\theta\)-regulators of an algebraic number: \(p\)-adic conjectures (Q2810696)

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 \(\mathcal R^G_p(\eta)\equiv 0\mod p\) is less than \(C_\infty(\eta)p^{-A}\) when \(p\to\infty\) where \(A=\log(\log p)/\log(c_0(\eta))-0(1)\), \(e^{-1}\leq C_\infty(\eta)\leq 1\) and \(c_0(\eta)=\max_{\sigma\in G}(|\eta^\sigma|)\). \item[(2)] Under the same assumption, the Borel-Cantelli heuristics imply the finiteness of the set of \(p\) such that \(\mathcal R^G_p(\eta)\equiv 0\mod p\).NEWLINENEWLINE\end{itemize}} Actually the assumption in (1) and (2) on the existence of a binomial probability law is confirmed by numerical studies for \(G=C_3,C_5,D_6\), and can be considered as heuristic.NEWLINENEWLINELet us say a few words of the author's technical approach. First replace \(\mathcal R^G_p(\eta)\) by a \(p\)-adic regulator \(\Delta^G_p(\eta)\) built on \textit{Fermat generalized quotients} \(\alpha_p(\eta)\) which are \(p\)-integers of \(K\) defined as follows: if \(n_p\) denotes the common residual degree of the \(p\)-places of \(K\), then \(\eta^{n_p-1}\) is of the form \(1+p\alpha_p(\eta)\), and \(\Delta^G_p:=\mathrm{Frob}^G(\alpha_p(\eta))\equiv\mathcal R^G_p\mod p\). To study the nullity mod \(p\) of \(\Delta^G_p(\eta)\), following the same heuristics as in [the author, ``Étude probabiliste des quotients de Fermat'', Funct. Approximatio, Comment. Math. 54, No. 1, 115--140 (2016; \url{euclid:facm/1458656166})], introduce \(\mathcal L\), the \(G\)-module of \(\mathbb Z[1/p][G]\) relations relative to \(\alpha_p(\eta)\), i.e., linear relations with coefficients \(u(\sigma)\in \mathbb Z[1/p]\) of the form \(\Sigma_{\sigma\in G}u(\sigma)\alpha_p(\eta)^{\sigma^{-1}}\). Let \(\theta\) be an irreducible \(p\)-adic character of \(G\) and \(f\) the residual degree of \(p\) in the field of values of all the absolutely irreducible characters \(\varphi\mid\theta\). Then, viewed inside \(\mathbb F_p[G]\), the \(G\)-module \(e_\theta\mathcal L\) is non null if and only if the local \(\theta\)-regulator \(\Delta^\theta_p(\eta)\) (defined naturally from \(\Delta^G_p(\eta)\)) is null mod \(p\), and in this case the \(\mathbb F_p\)-dimension of \(e_\theta\mathcal L\) is \(\delta f\varphi(1)\), with \(\delta\leq 1\). The obstruction to the application of the Borel-Cantelli heuristics lies in the cases called \textit{minimal \(p\)-divisibility}, i.e., when \(\Delta^\theta_p(\eta)\equiv 0\mod p\) for a unique \(\theta\) such that \(f=\delta=1\) and \(\mathrm{Reg}^\theta_p(\eta)\sim p\) (where \(\mathrm{Reg}^\theta_p(\eta)\) is defined naturally from \(\mathcal R^G_p(\eta)\)). It is this obstruction which is heuristically removed by the probabilistic assumption in the main theorem.