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1. CJM Online first

Gras, Georges
Les $\theta$-régulateurs locaux d'un nombre algébrique -- Conjectures $p$-adiques
Let $K/\mathbb{Q}$ be Galois and let $\eta\in K^\times$ be such that $\operatorname{Reg}_\infty (\eta) \ne 0$. We define the local $\theta$-regulators $\Delta_p^\theta(\eta) \in \mathbb{F}_p$ for the $\mathbb{Q}_p\,$-irreducible characters $\theta$ of $G=\operatorname{Gal}(K/\mathbb{Q})$. A linear representation ${\mathcal L}^\theta\simeq \delta \, V_\theta$ is associated with $\Delta_p^\theta (\eta)$ whose nullity is equivalent to $\delta \geq 1$. Each $\Delta_p^\theta (\eta)$ yields $\operatorname{Reg}_p^\theta (\eta)$ modulo $p$ in the factorization $\prod_{\theta}(\operatorname{Reg}_p^\theta (\eta))^{\varphi(1)}$ of $\operatorname{Reg}_p^G (\eta) := \frac{ \operatorname{Reg}_p(\eta)}{p^{[K : \mathbb{Q}\,]} }$ (normalized $p$-adic regulator). From $\operatorname{Prob}\big (\Delta_p^\theta(\eta) = 0 \ \& \ {\mathcal L}^\theta \simeq \delta \, V_\theta\big ) \leq p^{- f \delta^2}$ ($f \geq 1$ is a residue degree) and the Borel-Cantelli heuristic, we conjecture that, for $p$ large enough, $\operatorname{Reg}_p^G (\eta)$ is a $p$-adic unit or that $p^{\varphi(1)} \parallel \operatorname{Reg}_p^G (\eta)$ (a single $\theta$ with $f=\delta=1$); this obstruction may be lifted assuming the existence of a binomial probability law confirmed through numerical studies (groups $C_3$, $C_5$, $D_6$). This conjecture would imply that, for all $p$ large enough, Fermat quotients, normalized $p$-adic regulators are $p$-adic units and that number fields are $p$-rational. We recall some deep cohomological results that may strengthen such conjectures.

Keywords:$p$-adic regulators, Leopoldt-Jaulent conjecture, Frobenius group determinants, characters, Fermat quotient, Abelian $p$-ramification, probabilistic number theory
Categories:11F85, 11R04, 20C15, 11C20, 11R37, 11R27, 11Y40

2. CJM 2007 (vol 59 pp. 553)

Dasgupta, Samit
Computations of Elliptic Units for Real Quadratic Fields
Let $K$ be a real quadratic field, and $p$ a rational prime which is inert in $K$. Let $\alpha$ be a modular unit on $\Gamma_0(N)$. In an earlier joint article with Henri Darmon, we presented the definition of an element $u(\alpha, \tau) \in K_p^\times$ attached to $\alpha$ and each $\tau \in K$. We conjectured that the $p$-adic number $u(\alpha, \tau)$ lies in a specific ring class extension of $K$ depending on $\tau$, and proposed a ``Shimura reciprocity law" describing the permutation action of Galois on the set of $u(\alpha, \tau)$. This article provides computational evidence for these conjectures. We present an efficient algorithm for computing $u(\alpha, \tau)$, and implement this algorithm with the modular unit $\alpha(z) = \Delta(z)^2\Delta(4z)/\Delta(2z)^3.$ Using $p = 3, 5, 7,$ and $11$, and all real quadratic fields $K$ with discriminant $D < 500$ such that $2$ splits in $K$ and $K$ contains no unit of negative norm, we obtain results supporting our conjectures. One of the theoretical results in this paper is that a certain measure used to define $u(\alpha, \tau)$ is shown to be $\mathbf{Z}$-valued rather than only $\mathbf{Z}_p \cap \mathbf{Q}$-valued; this is an improvement over our previous result and allows for a precise definition of $u(\alpha, \tau)$, instead of only up to a root of unity.

Categories:11R37, 11R11, 11Y40

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