Plus and minus logarithms and Amice transform

Cédric Dion; Antonio Lei

doi:10.1016/j.crma.2017.09.012

Number theory

Plus and minus logarithms and Amice transform

Presented by: Jean-Pierre Serre

Cédric Dion ¹; Antonio Lei ¹

¹ Département de mathématiques et de statistique, Université Laval, pavillon Alexandre-Vachon, 1045, avenue de la Médecine, Québec, QC, G1V 0A6, Canada

Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 942-948.

Abstract
Résumé

We give a new description of Pollack's plus and minus p-adic logarithms $\log_{p}^{\pm}$ in terms of distributions. In particular, if $μ_{\pm}$ denote the pre-images of $\log_{p}^{\pm}$ under the Amice transform, we give explicit formulae for the values $μ_{\pm} (a + p^{n} Z_{p})$ for all $a \in Z_{p}$ and all integers $n \geq 1$ . Our formulae imply that the distribution $μ_{-}$ agrees with a distribution studied by Koblitz in 1977. Furthermore, we show that a similar description exists for Loeffler's two-variable analogues of these plus and minus logarithms.

Nous donnons une nouvelle description des logarithmes p-adiques plus et moins définis par Pollack en termes de distributions. En particulier, si $μ_{\pm}$ dénote la pré-image de $\log_{p}^{\pm}$ sous la transformation d'Amice, nous donnons des formules explicites pour les valeurs $μ_{\pm} (a + p^{n} Z_{p})$ pour tout $a \in Z_{p}$ et tout entier $n \geq 1$ . Nos formules impliquent que la distribution $μ_{-}$ correspond à une distribution étudiée par Koblitz en 1977. Par ailleurs, nous montrons qu'il existe une description similaire, due à Loeffler, pour des analogues à deux variables de ces logarithmes plus et moins.

Received: 2017-08-18
Accepted: 2017-09-20
Published online: 2017-09-01

DOI: 10.1016/j.crma.2017.09.012

Author's affiliations:

Cédric Dion ¹; Antonio Lei ¹

¹ Département de mathématiques et de statistique, Université Laval, pavillon Alexandre-Vachon, 1045, avenue de la Médecine, Québec, QC, G1V 0A6, Canada

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     author = {C\'edric Dion and Antonio Lei},
     title = {Plus and minus logarithms and {Amice} transform},
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     pages = {942--948},
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     volume = {355},
     number = {9},
     year = {2017},
     doi = {10.1016/j.crma.2017.09.012},
     language = {en},
}

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Cédric Dion; Antonio Lei. Plus and minus logarithms and Amice transform. Comptes Rendus. Mathématique, Volume 355 (2017) no. 9, pp. 942-948. doi : 10.1016/j.crma.2017.09.012. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.09.012/

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