Integral structures on $p$-adic Fourier theory

Bannai, Kenichi; Kobayashi, Shinichi

Annales de l'Institut Fourier, Tome 66 (2016) no. 2, pp. 521-550.

Dans cet article, nous donnons une construction explicite de la transformation de Fourier $p$-adique de Schneider et Teitelbaum, qui nous permet d’étudier son integralité. Comme application, pour toute extension finie $K$ de ${\mathbb{Q}}_p$ nous donnons une certaine base entière de l’espace de $K$-fonctions localement analytiques sur l’anneau des entiers ${𝒪}_K$ , en généralisant la base construite par Amice pour les fonctions localement analytiques sur ${\mathbb{Z}}_p$. Nous utilisons également notre résultat pour démontrer certaines relations de congruence étudiées initialement par Katz et Chellali entre nombres de Bernoulli-Hurwitz aux places non-ordinaires (c’est-à-dire supersingulières).

In this article, we give an explicit construction of the $p$-adic Fourier transform by Schneider and Teitelbaum, which allows for the investigation of the integral property. As an application, we give a certain integral basis of the space of $K$-locally analytic functions on the ring of integers ${𝒪}_K$ for any finite extension $K$ of ${\mathbb{Q}}_p$, generalizing the basis constructed by Amice for locally analytic functions on ${\mathbb{Z}}_p$. We also use our result to prove congruences of Bernoulli-Hurwitz numbers at non-ordinary (i.e. supersingular) primes originally investigated by Katz and Chellali.

Classification : 11S40
Keywords: $p$-adic distribution, $p$-adic Fourier theory, Amice transform, integrality, congruence, Lubin-Tate group, Bernoulli-Hurwitz number, $p$-adic periods
Mot clés : Distribution $p$-adique, Théorie de Fourier $p$-adique, transform d’Amice, intégralité, congruence, groupe de Lubin-Tate, nombre de Bernoulli-Hurwitz, périodes $p$-adiques

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