Integral structures on p-adic Fourier theory
Annales de l'Institut Fourier, Volume 66 (2016) no. 2, p. 521-550

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 ℚ p , generalizing the basis constructed by Amice for locally analytic functions on 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.

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 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 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).

Received : 2015-03-27
Accepted : 2015-06-11
Published online : 2016-02-17
DOI : https://doi.org/10.5802/aif.3018
Classification:  11S40
Keywords: p-adic distribution, p-adic Fourier theory, Amice transform, integrality, congruence, Lubin-Tate group, Bernoulli-Hurwitz number, p-adic periods 
@article{AIF_2016__66_2_521_0,
     author = {Bannai, Kenichi and Kobayashi, Shinichi},
     title = {Integral structures on $p$-adic Fourier theory},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {66},
     number = {2},
     year = {2016},
     pages = {521-550},
     doi = {10.5802/aif.3018},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2016__66_2_521_0}
}

Bannai, Kenichi; Kobayashi, Shinichi. Integral structures on $p$-adic Fourier theory. Annales de l'Institut Fourier, Volume 66 (2016) no. 2, pp. 521-550. doi : 10.5802/aif.3018. http://www.numdam.org/item/AIF_2016__66_2_521_0/

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