Integral structures on p-adic Fourier theory
Annales de l'Institut Fourier, Volume 66 (2016) no. 2, pp. 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 p , generalizing the basis constructed by Amice for locally analytic functions on 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 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 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).

Published online:
DOI: 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
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
Bannai, Kenichi 1; Kobayashi, Shinichi 2

1 Department of Mathematics, Keio University, 3-14-1 Hiyoshi, Kouhoku-ku, Yokohama 223-8522 (Japan)
2 Mathematical Institute, Tohoku University, 6-3 Aramaki-aza-Aoba, Aoba-ku, Sendai 980-8578 (Japan)
     author = {Bannai, Kenichi and Kobayashi, Shinichi},
     title = {Integral structures on $p$-adic {Fourier} theory},
     journal = {Annales de l'Institut Fourier},
     pages = {521--550},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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     year = {2016},
     doi = {10.5802/aif.3018},
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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.

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