no. 2
Integral structures on p-adic Fourier theory
Bannai, Kenichi1; Kobayashi, Shinichi2
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)
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 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:
Accepted:
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) 
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.3018/

[1] Amice, Yvette Interpolation p-adique, Bull. Soc. Math. France, Volume 92 (1964), pp. 117-180

[2] Bannai, Kenichi; Kobayashi, Shinichi Algebraic theta functions and the p-adic interpolation of Eisenstein-Kronecker numbers, Duke Math. J., Volume 153 (2010) no. 2, pp. 229-295 | DOI

[3] Boxall, John L. p-adic interpolation of logarithmic derivatives associated to certain Lubin-Tate formal groups, Ann. Inst. Fourier (Grenoble), Volume 36 (1986) no. 3, pp. 1-27 | DOI

[4] Chellali, Mustapha Congruences entre nombres de Bernoulli-Hurwitz dans le cas supersingulier, J. Number Theory, Volume 35 (1990) no. 2, pp. 157-179 | DOI

[5] Coleman, Robert F. Division values in local fields, Invent. Math., Volume 53 (1979) no. 2, pp. 91-116 | DOI

[6] Honda, Taira Formal groups and zeta-functions, Osaka J. Math., Volume 5 (1968), pp. 199-213

[7] Katz, Nicholas M. Divisibilities, congruences, and Cartier duality, J. Fac. Sci. Univ. Tokyo Sect. IA Math., Volume 28 (1981) no. 3, p. 667-678 (1982)

[8] Schneider, P.; Teitelbaum, J. p-adic Fourier theory, Doc. Math., Volume 6 (2001), p. 447-481 (electronic)

[9] de Shalit, Ehud Iwasawa theory of elliptic curves with complex multiplication, Perspectives in Mathematics, 3, Academic Press, Inc., Boston, MA, 1987, x+154 pages (p-adic L functions)

[10] Shiratani, Katsumi; Imada, Tsunehisa The exponential series of the Lubin-Tate groups and p-adic interpolation, Mem. Fac. Sci. Kyushu Univ. Ser. A, Volume 46 (1992) no. 2, pp. 351-365 | DOI

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