On the linear independence of p-adic L-functions modulo p
Annales de l'Institut Fourier, Volume 60 (2010) no. 5, pp. 1831-1855.

Let p3 be a prime. Let n such that n1, let χ 1 ,...,χ n be characters of conductor d not divided by p and let ω be the Teichmüller character. For all i between 1 and n, for all j between 0 and (p-3)/2, set

θi,j=χiω2j+1ifχi is odd ;χiω2jifχi is even .

Let K= p (χ 1 ,...,χ n ) and let π be a prime of the valuation ring 𝒪 K of K. For all i,j let f(T,θ i,j ) be the Iwasawa series associated to θ i,j and f(T,θ i,j ) ¯ its reduction modulo (π). Finally let 𝔽 p ¯ be an algebraic closure of 𝔽 p . Our main result is that if the characters χ i are all distinct modulo (π), then 1 and the series f(T,θ i,j ) ¯ are linearly independent over a certain field Ω that contains 𝔽 p ¯(T).

Soit p3 un nombre premier. Soit n tel que n1, soient χ 1 ,...,χ n des caractères de conducteur d premier à p ; notons ω le caractère de Teichmüller. Pour tout i entre 1 et n et pour tout j entre 0 et (p-3)/2, on pose

θi,j=χiω2j+1siχi est impair ;χiω2jsiχi est pair .

Soit K= p (χ 1 ,...,χ n ) et soit π un premier de l’anneau de valuation 𝒪 K de K. Pour tout i,j notons f(T,θ i,j ) la série d’Iwasawa associée à θ i,j et f(T,θ i,j ) ¯ sa réduction modulo (π). Finalement soit 𝔽 p ¯ une clôture algébrique de 𝔽 p . Nous montrons que si les caractères χ i sont distincts modulo (π), alors 1 et les séries f(T,θ i,j ) ¯, sont linéairement indépendantes sur un certain corps Ω qui contient 𝔽 p ¯(T).

DOI: 10.5802/aif.2573
Classification: 11R23, 11R18, 11S80, 11J72
Keywords: $p$-adic $L$-functions, $p$-adic Leopoldt transform, Iwasawa theory, irrationality
Anglès, Bruno 1; Ranieri, Gabriele 2

1 Université de Caen Laboratoire de mathématiques Nicolas Oresme CNRS UMR 6139 BP 5186 14032 Caen cedex (France)
2 Universität Basel Departement Matematik Rheinsprung 21 CH-4051 Basel (Switzerland)
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Anglès, Bruno; Ranieri, Gabriele. On the linear independence of $p$-adic $L$-functions modulo $p$. Annales de l'Institut Fourier, Volume 60 (2010) no. 5, pp. 1831-1855. doi : 10.5802/aif.2573. https://aif.centre-mersenne.org/articles/10.5802/aif.2573/

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[3] Sinnott, W. On the power series attached to p-adic L-functions, J. Reine Angew. Math., Volume 382 (1987), pp. 22-34 | DOI | MR | Zbl

[4] Washington, Lawrence C. Introduction to cyclotomic fields, Graduate Texts in Mathematics, 83, Springer-Verlag, New York, 1997 | MR | Zbl

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