On the linear independence of p-adic L-functions modulo p
Annales de l'Institut Fourier, Volume 60 (2010) no. 5, p. 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 : https://doi.org/10.5802/aif.2573
Classification:  11R23,  11R18,  11S80,  11J72
Keywords: p-adic L-functions, p-adic Leopoldt transform, Iwasawa theory, irrationality
@article{AIF_2010__60_5_1831_0,
     author = {Angl\`es, Bruno and Ranieri, Gabriele},
     title = {On the linear independence of $p$-adic $L$-functions modulo $p$},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {60},
     number = {5},
     year = {2010},
     pages = {1831-1855},
     doi = {10.5802/aif.2573},
     mrnumber = {2766231},
     zbl = {1219.11162},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2010__60_5_1831_0}
}
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. http://www.numdam.org/item/AIF_2010__60_5_1831_0/

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