On the linear independence of $p$-adic $L$-functions modulo $p$  [ Sur l’indépendance linéaire des fonctions $L$ $p$-adiques modulo $p$ ]
Annales de l'Institut Fourier, Tome 60 (2010) no. 5, pp. 1831-1855.

Soit $p\ge 3$ un nombre premier. Soit $n\in ℕ$ tel que $n\ge 1$, soient ${\chi }_{1},...,{\chi }_{n}$ des caractères de conducteur $d$ premier à $p$ ; notons $\omega$ le caractère de Teichmüller. Pour tout $i$ entre $1$ et $n$ et pour tout $j$ entre $0$ et $\left(p-3\right)/2$, on pose

 ${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_{i}{\omega }^{2j+1}\hfill & \phantom{\rule{0.277778em}{0ex}}\text{si}\phantom{\rule{4pt}{0ex}}{\chi }_{i}\phantom{\rule{0.277778em}{0ex}}\mathrm{est}\phantom{\rule{4pt}{0ex}}\mathrm{impair}\phantom{\rule{4pt}{0ex}};\hfill \\ {\chi }_{i}{\omega }^{2j}\hfill & \phantom{\rule{0.277778em}{0ex}}\text{si}\phantom{\rule{4pt}{0ex}}{\chi }_{i}\phantom{\rule{0.277778em}{0ex}}\mathrm{est}\phantom{\rule{4pt}{0ex}}\mathrm{pair}.\hfill \end{array}\right\$

Soit $K={ℚ}_{p}\left({\chi }_{1},...,{\chi }_{n}\right)$ et soit $\pi$ un premier de l’anneau de valuation ${𝒪}_{K}$ de $K$. Pour tout $i,j$ notons $f\left(T,{\theta }_{i,j}\right)$ la série d’Iwasawa associée à ${\theta }_{i,j}$ et $\overline{f\left(T,{\theta }_{i,j}\right)}$ sa réduction modulo $\left(\pi \right)$. Finalement soit $\overline{{𝔽}_{p}}$ une clôture algébrique de ${𝔽}_{p}$. Nous montrons que si les caractères ${\chi }_{i}$ sont distincts modulo $\left(\pi \right)$, alors $1$ et les séries $\overline{f\left(T,{\theta }_{i,j}\right)}$, sont linéairement indépendantes sur un certain corps $\Omega$ qui contient $\overline{{𝔽}_{p}}\left(T\right)$.

Let $p\ge 3$ be a prime. Let $n\in ℕ$ such that $n\ge 1$, let ${\chi }_{1},...,{\chi }_{n}$ be characters of conductor $d$ not divided by $p$ and let $\omega$ be the Teichmüller character. For all $i$ between $1$ and $n$, for all $j$ between $0$ and $\left(p-3\right)/2$, set

 ${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_{i}{\omega }^{2j+1}\hfill & \phantom{\rule{0.277778em}{0ex}}\text{if}\phantom{\rule{4pt}{0ex}}{\chi }_{i}\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{4pt}{0ex}}\mathrm{odd};\hfill \\ {\chi }_{i}{\omega }^{2j}\hfill & \phantom{\rule{0.277778em}{0ex}}\text{if}\phantom{\rule{4pt}{0ex}}{\chi }_{i}\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{4pt}{0ex}}\mathrm{even}.\hfill \end{array}\right\$

Let $K={ℚ}_{p}\left({\chi }_{1},...,{\chi }_{n}\right)$ and let $\pi$ be a prime of the valuation ring ${𝒪}_{K}$ of $K$. For all $i,j$ let $f\left(T,{\theta }_{i,j}\right)$ be the Iwasawa series associated to ${\theta }_{i,j}$ and $\overline{f\left(T,{\theta }_{i,j}\right)}$ its reduction modulo $\left(\pi \right)$. Finally let $\overline{{𝔽}_{p}}$ be an algebraic closure of ${𝔽}_{p}$. Our main result is that if the characters ${\chi }_{i}$ are all distinct modulo $\left(\pi \right)$, then $1$ and the series $\overline{f\left(T,{\theta }_{i,j}\right)}$ are linearly independent over a certain field $\Omega$ that contains $\overline{{𝔽}_{p}}\left(T\right)$.

DOI : https://doi.org/10.5802/aif.2573
Classification : 11R23,  11R18,  11S80,  11J72
Mots clés : fonctions $L$ $p$-adiques, transformée de Leopoldt $p$-adique, théorie d’Iwasawa, irrationalité
@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},
pages = {1831--1855},
publisher = {Association des Annales de l'institut Fourier},
volume = {60},
number = {5},
year = {2010},
doi = {10.5802/aif.2573},
mrnumber = {2766231},
zbl = {1219.11162},
language = {en},
url = {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, Tome 60 (2010) no. 5, pp. 1831-1855. doi : 10.5802/aif.2573. http://www.numdam.org/item/AIF_2010__60_5_1831_0/`

[1] Anglès, Bruno On the $p$-adic Leopoldt transform of a power series, Acta Arith., Volume 134 (2008) no. 4, pp. 349-367 | Article | MR 2449158 | Zbl pre05354502

[2] Lang, Serge Cyclotomic fields I and II, Graduate Texts in Mathematics, Volume 121, Springer-Verlag, New York, 1990 (With an appendix by Karl Rubin) | MR 1029028

[3] Sinnott, W. On the power series attached to $p$-adic $L$-functions, J. Reine Angew. Math., Volume 382 (1987), pp. 22-34 | Article | MR 921164 | Zbl 0621.12015

[4] Washington, Lawrence C. Introduction to cyclotomic fields, Graduate Texts in Mathematics, Volume 83, Springer-Verlag, New York, 1997 | MR 1421575 | Zbl 0966.11047