On the linear independence of $p$-adic $L$-functions modulo $p$

Anglès, Bruno; Ranieri, Gabriele

Annales de l'Institut Fourier, Tome 60 (2010) no. 5, pp. 1831-1855.

Soit $p\ge 3$ un nombre premier. Soit $n\in \mathbb{N}$ 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 $(p-3)/2$, on pose

$${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_i{\omega }^{2j+1}\hfill & \text{si}{\chi }_i\mathrm{est}\mathrm{impair};\hfill \\ {\chi }_i{\omega }^{2j}\hfill & \text{si}{\chi }_i\mathrm{est}\mathrm{pair}.\hfill \end{array}\right.$$

Soit $K={\mathbb{Q}}_p({\chi }_1,...,{\chi }_n)$ et soit $\pi$ un premier de l’anneau de valuation ${𝒪}_K$ de $K$. Pour tout $i,j$ notons $f(T,{\theta }_{i,j})$ la série d’Iwasawa associée à ${\theta }_{i,j}$ et $\overline{f(T,{\theta }_{i,j})}$ 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 $\overline{f(T,{\theta }_{i,j})}$, sont linéairement indépendantes sur un certain corps $\Omega$ qui contient $\overline{{𝔽}_p}\left(T\right)$.

Let $p\ge 3$ be a prime. Let $n\in \mathbb{N}$ 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 $(p-3)/2$, set

$${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_i{\omega }^{2j+1}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{odd};\hfill \\ {\chi }_i{\omega }^{2j}\hfill & \text{if}{\chi }_i\mathrm{is}\mathrm{even}.\hfill \end{array}\right.$$

Let $K={\mathbb{Q}}_p({\chi }_1,...,{\chi }_n)$ and let $\pi$ be a prime of the valuation ring ${𝒪}_K$ of $K$. For all $i,j$ let $f(T,{\theta }_{i,j})$ be the Iwasawa series associated to ${\theta }_{i,j}$ and $\overline{f(T,{\theta }_{i,j})}$ 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 $\overline{f(T,{\theta }_{i,j})}$ are linearly independent over a certain field $\Omega$ that contains $\overline{{𝔽}_p}\left(T\right)$.

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

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

[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