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

Anglès, Bruno, and Ranieri, Gabriele. "On the linear independence of $p$-adic $L$-functions modulo $p$." Annales de l’institut Fourier 60.5 (2010): 1831-1855. <http://eudml.org/doc/116322>.

@article{Anglès2010,
abstract = {Let $p \ge 3$ be a prime. Let $n \in \mathbb\{N\}$ such that $n \ge 1$, let $\chi _1, \ldots , \chi _n$ 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\[ \theta \_\{i, j\} = \{\left\lbrace \begin\{array\}\{ll\} \chi \_i \omega ^\{2 j + 1\}&\;\hbox\{if\} \ \chi \_i\;\{\rm is\} \ \{\rm odd\}; \\ \chi \_i \omega ^\{2 j\}&\;\hbox\{if\} \ \chi \_i\;\{\rm is\} \ \{\rm even\}. \end\{array\}\right.\} \]Let $K = \mathbb\{Q\}_p ( \chi _1, \ldots , \chi _n )$ and let $\pi $ be a prime of the valuation ring $\mathcal\{O\}_K$ 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 $( \pi )$. Finally let $\overline\{\mathbb\{F\}_p\}$ be an algebraic closure of $\mathbb\{F\}_p$. Our main result is that if the characters $\chi _i$ are all distinct modulo $( \pi )$, then $1$ and the series $\overline\{f ( T, \theta _\{i, j\} )\}$ are linearly independent over a certain field $\Omega $ that contains $\overline\{\mathbb\{F\}_p\} ( T )$.},
affiliation = {Université de Caen Laboratoire de mathématiques Nicolas Oresme CNRS UMR 6139 BP 5186 14032 Caen cedex (France); Universität Basel Departement Matematik Rheinsprung 21 CH-4051 Basel (Switzerland)},
author = {Anglès, Bruno, Ranieri, Gabriele},
journal = {Annales de l’institut Fourier},
keywords = {$p$-adic $L$-functions; $p$-adic Leopoldt transform; Iwasawa theory; irrationality; -adic -functions; -adic Leopoldt transform},
language = {eng},
number = {5},
pages = {1831-1855},
publisher = {Association des Annales de l’institut Fourier},
title = {On the linear independence of $p$-adic $L$-functions modulo $p$},
url = {http://eudml.org/doc/116322},
volume = {60},
year = {2010},
}

TY - JOUR
AU - Anglès, Bruno
AU - Ranieri, Gabriele
TI - On the linear independence of $p$-adic $L$-functions modulo $p$
JO - Annales de l’institut Fourier
PY - 2010
PB - Association des Annales de l’institut Fourier
VL - 60
IS - 5
SP - 1831
EP - 1855
AB - Let $p \ge 3$ be a prime. Let $n \in \mathbb{N}$ such that $n \ge 1$, let $\chi _1, \ldots , \chi _n$ 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\[ \theta _{i, j} = {\left\lbrace \begin{array}{ll} \chi _i \omega ^{2 j + 1}&\;\hbox{if} \ \chi _i\;{\rm is} \ {\rm odd}; \\ \chi _i \omega ^{2 j}&\;\hbox{if} \ \chi _i\;{\rm is} \ {\rm even}. \end{array}\right.} \]Let $K = \mathbb{Q}_p ( \chi _1, \ldots , \chi _n )$ and let $\pi $ be a prime of the valuation ring $\mathcal{O}_K$ 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 $( \pi )$. Finally let $\overline{\mathbb{F}_p}$ be an algebraic closure of $\mathbb{F}_p$. Our main result is that if the characters $\chi _i$ are all distinct modulo $( \pi )$, then $1$ and the series $\overline{f ( T, \theta _{i, j} )}$ are linearly independent over a certain field $\Omega $ that contains $\overline{\mathbb{F}_p} ( T )$.
LA - eng
KW - $p$-adic $L$-functions; $p$-adic Leopoldt transform; Iwasawa theory; irrationality; -adic -functions; -adic Leopoldt transform
UR - http://eudml.org/doc/116322
ER -