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

Bruno Anglès[1]; Gabriele Ranieri[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)
• Volume: 60, Issue: 5, page 1831-1855
• ISSN: 0373-0956

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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${\theta }_{i,j}=\left\{\begin{array}{cc}{\chi }_{i}{\omega }^{2j+1}\hfill & \phantom{\rule{0.277778em}{0ex}}\text{if}\phantom{\rule{4pt}{0ex}}{\chi }_{i}\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{4pt}{0ex}}\mathrm{odd};\hfill \\ {\chi }_{i}{\omega }^{2j}\hfill & \phantom{\rule{0.277778em}{0ex}}\text{if}\phantom{\rule{4pt}{0ex}}{\chi }_{i}\phantom{\rule{0.277778em}{0ex}}\mathrm{is}\phantom{\rule{4pt}{0ex}}\mathrm{even}.\hfill \end{array}\right$/extract_itex]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 $\overline{{𝔽}_{p}}\left(T\right)$. How to cite top 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 -

References

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1. Bruno Anglès, On the $p$-adic Leopoldt transform of a power series, Acta Arith. 134 (2008), 349-367 Zbl1230.11131MR2449158
2. Serge Lang, Cyclotomic fields I and II, 121 (1990), Springer-Verlag, New York Zbl0704.11038MR1029028
3. W. Sinnott, On the power series attached to $p$-adic $L$-functions, J. Reine Angew. Math. 382 (1987), 22-34 Zbl0621.12015MR921164
4. Lawrence C. Washington, Introduction to cyclotomic fields, 83 (1997), Springer-Verlag, New York Zbl0966.11047MR1421575

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