PAdicCyclotomicCharacter 2005-12-06 18:23:14 2005-12-09 15:34:44 Definition Galois representation % this is the default PlanetMath preamble. as your knowledge % of TeX increases, you will probably want to edit this, but % it should be fine as is for beginners. % almost certainly you want these \usepackage{amssymb} \usepackage{amsmath} \usepackage{amsthm} \usepackage{amsfonts} % used for TeXing text within eps files %\usepackage{psfrag} % need this for including graphics (\includegraphics) %\usepackage{graphicx} % for neatly defining theorems and propositions %\usepackage{amsthm} % making logically defined graphics %\usepackage{xypic} % there are many more packages, add them here as you need them % define commands here \newtheorem{thm}{Theorem} \newtheorem{defn}{Definition} \newtheorem{prop}{Proposition} \newtheorem{lemma}{Lemma} \newtheorem{cor}{Corollary} \theoremstyle{definition} \newtheorem{exa}{Example} % Some sets \newcommand{\Nats}{\mathbb{N}} \newcommand{\Ints}{\mathbb{Z}} \newcommand{\Reals}{\mathbb{R}} \newcommand{\Complex}{\mathbb{C}} \newcommand{\Rats}{\mathbb{Q}} \newcommand{\Gal}{\operatorname{Gal}} \newcommand{\Cl}{\operatorname{Cl}} Let $G_{\Rats}=\Gal(\overline{\Rats}/\Rats)$ be the absolute Galois group of $\Rats$. The purpose of this entry is to define, for every prime $p$, a Galois representation: $$\chi_p : G_{\Rats} \longrightarrow \Ints_p^\times$$ where $\Ints_p^\times$ is the group of units of $\Ints_p$, the $p$-adic integers. $\chi_p$ is a $\Ints_p^\times$ valued character, usually called the {\it cyclotomic character} of $G_{\Rats}$, or the $p$-adic cyclotomic Galois representation of $G_{\Rats}$. Here is the construction: For each $n\geq 1$, let $\zeta_{p^n}$ be a primitive $p^n$-th root of unity and let $K_n=\Rats(\zeta_{p^n})$ be the corresponding cyclotomic extension of $\Rats$. By the basic theory of cyclotomic extensions, we know that $$\Gal(K_n/\Rats)\cong (\Ints/p^n\Ints)^\times.$$ Moreover, the restriction map $\Gal(K_{n+1}/\Rats)\to \Gal(K_n/\Rats)$ is given by reduction modulo $p^n$ from $(\Ints/p^{n+1}\Ints)^\times$ to $(\Ints/p^n\Ints)^\times$. Therefore, for each $n$ we can construct a representation: $$\chi_{p,n} : G_{\Rats} \to \Gal(K_n/\Rats) \to (\Ints/p^n\Ints)^\times$$ where the first map is simply restriction to $K_n$ and the second map is an isomorphism. By the remarks above, the representations $\chi_{p,n}$ are coherent in a strong sense, i.e. $$\chi_{p,n+1}(\sigma) \equiv \chi_{p,n}(\sigma) \mod p^n.$$ Therefore, one can construct a ``big'' Galois representation: $$\chi_p : G_{\Rats} \longrightarrow \Ints_p^\times$$ by requiring $\chi(\sigma) \equiv \chi_{p,n}(\sigma) \mod p^n$, for every $n\geq 1$. One can rephrase the above definition as follows. Let $\sigma\in G_{\Rats}$. We need to define a group homomorphism $\chi_p:G_{\Rats} \to \Ints_p^\times$, so we need to first define $\chi_p(\sigma)$ and then check that it is a homomorphism. By the theory, $\sigma(\zeta_{p^n})$ is another primitive $p^n$-th root of unity, thus $$\sigma(\zeta_{p^n})=\zeta_{p^n}^{t_n}$$ for some integer $1\leq t_n \leq p^n-1$ with $\gcd(t_n,p)=1$ (so $t_n$ is a unit modulo $p^n$). Moreover, $$\sigma(\zeta_{p^{n-1}})=\sigma(\zeta_{p^{n}}^p)=\zeta_{p^n}^{pt_n}=\zeta_{p^{n-1}}^{t_n}$$ Therefore, $t_n \equiv t_{n-1}$ modulo $p^{n-1}$. Thus, we may define: $$\chi_p(\sigma) = \varprojlim t_n \in \Ints_p$$ and as we have shown, $\chi_p(\sigma)$ is a unit of $\Ints_p$. Finally, the reader should check that $\chi_p$ is a group homomorphism.