# $p$-adic cyclotomic character

Let $G_{\mathbb{Q}}=\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$. The purpose of this entry is to define, for every prime $p$, a Galois representation:

 $\chi_{p}:G_{\mathbb{Q}}\longrightarrow\mathbb{Z}_{p}^{\times}$

where $\mathbb{Z}_{p}^{\times}$ is the group of units of $\mathbb{Z}_{p}$, the $p$-adic integers. $\chi_{p}$ is a $\mathbb{Z}_{p}^{\times}$ valued character, usually called the cyclotomic character of $G_{\mathbb{Q}}$, or the $p$-adic cyclotomic Galois representation of $G_{\mathbb{Q}}$. Here is the construction:

For each $n\geq 1$, let $\zeta_{p^{n}}$ be a primitive $p^{n}$-th root of unity and let $K_{n}=\mathbb{Q}(\zeta_{p^{n}})$ be the corresponding cyclotomic extension of $\mathbb{Q}$. By the basic theory of cyclotomic extensions, we know that

 $\operatorname{Gal}(K_{n}/\mathbb{Q})\cong(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}.$

Moreover, the restriction map $\operatorname{Gal}(K_{n+1}/\mathbb{Q})\to\operatorname{Gal}(K_{n}/\mathbb{Q})$ is given by reduction modulo $p^{n}$ from $(\mathbb{Z}/p^{n+1}\mathbb{Z})^{\times}$ to $(\mathbb{Z}/p^{n}\mathbb{Z})^{\times}$.

Therefore, for each $n$ we can construct a representation:

 $\chi_{p,n}:G_{\mathbb{Q}}\to\operatorname{Gal}(K_{n}/\mathbb{Q})\to(\mathbb{Z}% /p^{n}\mathbb{Z})^{\times}$

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:

 $\chi_{p}:G_{\mathbb{Q}}\longrightarrow\mathbb{Z}_{p}^{\times}$

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 $\sigma\in G_{\mathbb{Q}}$. We need to define a group homomorphism $\chi_{p}:G_{\mathbb{Q}}\to\mathbb{Z}_{p}^{\times}$, 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:

 $\chi_{p}(\sigma)=\varprojlim t_{n}\in\mathbb{Z}_{p}$

and as we have shown, $\chi_{p}(\sigma)$ is a unit of $\mathbb{Z}_{p}$. Finally, the reader should check that $\chi_{p}$ is a group homomorphism.

Title $p$-adic cyclotomic character PadicCyclotomicCharacter 2013-03-22 15:36:16 2013-03-22 15:36:16 alozano (2414) alozano (2414) 6 alozano (2414) Definition msc 11R34 msc 11R32 msc 11R04 $p$-adic cyclotomic Galois representation cyclotomic character