$p$-adic cyclotomic character

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

$${\chi }_p:G_{\mathbb{Q}}⟶{\mathbb{Z}}_p^{\times }$$

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 $n\ge 1$, 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

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

Moreover, the restriction map $\mathrm{Gal}(K_{n+1}/\mathbb{Q})\to \mathrm{Gal}(K_n/\mathbb{Q})$ 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:

$${\chi }_{p,n}:G_{\mathbb{Q}}\to \mathrm{Gal}(K_n/\mathbb{Q})\to {(\mathbb{Z}/p^n\mathbb{Z})}^{\times }$$

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:

$${\chi }_p:G_{\mathbb{Q}}⟶{\mathbb{Z}}_p^{\times }$$

by requiring $\chi (\sigma )\equiv {\chi }_{p,n}(\sigma )mod{p}^n$, for every $n\ge 1$.

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 $1\le t_n\le p^n-1$ with $\gcd (t_n,p)=1$ (so $t_n$ is a unit modulo $p^n$). Moreover,

$$\sigma ({\zeta }_{p^{n-1}})=\sigma ({\zeta }_{p^n}^p)={\zeta }_{p^n}^{p{t}_n}={\zeta }_{p^{n-1}}^{t_n}$$

Therefore, $t_n\equiv t_{n-1}$ modulo $p^{n-1}$. Thus, we may define:

$${\chi }_p(\sigma )=\underleftarrow{\lim }{t}_n\in {\mathbb{Z}}_p$$

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.