Cyclotomic character

In number theory, a cyclotomic character is a character of a Galois group giving the Galois action on a group of roots of unity. As a one-dimensional representation over a ring $R$, its representation space is generally denoted by $R(1)$ (that is, it is a representation $χ : G → Aut_{R}(R(1)) ≈ GL(1, R)$).

p-adic cyclotomic character
Fix $p$ a prime, and let $G_{Q}$ denote the absolute Galois group of the rational numbers. The roots of unity $$\mu_{p^n} = \left\{ \zeta \in \bar\mathbf{Q}^\times \mid \zeta^{p^n} = 1 \right\}$$ form a cyclic group of order $$p^n$$, generated by any choice of a primitive $p^{n}$th root of unity $ζ_{p^{n}}|undefined$.

Since all of the primitive roots in $$\mu_{p^n}$$ are Galois conjugate, the Galois group $$G_\mathbf{Q}$$ acts on $$\mu_{p^n}$$ by automorphisms. After fixing a primitive root of unity $$\zeta_{p^n}$$ generating $$\mu_{p^n}$$, any element of $$\mu_{p^n}$$ can be written as a power of $$\zeta_{p^n}$$, where the exponent is a unique element in $$(\mathbf{Z}/p^n\mathbf{Z})^\times$$. One can thus write

$$\sigma.\zeta := \sigma(\zeta) = \zeta_{p^n}^{a(\sigma, n)}$$

where $$a(\sigma,n) \in (\mathbf{Z}/p^n \mathbf{Z})^\times$$ is the unique element as above, depending on both $$\sigma$$ and $$p$$. This defines a group homomorphism called the mod $p^{n}$ cyclotomic character:

$$\begin{align}{\chi_{p^n}}:G_{\mathbf{Q}} &\to (\mathbf{Z}/p^n\mathbf{Z})^{\times} \\ \sigma &\mapsto a(\sigma, n), \end{align}$$ which is viewed as a character since the action corresponds to a homomorphism $$G_{\mathbf Q} \to \mathrm{Aut}(\mu_{p^n}) \cong (\mathbf{Z}/p^n\mathbf{Z})^\times \cong \mathrm{GL}_1(\mathbf{Z}/p^n\mathbf{Z})$$.

Fixing $$p$$ and $$\sigma$$ and varying $$n$$, the $$a(\sigma, n)$$ form a compatible system in the sense that they give an element of the inverse limit $$\varprojlim_n (\mathbf{Z}/p^n\mathbf{Z})^\times \cong \mathbf{Z}_p^\times,$$the units in the ring of p-adic integers. Thus the $${\chi_{p^n}}$$ assemble to a group homomorphism called $p$-adic cyclotomic character:

$$\begin{align} \chi_p:G_{\mathbf Q} &\to \mathbf{Z}_p^\times \cong \mathrm{GL_1}(\mathbf{Z}_p) \\ \sigma &\mapsto (a(\sigma, n))_n \end{align}$$ encoding the action of $$G_{\mathbf Q}$$ on all $p$-power roots of unity $$\mu_{p^n}$$ simultaneously. In fact equipping $$G_{\mathbf Q}$$ with the Krull topology and $$\mathbf{Z}_p$$ with the $p$-adic topology makes this a continuous representation of a topological group.

As a compatible system of $ℓ$-adic representations
By varying $ℓ$ over all prime numbers, a compatible system of ℓ-adic representations is obtained from the $ℓ$-adic cyclotomic characters (when considering compatible systems of representations, the standard terminology is to use the symbol $ℓ$ to denote a prime instead of $p$). That is to say, $χ = { χ_{ℓ} }_{ℓ}$ is a "family" of $ℓ$-adic representations


 * $$\chi_\ell:G_\mathbf{Q}\rightarrow\operatorname{GL}_1(\mathbf{Z}_\ell)$$

satisfying certain compatibilities between different primes. In fact, the $χ_{ℓ}$ form a strictly compatible system of ℓ-adic representations.

Geometric realizations
The $p$-adic cyclotomic character is the $p$-adic Tate module of the multiplicative group scheme $G_{m,Q}$ over $Q$. As such, its representation space can be viewed as the inverse limit of the groups of $p^{n}$th roots of unity in $\overline{Q}$.

In terms of cohomology, the $p$-adic cyclotomic character is the dual of the first $p$-adic étale cohomology group of $G_{m}$. It can also be found in the étale cohomology of a projective variety, namely the projective line: it is the dual of $H^{2}_{ét}(P^{1} )$.

In terms of motives, the $p$-adic cyclotomic character is the $p$-adic realization of the Tate motive $Z(1)$. As a Grothendieck motive, the Tate motive is the dual of $H^{2}( P^{1} )$.

Properties
The $p$-adic cyclotomic character satisfies several nice properties.


 * It is unramified at all primes $ℓ ≠ p$ (i.e. the inertia subgroup at $ℓ$ acts trivially).
 * If $Frob_{ℓ}$ is a Frobenius element for $ℓ ≠ p$, then $χ_{p}(Frob_{ℓ}) = ℓ$
 * It is crystalline at $p$.