Cyclotomic character
# 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 $p^nth$ root of unity $ζ_n.$

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:

${\begin{aligned}{\chi _{p^{n}}}:G_{\mathbf {Q} }&\to (\mathbf {Z} /p^{n}\mathbf {Z} )^{\times }\\\sigma &\mapsto a(\sigma ,n),\end{aligned}}$

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:

${\begin{aligned}\chi _{p}:G_{\mathbf {Q} }&\to \mathbf {Z} _{p}^{\times }\cong \mathrm {GL_{1}} (\mathbf {Z} _{p})\\\sigma &\mapsto (a(\sigma ,n))_{n}\end{aligned}}$

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 $p^nth$ roots of unity in 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.
