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)).

Fix p a prime, and let G_Q denote the absolute Galois group of the rational numbers. The roots of unity $${\displaystyle \mu _{p^{n}}=\left\{\zeta \in {\bar {\mathbf {Q} }}^{\times }\mid \zeta ^{p^{n}}=1\right\}}$$ form a cyclic group of order ${\displaystyle p^{n}}$, generated by any choice of a primitive pⁿth root of unity ζ_pⁿ.

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

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

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

$${\displaystyle {\begin{aligned}{\chi _{p^{n}}}:G_{\mathbf {Q} }&\to (\mathbf {Z} /p^{n}\mathbf {Z} )^{\times }\\\sigma &\mapsto a(\sigma ,n),\end{aligned}}}$$ which is viewed as a character since the action corresponds to a homomorphism ${\displaystyle G_{\mathbf {Q} }\to \mathrm {Aut} (\mu _{p^{n}})\cong (\mathbf {Z} /p^{n}\mathbf {Z} )^{\times }\cong \mathrm {GL} _{1}(\mathbf {Z} /p^{n}\mathbf {Z} )}$.

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

$${\displaystyle {\begin{aligned}\chi _{p}:G_{\mathbf {Q} }&\to \mathbf {Z} _{p}^{\times }\cong \mathrm {GL_{1}} (\mathbf {Z} _{p})\\\sigma &\mapsto (a(\sigma ,n))_{n}\end{aligned}}}$$ encoding the action of ${\displaystyle G_{\mathbf {Q} }}$ on all p-power roots of unity ${\displaystyle \mu _{p^{n}}}$ simultaneously. In fact equipping ${\displaystyle G_{\mathbf {Q} }}$ with the Krull topology and ${\displaystyle \mathbf {Z} _{p}}$ with the p-adic topology makes this a continuous representation of a topological group.

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

${\displaystyle \chi _{\ell }:G_{\mathbf {Q} }\rightarrow \operatorname {GL} _{1}(\mathbf {Z} _{\ell })}$

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

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 pⁿth 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 H²_ét(P¹ ).

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 H²( P¹ ).^{[1][clarification needed]}

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.

• Tate twist