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 $${\mu }_{p^n}=\{\zeta \in {\overline{\mathbf{𝐐}}}^{\times }\mid {\zeta }^{p^n}=1\}$$ form a cyclic group of order $p^n$, generated by any choice of a primitive pⁿth 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 pⁿ cyclotomic character:

$$\begin{array}{rl}{\chi }_{p^n}:G_{\mathbf{𝐐}}& \to (\mathbf{𝐙}/p^n\mathbf{𝐙}{)}^{\times }\\ \sigma & \mapsto a(\sigma ,n),\end{array}$$ which is viewed as a character since the action corresponds to a homomorphism $G_{\mathbf{𝐐}}\to \mathrm{A}\mathrm{u}\mathrm{t}({\mu }_{p^n})\cong (\mathbf{𝐙}/p^n\mathbf{𝐙}{)}^{\times }\cong {\mathrm{G}\mathrm{L}}_1(\mathbf{𝐙}/p^n\mathbf{𝐙})$.

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 $$\underset{n}{\underleftarrow{\lim }}(\mathbf{𝐙}/p^n\mathbf{𝐙}{)}^{\times }\cong {\mathbf{𝐙}}_p^{\times },$$the units in the ring of p-adic integers. Thus the ${\chi }_{p^n}$ assemble to a group homomorphism called p-adic cyclotomic character:

$$\begin{array}{rl}{\chi }_p:G_{\mathbf{𝐐}}& \to {\mathbf{𝐙}}_p^{\times }\cong \mathrm{G}{\mathrm{L}}_{\mathrm{1}}({\mathbf{𝐙}}_p)\\ \sigma & \mapsto (a(\sigma ,n){)}_n\end{array}$$ 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

${\chi }_{\ell }:G_{\mathbf{𝐐}}\to {\mathrm{GL}}_1({\mathbf{𝐙}}_{\ell })$

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 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 ).<ref>Section 3 of Deligne, Pierre (1979), "Valeurs de fonctions L et périodes d'intégrales", in Borel, Armand; Casselman, William (in French), Automorphic Forms, Representations, and L-Functions, Proceedings of the Symposium in Pure Mathematics, 33, Providence, RI: AMS, p. 325, ISBN 0-8218-1437-0, https://www.ams.org/online_bks/pspum332/pspum332-ptIV-8.pdf 

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.

See also

• Tate twist