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 [math]\displaystyle{ \mu_{p^n} = \left\{ \zeta \in \bar\mathbf{Q}^\times \mid \zeta^{p^n} = 1 \right\} }[/math] form a cyclic group of order [math]\displaystyle{ p^n }[/math], generated by any choice of a primitive pⁿth root of unity ζ_n.

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

[math]\displaystyle{ \sigma.\zeta := \sigma(\zeta) = \zeta_{p^n}^{a(\sigma, n)} }[/math]

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

[math]\displaystyle{ \begin{align}{\chi_{p^n}}:G_{\mathbf{Q}} &\to (\mathbf{Z}/p^n\mathbf{Z})^{\times} \\ \sigma &\mapsto a(\sigma, n), \end{align} }[/math] which is viewed as a character since the action corresponds to a homomorphism [math]\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}) }[/math].

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

[math]\displaystyle{ \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} }[/math] encoding the action of [math]\displaystyle{ G_{\mathbf Q} }[/math] on all p-power roots of unity [math]\displaystyle{ \mu_{p^n} }[/math] simultaneously. In fact equipping [math]\displaystyle{ G_{\mathbf Q} }[/math] with the Krull topology and [math]\displaystyle{ \mathbf{Z}_p }[/math] 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

[math]\displaystyle{ \chi_\ell:G_\mathbf{Q}\rightarrow\operatorname{GL}_1(\mathbf{Z}_\ell) }[/math]

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