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 $χ : G \to Aut R (R (1)) \approx GL(1, R)$ ).

p-adic cyclotomic character

If $p$ is a prime, and $G$ is the absolute Galois group of the rational numbers, the $p$ -adic cyclotomic character is a group homomorphism

\chi _{p}:G\rightarrow \mathbf {Z} _{p}^{\times }

where $Z p \times$ is the group of units of the ring of p-adic integers. This homomorphism is defined as follows. Let $ζ n$ be a primitive $p n$ root of unity. Every $p n$ root of unity is a power of $ζ n$ uniquely defined as an element of the ring of integers modulo $p n$ . Primitive roots of unity correspond to the invertible elements, i.e. to $(Z / p n) \times$ . An element $g$ of the Galois group $G$ sends $ζ n$ to another primitive $p n$ root of unity

\theta =\zeta _{n}^{a_{g,n}}

where $a g, n \in (Z / p n) \times$ . For a given $g$ , as $n$ varies, the $a g, n$ form a compatible system in the sense that they give an element of the inverse limit of the $(Z / p n) \times$ , which is $Z p \times$ . Therefore, the $p$ -adic cyclotomic character sends $g$ to the system $(a g, n) n$ , thus encoding the action of $g$ on all $p$ -power roots of unity.

In fact, $\chi _{p}$ is a continuous homomorphism (where the topology on $G$ is the Krull topology, and that on $Z p \times$ is the $p$ -adic topology).

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 $Q$ .

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 $H 2 (P 1)$ .[1]

Properties

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

It is unramified at all primes $ℓ \neq p$ (i.e. the inertia subgroup at $ℓ$ acts trivially).
If $Frob ℓ$ is a Frobenius element for $ℓ \neq p$ , then $χ p (Frob ℓ) = ℓ$
It is crystalline at $p$ .

References

Section 3 of Deligne, Pierre (1979), "Valeurs de fonctions L et périodes d'intégrales" (PDF), in Borel, Armand; Casselman, William (eds.), Automorphic Forms, Representations, and L-Functions, Proceedings of the Symposium in Pure Mathematics (in French), 33.2, Providence, RI: AMS, p. 325, ISBN 0-8218-1437-0, MR 0546622, Zbl 0449.10022

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[1] Section 3 of Deligne, Pierre (1979), "Valeurs de fonctions L et périodes d'intégrales" (PDF), in Borel, Armand; Casselman, William (eds.), Automorphic Forms, Representations, and L-Functions, Proceedings of the Symposium in Pure Mathematics (in French), 33.2, Providence, RI: AMS, p. 325, ISBN 0-8218-1437-0, MR 0546622, Zbl 0449.10022