# 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

*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

where **Z**_{p}^{×} 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})^{×}. An element *g* of the Galois group *G* sends *ζ*_{n} to another primitive *p*^{n} root of unity

where *a*_{g,n} ∈ (**Z**/*p*^{n})^{×}. For a given *g*, as *n* varies, the *a*_{g,n} form a comptatible system in the sense that they give an element of the inverse limit of the (**Z**/*p*^{n})^{×}, which is **Z**_{p}^{×}. 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, is a continuous homomorphism (where the topology on *G* is the Krull topology, and that on **Z**_{p}^{×} 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

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 ≠
*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

## 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