Cyclotomic character

In number theory, a cyclotomic character is a character of a Galois group giving the Galois action (mathematics)|action] on a group of roots of unity. As a one-dimensional representation over a ring, its representation space is generally denoted by .

''p''-adic cyclotomic character

Fix a prime, and let denote the absolute Galois group of the rational numbers.
The roots of unity form a cyclic group of order, generated by any choice of a primitive th root of unity.
Since all of the primitive roots in are Galois conjugate, the Galois group acts on by automorphisms. After fixing a primitive root of unity generating, any element can be written as a power of, where the exponent is a unique element in, which is a unit if is also primitive. One can thus write, for,
where is the unique element as above, depending on both and. This defines a group homomorphism called the mod cyclotomic character:
which is viewed as a character since the action corresponds to a homomorphism.
Fixing and and varying, the form a compatible system in the sense that they give an element of the inverse limit the units in the ring of p-adic integers. Thus the assemble to a group homomorphism called -adic cyclotomic character:
encoding the action of on all -power roots of unity simultaneously. In fact equipping with the Krull topology and with the -adic topology makes this a continuous representation of a topological group.

By varying over all prime numbers, a compatible system of ℓ-adic representations is obtained from the -adic cyclotomic characters. That is to say, is a "family" of -adic representations
satisfying certain compatibilities between different primes. In fact, the form a strictly compatible system of ℓ-adic representations.

The -adic cyclotomic character is the -adic Tate module of the [Algebraic torus|multiplicative group scheme] over. As such, its representation space can be viewed as the inverse limit of the groups of th roots of unity in.
In terms of cohomology, the -adic cyclotomic character is the dual of the first -adic étale cohomology group of. It can also be found in the étale cohomology of a projective variety, namely the projective line: it is the dual of.
In terms of motives, the -adic cyclotomic character is the -adic realization of the Tate motive. As a Grothendieck motive, the Tate motive is the dual of.

The -adic cyclotomic character satisfies several nice properties.