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 , its representation space is generally denoted by (that is, it is a representation ).
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 âÂÂ-adic representations is obtained from the -adic cyclotomic characters (when considering compatible systems of representations, the standard terminology is to use the symbol to denote a prime instead of ). 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 âÂÂ-adic representations.
The -adic cyclotomic character is the -adic Tate module of the 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.