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 → AutR(R(1)) ≈ GL(1, R) ).

p-adic cyclotomic character

Fix p a prime, and let GQ denote the absolute Galois group of the rational numbers. The roots of unity form a cyclic group of order p^n, generated by any choice of a primitive pn th root of unity ζpn . Since all of the primitive roots in \mu_{p^n} are Galois conjugate, the Galois group acts on \mu_{p^n} by automorphisms. After fixing a primitive root of unity \zeta_{p^n} generating \mu_{p^n}, any element of \mu_{p^n} can be written as a power of \zeta_{p^n}, where the exponent is a unique element in. One can thus write where is the unique element as above, depending on both \sigma and p. This defines a group homomorphism called the '''mod pn cyclotomic character''': which is viewed as a character since the action corresponds to a homomorphism. Fixing p and \sigma and varying n, 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 ''' p -adic cyclotomic character''': encoding the action of on all p -power roots of unity \mu_{p^n} simultaneously. In fact equipping with the Krull topology and with the [[p-adic| 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 = { χℓ }ℓ is a "family" of ℓ -adic representations satisfying certain compatibilities between different primes. In fact, the χℓ 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 Gm,Q over Q . As such, its representation space can be viewed as the inverse limit of the groups of pn 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 Gm . It can also be found in the étale cohomology of a projective variety, namely the projective line: it is the dual of H2ét(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 ) .

Properties

The p -adic cyclotomic character satisfies several nice properties. ℓ ≠ p (i.e. the inertia subgroup at ℓ acts trivially). Frobℓ is a Frobenius element for ℓ ≠ p , then χp(Frobℓ) = ℓ p .