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Concept
Cyclotomic character
Summary
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)).
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 , generated by any choice of a primitive pnth root of unity ζn.
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 of can be written as a power of , where the exponent is a unique element in . One can thus write
where is the unique element as above, depending on both and . 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 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 p-adic cyclotomic character:
encoding the action of on all p-power roots of unity simultaneously. In fact equipping with the Krull topology and with the p-adic topology makes this a continuous representation of a topological group.
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.
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 pnth roots of unity in .
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