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

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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 ζpn.

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.

As a compatible system of -adic representations

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By varying over all prime numbers, a compatible system of ℓ-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 p). That is to say, χ = { χ } is a "family" of -adic representations

 

satisfying certain compatibilities between different primes. In fact, the χ form a strictly compatible system of ℓ-adic representations.

Geometric realizations

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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 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 ).[1][clarification needed]

Properties

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The p-adic cyclotomic character satisfies several nice properties.

See also

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References

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  1. ^ 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), vol. 33, Providence, RI: AMS, p. 325, ISBN 0-8218-1437-0, MR 0546622, Zbl 0449.10022