Cyclotomic character explained

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).

p-adic cyclotomic character

Fix a prime, and let denote the absolute Galois group of the rational numbers. The roots of unity $$\backslash mu\_\; =\; \backslash left\backslash$$ form a cyclic group of order

p ^{n} , generated by any choice of a primitive th root of unity .

Since all of the primitive roots in

\mu _{p ^{n}} are Galois conjugate, the Galois group G _{Q} 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 ( Z /p ^{nZ)} ^{x} . One can thus write

$$\backslash sigma.\backslash zeta\; :=\; \backslash sigma(\backslash zeta)\; =\; \backslash zeta\_^$$

where

a(\sigma,n) \in ( Z /p ^{n} Z ) ^{x} is the unique element as above, depending on both \sigma and p . This defines a group homomorphism called the mod cyclotomic character:

which is viewed as a character since the action corresponds to a homomorphism

G _{Q} \to Aut (\mu _{p ^{n}} ) \cong ( Z /p ^{nZ)} ^{x} \cong GL ^{n Z )} _{1 ( Z /p} .

Fixing

p and \sigma and varying n , the a(\sigma, n) form a compatible system in the sense that they give an element of the inverse limit $$\backslash varprojlim\_n\; (\backslash mathbf/p^n\backslash mathbf)^\backslash times\; \backslash cong\; \backslash mathbf\_p^\backslash times,$$the units in the ring of p-adic integers. Thus the {\chi _{p ^{n}} } assemble to a group homomorphism called -adic cyclotomic character:

encoding the action of

G _{Q} on all -power roots of unity \mu _{p ^{n}} simultaneously. In fact equipping G _{Q} with the Krull topology and Z _{p} with the -adic topology makes this a continuous representation of a topological group.

As a compatible system of -adic representations

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). That is to say, is a "family" of -adic representations

\chi _{\ell:G} _{Q → \operatorname{GL}} _{1(Z} _{\ell)}

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

Geometric realizations

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 .^[1]

Properties

The -adic cyclotomic character satisfies several nice properties.

• It is unramified at all primes (i.e. the inertia subgroup at acts trivially).
• If is a Frobenius element for, then
• It is crystalline at .

See also

• Tate twist

Notes and References

1. Section 3 of