Cyclotomic Character

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 $\mu_ = \left\$ 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 $\mu_$ are Galois conjugate, the Galois group $G_\mathbf$ acts on $\mu_$ by automorphisms. After fixing a primitive root of unity $\zeta_$ generating $\mu_$, any element of $\mu_$ can be written as a power of $\zeta_$, where the exponent is a unique element in $(\mathbf/p^n\mathbf)^\times$. One can thus write

$\sigma.\zeta := \sigma(\zeta) = \zeta_^$

where $a(\sigma,n) \in (\mathbf/p^n \mathbf)^\times$ is the unique element as above, depending on both $\sigma$ and $p$. This defines a group homomorphism called the mod cyclotomic character:

$\begin:G_ &\to (\mathbf/p^n\mathbf)^ \\ \sigma &\mapsto a(\sigma, n), \end$
which is viewed as a character since the action corresponds to a homomorphism $G_ \to \mathrm(\mu_) \cong (\mathbf/p^n\mathbf)^\times \cong \mathrm_1(\mathbf/p^n\mathbf)$.

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 $\varprojlim_n (\mathbf/p^n\mathbf)^\times \cong \mathbf_p^\times,$the units in the ring of p-adic integers. Thus the $$ assemble to a group homomorphism called -adic cyclotomic character:

$\begin \chi_p:G_ &\to \mathbf_p^\times \cong \mathrm(\mathbf_p) \\ \sigma &\mapsto (a(\sigma, n))_n \end$
encoding the action of $G_$ on all -power roots of unity $\mu_$ simultaneously. In fact equipping $G_$ with the Krull topology and $\mathbf_p$ 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 â„“-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

:$\chi_\ell:G_\mathbf\rightarrow\operatorname_1(\mathbf_\ell)$

satisfying certain compatibilities between different primes. In fact, the form a strictly compatible system of â„“-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 .Section 3 of

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

References

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Algebraic number theory