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 Template:Math, its representation space is generally denoted by Template:Math (that is, it is a representation Template:Math).

p-adic cyclotomic character

Fix Template:Math a prime, and let Template:Math denote the absolute Galois group of the rational numbers. The roots of unity $${\mu }_{p^n}=\{\zeta \in {\overline{𝐐}}^{\times }\mid {\zeta }^{p^n}=1\}$$ form a cyclic group of order $p^n$, generated by any choice of a primitive Template:Mathth root of unity Template:Math.

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 $(𝐙/p^n𝐙{)}^{\times }$. One can thus write

$$\sigma .\zeta :=\sigma (\zeta )={\zeta }_{p^n}^{a(\sigma ,n)}$$

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

$$\begin{array}{cc}{\chi }_{p^n}:G_{𝐐}& \to (𝐙/p^n𝐙{)}^{\times }\\ \sigma & \mapsto a(\sigma ,n),\end{array}$$ which is viewed as a character since the action corresponds to a homomorphism $G_{𝐐}\to \mathrm{A}\mathrm{u}\mathrm{t}({\mu }_{p^n})\cong (𝐙/p^n𝐙{)}^{\times }\cong {\mathrm{G}\mathrm{L}}_1(𝐙/p^n𝐙)$.

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 $$\underset{n}{\underleftarrow{\lim }}(𝐙/p^n𝐙{)}^{\times }\cong {𝐙}_p^{\times },$$the units in the ring of p-adic integers. Thus the ${\chi }_{p^n}$ assemble to a group homomorphism called Template:Math-adic cyclotomic character:

$$\begin{array}{cc}{\chi }_p:G_{𝐐}& \to {𝐙}_p^{\times }\cong \mathrm{G}{\mathrm{L}}_{\mathrm{1}}({𝐙}_p)\\ \sigma & \mapsto (a(\sigma ,n){)}_n\end{array}$$ encoding the action of $G_{𝐐}$ on all Template:Math-power roots of unity ${\mu }_{p^n}$ simultaneously. In fact equipping $G_{𝐐}$ with the Krull topology and ${𝐙}_p$ with the [[p-adic|Template:Math-adic]] topology makes this a continuous representation of a topological group.

As a compatible system of Template:Math-adic representations

By varying Template:Math over all prime numbers, a compatible system of ℓ-adic representations is obtained from the Template:Math-adic cyclotomic characters (when considering compatible systems of representations, the standard terminology is to use the symbol Template:Math to denote a prime instead of Template:Math). That is to say, Template:Math is a "family" of Template:Math-adic representations

${\chi }_{\ell }:G_{𝐐}\to {\mathrm{GL}}_1({𝐙}_{\ell })$

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

Geometric realizations

The Template:Math-adic cyclotomic character is the Template:Math-adic Tate module of the multiplicative group scheme Template:Math over Template:Math. As such, its representation space can be viewed as the inverse limit of the groups of Template:Mathth roots of unity in Template:Math.

In terms of cohomology, the Template:Math-adic cyclotomic character is the dual of the first Template:Math-adic étale cohomology group of Template:Math. It can also be found in the étale cohomology of a projective variety, namely the projective line: it is the dual of Template:Math.

In terms of motives, the Template:Math-adic cyclotomic character is the Template:Math-adic realization of the Tate motive Template:Math. As a Grothendieck motive, the Tate motive is the dual of Template:Math.Template:Clarify

Properties

The Template:Math-adic cyclotomic character satisfies several nice properties.

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

See also

• Tate twist