Cyclotomic character

p-adic cyclotomic character

Fix p a prime, and let $G_{Q}$ denote the absolute Galois group of the rational numbers. The roots of unity $μ_{p^{n}} = {ζ \in {\bar{Q}}^{\times} ∣ ζ^{p^{n}} = 1}$ form a cyclic group of order $p^{n}$ , generated by any choice of a primitive pⁿth root of unity ζ_pⁿ.

Since all of the primitive roots in $μ_{p^{n}}$ are Galois conjugate, the Galois group $G_{Q}$ acts on $μ_{p^{n}}$ by automorphisms. After fixing a primitive root of unity $ζ_{p^{n}}$ generating $μ_{p^{n}}$ , any element $ζ \in μ_{p^{n}}$ can be written as a power of $ζ_{p^{n}}$ , where the exponent is a unique element in $Z / p^{n} Z$ , which is a unit if $ζ$ is also primitive. One can thus write, for $σ \in G_{Q}$ ,

$σ . ζ := σ (ζ) = ζ^{a (σ, n)}$

with $a (σ, n) \in (Z / p^{n} Z)^{\times}$ independent of $ζ$ , but depending on both $σ$ and $p$ . This defines a group homomorphism called the mod pⁿ cyclotomic character:

$\begin{matrix} χ_{p^{n}} : G_{Q} & \to (Z / p^{n} Z)^{\times} \\ σ & \mapsto a (σ, n), \end{matrix}$ which is viewed as a character since the action corresponds to a homomorphism $G_{Q} \to A u t (μ_{p^{n}}) ≅ (Z / p^{n} Z)^{\times} ≅ {G L}_{1} (Z / p^{n} Z)$ .

Fixing $p$ and $σ$ and varying $n$ , the $a (σ, n)$ form a compatible system in the sense that they give an element of the inverse limit $\underset{n}{\lim_{\leftarrow}} (Z / p^{n} Z)^{\times} ≅ Z_{p}^{\times},$ the units in the ring of p-adic integers. Thus the $χ_{p^{n}}$ assemble to a group homomorphism called p-adic cyclotomic character:

$\begin{matrix} χ_{p} : G_{Q} & \to Z_{p}^{\times} ≅ G L_{1} (Z_{p}) \\ σ & \mapsto (a (σ, n))_{n} \end{matrix}$ encoding the action of $G_{Q}$ on all p-power roots of unity $μ_{p^{n}}$ simultaneously. In fact equipping $G_{Q}$ with the Krull topology and $Z_{p}$ with the p-adic topology makes this a continuous representation of a topological group.

3 more chapters are free with a Lably account.

p-adic cyclotomic character

As a compatible system of ℓ-adic representations

Geometric realizations

Properties

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