PlanetMath: $p$-adic cyclotomic character

(more info)

Math for the people, by the people.

Main Menu

sections Encyclopædia
Papers
Books
Expositions

meta Requests (211)
Orphanage
Unclass'd (1)
Unproven (472)
Corrections (36)
Classification

talkback Polls
Forums
Feedback
Bug Reports

downloads Snapshots
PM Book

information News
Docs
Wiki
ChangeLog
TODO List
Legalese
About

Entry average rating: No information on entry rating

-adic cyclotomic character

(Definition)

Let $G_{\mathbb{Q}}=\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})$ be the absolute Galois group of $\mathbb{Q}$ . The purpose of this entry is to define, for every prime , a Galois representation:

$\displaystyle \chi_p : G_{\mathbb{Q}} \longrightarrow \mathbb{Z}_p^\times$

where $\mathbb{Z}_p^\times$ is the group of units of $\mathbb{Z}_p$ , the -adic integers. $\chi_p$ is a $\mathbb{Z}_p^\times$ valued character, usually called the cyclotomic character of $G_{\mathbb{Q}}$ , or the -adic cyclotomic Galois representation of $G_{\mathbb{Q}}$ . Here is the construction:

For each $n\geq 1$ , let $\zeta_{p^n}$ be a primitive -th root of unity and let $K_n=\mathbb{Q}(\zeta_{p^n})$ be the corresponding cyclotomic extension of $\mathbb{Q}$ . By the basic theory of cyclotomic extensions, we know that

$\displaystyle \operatorname{Gal}(K_n/\mathbb{Q})\cong (\mathbb{Z}/p^n\mathbb{Z})^\times.$

Moreover, the restriction map $\operatorname{Gal}(K_{n+1}/\mathbb{Q})\to \operatorname{Gal}(K_n/\mathbb{Q})$ is given by reduction modulo from $(\mathbb{Z}/p^{n+1}\mathbb{Z})^\times$ to $(\mathbb{Z}/p^n\mathbb{Z})^\times$ .

Therefore, for each we can construct a representation:

$\displaystyle \chi_{p,n} : G_{\mathbb{Q}} \to \operatorname{Gal}(K_n/\mathbb{Q}) \to (\mathbb{Z}/p^n\mathbb{Z})^\times$

where the first map is simply restriction to

and the second map is an isomorphism. By the remarks above, the representations $\chi_{p,n}$ are coherent in a strong sense, i.e.

$\displaystyle \chi_{p,n+1}(\sigma) \equiv \chi_{p,n}(\sigma) \mod p^n.$

Therefore, one can construct a “big” Galois representation:

$\displaystyle \chi_p : G_{\mathbb{Q}} \longrightarrow \mathbb{Z}_p^\times$

by requiring $\chi(\sigma) \equiv \chi_{p,n}(\sigma) \mod p^n$ , for every $n\geq 1$ .

One can rephrase the above definition as follows. Let $\sigma\in G_{\mathbb{Q}}$ . We need to define a group homomorphism $\chi_p:G_{\mathbb{Q}} \to \mathbb{Z}_p^\times$ , so we need to first define $\chi_p(\sigma)$ and then check that it is a homomorphism. By the theory, $\sigma(\zeta_{p^n})$ is another primitive -th root of unity, thus

$\displaystyle \sigma(\zeta_{p^n})=\zeta_{p^n}^{t_n}$

for some integer $1\leq t_n \leq p^n-1$ with $\gcd(t_n,p)=1$ (so

is a unit modulo

). Moreover,

$\displaystyle \sigma(\zeta_{p^{n-1}})=\sigma(\zeta_{p^{n}}^p)=\zeta_{p^n}^{pt_n}=\zeta_{p^{n-1}}^{t_n}$

Therefore, $t_n \equiv t_{n-1}$ modulo $p^{n-1}$ . Thus, we may define:

$\displaystyle \chi_p(\sigma) = \varprojlim t_n \in \mathbb{Z}_p$

and as we have shown, $\chi_p(\sigma)$ is a unit of $\mathbb{Z}_p$ . Finally, the reader should check that $\chi_p$ is a group homomorphism.

-adic cyclotomic character" is owned by alozano.

(view preamble)

Other names:

-adic cyclotomic Galois representation, cyclotomic character

This object's parent.

Attachments:: fundamental character of level for the inertia group at (Definition) by alozano

Log in to rate this entry.
(view current ratings)

Cross-references: unit, group homomorphism, isomorphism, representation, reduction, map, restriction, theory, cyclotomic extension, root of unity, primitive, character, integers, group of units, Galois representation, prime, absolute Galois group
There is 1 reference to this entry.

This is version 3 of $p$

-adic cyclotomic character, born on 2005-12-06, modified 2005-12-09.
Object id is 7521, canonical name is PAdicCyclotomicCharacter.
Accessed 2220 times total.

Classification:

AMS MSC:	11R04 (Number theory :: Algebraic number theory: global fields :: Algebraic numbers; rings of algebraic integers)
	11R32 (Number theory :: Algebraic number theory: global fields :: Galois theory)
	11R34 (Number theory :: Algebraic number theory: global fields :: Galois cohomology)

Pending Errata and Addenda

None.

Discussion

forum policy

No messages.

Interact