PlanetMath: $p$-adic cyclotomic character

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-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.

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Other names:

-adic cyclotomic Galois representation, cyclotomic character

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

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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
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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 2236 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)

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