fundamental character of level n for the inertia group at p

Let p>2 be a prime, fix algebraic closuresMathworldPlanetmath ¯ and p¯, and fix an embeddingPlanetmathPlanetmath of ¯p¯. This embedding corresponds with an inclusion of the absolute Galois groups:


Let Ip be the inertia subgroupMathworldPlanetmathPlanetmath of Gal(p¯/p) which we regard as a subgroup of Gal(¯/) via the injection above (for more information on the inertia subgroup at p, Ip, see the entry on Galois representationsMathworldPlanetmath). Let 𝔽pn be the finite fieldMathworldPlanetmath of pn elements. The purpose of this entry is to define 𝔽pn-valued charactersPlanetmathPlanetmath Ψn, for every n1:


which we will refer to as the fundamental character of level n of Ip.

Definition 1.

Let χp:Gal(Q¯/Q)Zp× be the p-adic cyclotomic character and let χp¯ be the reductionPlanetmathPlanetmathPlanetmath of χp modulo p. The fundamental character of level 1 is Ψ1=χp¯|Ip, i.e. Ψ1 is the restrictionPlanetmathPlanetmath of the p-adic cyclotomic character χp to Ip, composed with reduction modulo p.

Next, we define the fundamental characters in more generality. Let Kn/p be the unique unramified field extension of degree n (it is unique by local fieldMathworldPlanetmath theory). The residue field of Kn is the field kn=𝔽pn (because k must be an extensionPlanetmathPlanetmathPlanetmath of degree n of 𝔽p).

Lemma 1.

The field Kn contains all (pn-1)th roots of unityMathworldPlanetmath.


Clearly, the polynomialPlanetmathPlanetmath xpn-1-1=0 has pn-1 distinct roots in kn=𝔽pn. Using Hensel’s lemma, one can check that each root in kn lifts to an element of Kn. ∎

Let Kn=Kn((-p)1pn-1). By the lemma, the (pn-1)th roots of unity are contained in Kn. Therefore, the extension Kn/Kn is Galois. Moreover, by Kummer theory one has:


Notice that the fact that Kn/p is unramified implies that the inertia group Ip injects into Gal(p¯/Kn)Gal(p¯/p). Therefore there is a map:

IpGal(p¯/Kn)Gal(Kn/Kn)𝔽pn× (1)

where the second map is simply given by restriction to Kn.

Definition 2.

The fundamental character of level n1 is the map Ψn:IpFpn× given by Eq. (1).

Note from the author: I would like to thank Eknath Ghate for explaining this to me.

Title fundamental character of level n for the inertia group at p
Canonical name FundamentalCharacterOfLevelNForTheInertiaGroupAtP
Date of creation 2013-03-22 15:36:26
Last modified on 2013-03-22 15:36:26
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 4
Author alozano (2414)
Entry type Definition
Classification msc 11R04
Classification msc 11R32
Classification msc 11R34