p-adic cyclotomic character

Let G=Gal(¯/) be the absolute Galois group of . The purpose of this entry is to define, for every prime p, a Galois representationMathworldPlanetmath:


where p× is the group of units of p, the p-adic integers. χp is a p× valued characterPlanetmathPlanetmath, usually called the cyclotomic character of G, or the p-adic cyclotomic Galois representation of G. Here is the construction:

For each n1, let ζpn be a primitive pn-th root of unityMathworldPlanetmath and let Kn=(ζpn) be the corresponding cyclotomic extension of . By the basic theory of cyclotomic extensions, we know that


Moreover, the restriction map Gal(Kn+1/)Gal(Kn/) is given by reductionPlanetmathPlanetmath modulo pn from (/pn+1)× to (/pn)×.

Therefore, for each n we can construct a representation:


where the first map is simply restriction to Kn and the second map is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath. By the remarks above, the representations χp,n are coherent in a strong sense, i.e.


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


by requiring χ(σ)χp,n(σ)modpn, for every n1.

One can rephrase the above definition as follows. Let σG. We need to define a group homomorphism χp:Gp×, so we need to first define χp(σ) and then check that it is a homomorphismMathworldPlanetmathPlanetmathPlanetmath. By the theory, σ(ζpn) is another primitive pn-th root of unity, thus


for some integer 1tnpn-1 with gcd(tn,p)=1 (so tn is a unit modulo pn). Moreover,


Therefore, tntn-1 modulo pn-1. Thus, we may define:


and as we have shown, χp(σ) is a unit of p. Finally, the reader should check that χp is a group homomorphism.

Title p-adic cyclotomic character
Canonical name PadicCyclotomicCharacter
Date of creation 2013-03-22 15:36:16
Last modified on 2013-03-22 15:36:16
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Definition
Classification msc 11R34
Classification msc 11R32
Classification msc 11R04
Synonym p-adic cyclotomic Galois representation
Synonym cyclotomic character