The tame character of the inertia group #
Let K be a nonarchimedean local field with residue characteristic p and residue field of order
q, let K^{alg} be its algebraic closure, and let I_K ≤ G_K = Gal(K^{alg}/K) be the inertia
subgroup, the automorphisms fixing the maximal unramified extension K^{ur}. For m prime to p,
every m-th root of unity of K^{alg} lies in K^{ur}, since m divides q ^ φ(m) − 1; so
inertia fixes it. Hence for a ∈ Kˣ and any α ∈ K^{alg} with α ^ m = a,
σ ↦ σ(α) / α
is a homomorphism I_K → μ_m(K^{alg}) that does not depend on the choice of the root α: two
roots differ by an m-th root of unity, which σ fixes. These characters are compatible along
the power maps μ_m → μ_n, ζ ↦ ζ ^ (m / n), for n ∣ m, because α ^ (m / n) is an n-th
root of a, and they are locally constant for the Krull topology. Together they form the
continuous homomorphism
tameKummerCharacter K a : I_K →ₜ* ℤ̂^{(p')}(1) = lim_{p ∤ m} μ_m(K^{alg})
into the prime-to-p Tate module TauCeti.PrimeToPTateModule. It is multiplicative in a and
trivial on the units 𝒪[K]ˣ: a unit is a (q − 1)-st root of unity times a principal unit, the
former has roots of unity of order prime to p as its m-th roots, and the latter is an m-th
power in K. So all uniformizers π give the same character, the tame character
inertiaTameCharacter K : I_K →ₜ* ℤ̂^{(p')}(1), σ ↦ (σ(π^{1/m})/π^{1/m})_m,
independent of the uniformizer and of the chosen roots. It is surjective, because X ^ m − π stays
irreducible over K^{ur}, so that I_K = Gal(K^{alg}/K^{ur}) moves π^{1/m} to each of its
conjugates ζ π^{1/m}, and I_K is compact. Its kernel is the wild inertia group P_K, the
automorphisms fixing all the π^{1/m} over K^{ur}. So it identifies the tame inertia group
I_K/P_K with ℤ̂^{(p')}(1), as topological groups. It is equivariant for conjugation by
G_K, which acts on ℤ̂^{(p')}(1) through its action on the roots of unity: this is the twist
(1).
Main definitions #
TauCeti.inertiaKummerCharacter K m hm a: formprime topanda ∈ Kˣ, the characterσ ↦ σ(α)/αofI_Kwith values inμ_m(K^{alg}), for anym-th rootαofa.TauCeti.tameKummerCharacter K a: the compatible family of these characters, a continuous homomorphismI_K →ₜ* PrimeToPTateModule p K^{alg}.TauCeti.inertiaTameCharacter K: the tame Kummer character of a uniformizer.TauCeti.quotientWildInertiaSubgroupEquiv K: the isomorphism of topological groupsI_K/P_K ≃ₜ* ℤ̂^{(p')}(1)induced by the tame character.
Main results #
TauCeti.coe_inertiaKummerCharacter_apply: the value atσisσ(α)/αfor everym-th rootαofa.TauCeti.inertiaKummerCharacter_eq_one_iff: the character is trivial exactly when inertia fixes the roots ofX ^ m − a.TauCeti.inertiaKummerCharacter_mul,TauCeti.tameKummerCharacter_mul: the characters are multiplicative ina.TauCeti.inertiaKummerCharacter_pow_div: the compatibility along the power maps.TauCeti.inertiaKummerCharacter_eq_one_of_mem_unitFiltration_zero,TauCeti.tameKummerCharacter_eq_one_of_mem_unitFiltration_zero: the characters of a unit are trivial.TauCeti.inertiaTameCharacter_eq_tameKummerCharacter,TauCeti.coe_proj_inertiaTameCharacter_apply: the tame character is the tame Kummer character of any uniformizerπ, with level-mcomponentσ(α)/αfor any rootαofX ^ m − π.TauCeti.inertiaKummerCharacter_surjective,TauCeti.proj_inertiaTameCharacter_surjective,TauCeti.inertiaTameCharacter_surjective: the Kummer character of a uniformizer and the tame character are surjective at each level, and the tame character is surjective.TauCeti.inertiaTameCharacter_eq_one_iff,TauCeti.ker_inertiaTameCharacter: the kernel of the tame character is the wild inertia subgroup.TauCeti.inertiaTameCharacter_conj: the tame character isG_K-equivariant.
References #
- J.-P. Serre, Corps Locaux, Chapter IV, §2, Proposition 7 and its corollaries.
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, (7.5.2).
The Kummer character at a finite level #
The Kummer character of a on inertia at level m. For m prime to the residue
characteristic p of K and a ∈ Kˣ, the character σ ↦ σ(α)/α of the inertia subgroup with
values in the m-th roots of unity of K^{alg}, where α is any root of X ^ m − a
(TauCeti.coe_inertiaKummerCharacter_apply).
Equations
- TauCeti.inertiaKummerCharacter K m hm a = { toFun := fun (σ : ↥(TauCeti.inertiaSubgroup K)) => rootsOfUnity.mkOfPowEq (TauCeti.kummerRatio✝ K m hm a ↑σ) ⋯, map_one' := ⋯, map_mul' := ⋯ }
Instances For
The Kummer character is σ(α)/α for every root α of X ^ m − a: the value does not
depend on the choice of the root.
An element of inertia moves every root α of X ^ m − a by the value of the Kummer
character: σ(α) = χ_a(σ) · α.
The Kummer character is trivial at σ exactly when σ fixes a root of X ^ m − a, and then
it fixes all of them.
The Kummer character is trivial exactly when inertia fixes a root of X ^ m − a, and then it
fixes all of them.
The Kummer character is multiplicative in a: a product of roots is a root of the product.
The Kummer character of one is trivial.
The Kummer character of an inverse is the inverse Kummer character.
The Kummer character of a natural power is the corresponding power of the Kummer character.
Compatibility along the power maps. For n ∣ m, the level-m Kummer character raised to
the power m / n is the level-n Kummer character, since α ^ (m / n) is an n-th root of a
whenever α is an m-th root.
The Kummer character is locally constant for the Krull topology: σ ↦ σ(α) only depends on
the restriction of σ to the finite extension K(α).
The Kummer character of a unit is trivial. For u ∈ U(K,0) = 𝒪[K]ˣ and m prime to p,
inertia fixes the m-th roots of u, which are unramified by
TauCeti.mem_maximalUnramifiedExtension_of_pow_eq: writing u = ζ v with ζ a (q - 1)-st root
of unity and v a principal unit, the roots of ζ are roots of unity of order prime to p, and
v is an m-th power in K.
Independence of the uniformizer. Any two uniformizers of K have the same Kummer character
on inertia, since their ratio is a unit.
The tame Kummer character with values in the Tate module #
The tame Kummer character of a ∈ Kˣ: the continuous homomorphism from the inertia
subgroup of K to the prime-to-p Tate module ℤ̂^{(p')}(1) = lim_{p ∤ m} μ_m(K^{alg}),
σ ↦ (σ(α_m)/α_m)_m for any roots α_m of X ^ m − a. Its components are the characters
TauCeti.inertiaKummerCharacter K m hm a. At a uniformizer a = π it is the tame character of
K.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The component of the tame Kummer character at level m is the Kummer character at level
m.
The tame Kummer character is multiplicative in a.
The tame Kummer character of one is trivial.
The tame Kummer character of an inverse is the inverse tame Kummer character.
The tame Kummer character of a natural power is the corresponding power of the tame Kummer character.
The tame Kummer character of a unit is trivial.
Any two uniformizers of K have the same tame Kummer character.
The tame character #
The tame character of K: the continuous homomorphism I_K →ₜ* ℤ̂^{(p')}(1),
σ ↦ (σ(π^{1/m})/π^{1/m})_m, for a uniformizer π of K and any m-th roots π^{1/m} of it.
It depends neither on π (TauCeti.inertiaTameCharacter_eq_tameKummerCharacter) nor on the roots
(TauCeti.coe_proj_inertiaTameCharacter_apply).
Equations
Instances For
The tame character is the tame Kummer character of any uniformizer.
The tame character at level m. For a uniformizer π of K, m prime to p, and any
root α of X ^ m − π, the level-m component of the tame character at σ ∈ I_K is σ(α)/α.
Surjectivity #
The Kummer character of a uniformizer is surjective at each level. For a uniformizer π
and m prime to p, every m-th root of unity ζ is σ(α)/α for some σ ∈ I_K, where α is
a root of X ^ m − π: as X ^ m − π is irreducible over K^{ur}
(TauCeti.X_pow_sub_C_irreducible_maximalUnramifiedExtension), Gal(K^{alg}/K^{ur}) = I_K
carries α to its conjugate ζ α.
The tame character is surjective at each level: every m-th root of unity, for m prime
to p, is the level-m component of the tame character of some element of inertia.
The tame character is surjective: I_K → ℤ̂^{(p')}(1) is onto, since it is onto at each
finite level (TauCeti.proj_inertiaTameCharacter_surjective) and I_K is compact.
The kernel is wild inertia #
The kernel of the tame character is wild inertia: σ ∈ I_K has trivial tame character
exactly when it lies in P_K, that is, when it fixes every m-th root of a uniformizer for
p ∤ m (TauCeti.mem_wildInertiaSubgroup_iff_of_isUniformizer).
The kernel of the tame character is the wild inertia subgroup P_K, viewed inside I_K.
Tame inertia #
Tame inertia is the prime-to-p Tate module: the tame character induces an isomorphism
of topological groups I_K / P_K ≃ₜ* ℤ̂^{(p')}(1) from the tame inertia group, the quotient of
the inertia subgroup by the wild inertia subgroup.
Equations
- One or more equations did not get rendered due to their size.
Instances For
The isomorphism I_K / P_K ≃ₜ* ℤ̂^{(p')}(1) sends the class of σ to its tame character.
The inverse isomorphism ℤ̂^{(p')}(1) ≃ₜ* I_K / P_K sends the tame character of σ to the
class of σ.
Equivariance #
The tame character is G_K-equivariant: conjugating an element of inertia by g ∈ G_K
applies g to its tame character, for the action of G_K on ℤ̂^{(p')}(1) through the roots of
unity of K^{alg}. This is the twist (1) in I_K / P_K ≅ ℤ̂^{(p')}(1).