The local Weil group inside the abelianized absolute Galois group #
Let K be a nonarchimedean local field. The inclusion W_K → G_K induces a continuous
homomorphism
W_K^{ab} → G_K^{ab}.
This file identifies its image without choosing a Frobenius lift: it consists exactly of the
classes whose unramified coordinate in ℤ̂ is an integer. It also descends the Weil degree to
W_K^{ab} and proves that the resulting square with ℤ → ℤ̂ commutes.
The map is injective. The commutators of W_K lie in inertia, on which the Weil topology is the
profinite topology of G_K, and W_K is dense in G_K, so the closed commutator subgroup of
W_K is the trace on W_K of the closed commutator subgroup of G_K. Hence W_K^{ab} is
isomorphic, as an abstract group, to the classes in G_K^{ab} with integral unramified
coordinate, and its topology is the one induced by x ↦ (x, deg x) into G_K^{ab} × ℤ, exactly
as for W_K itself. These are the inputs about W_K^{ab} itself to the local Weil reciprocity
isomorphism Kˣ ≃ₜ* W_K^{ab}.
Main definitions #
TauCeti.ClassFieldTheory.weilToAbsoluteAbelianization: the continuous mapW_K^{ab} → G_K^{ab}induced byW_K → G_K.TauCeti.ClassFieldTheory.weilDegreeAbelianization: the Weil degree descended toW_K^{ab}.TauCeti.ClassFieldTheory.integralUnramifiedSubgroup: the subgroup ofG_K^{ab}whose unramified coordinate is integral.TauCeti.ClassFieldTheory.weilToIntegralUnramified: the induced surjection fromW_K^{ab}onto that subgroup.TauCeti.ClassFieldTheory.weilAbelianizationEquivIntegralUnramified: the same map, as an isomorphism of groups.
Main results #
TauCeti.ClassFieldTheory.unramifiedCoordinate_weilToAbsoluteAbelianization: the unramified coordinate of a Weil class is the image of its degree inℤ̂.TauCeti.ClassFieldTheory.range_weilToAbsoluteAbelianization: the image ofW_K^{ab}is preciselyintegralUnramifiedSubgroup K.TauCeti.ClassFieldTheory.injective_weilToAbsoluteAbelianization: the mapW_K^{ab} → G_K^{ab}is injective.TauCeti.ClassFieldTheory.ker_weilDegreeAbelianization: the classes of degree zero inW_K^{ab}are the classes of inertia.isEmbedding_weilToAbsoluteAbelianization_prod_weilDegreeAbelianization, in the same namespace:x ↦ (x, deg x)is a topological embeddingW_K^{ab} → G_K^{ab} × ℤ.TauCeti.ClassFieldTheory.denseRange_weilToAbsoluteAbelianization: the image is dense inG_K^{ab}.
References #
- A. Weil, Sur la théorie du corps de classes, J. Math. Soc. Japan 3 (1951).
- J. Tate, Number theoretic background, in Automorphic forms, representations and L-functions, Proc. Sympos. Pure Math. 33, Part 2 (1979), §1.4.
The continuous homomorphism W_K^{ab} → G_K^{ab} induced by the inclusion W_K → G_K.
Instances For
The induced map on topological abelianizations sends the class of w to the class of its
image in the absolute Galois group.
The Weil degree descended to the topological abelianization of W_K.
Instances For
The abelianized Weil degree agrees with the degree on representatives.
Compatibility of degree and the unramified coordinate. The square formed by
W_K^{ab} → G_K^{ab}, the abelianized Weil degree, and ℤ → ℤ̂ commutes.
The subgroup of G_K^{ab} consisting of classes with integral unramified coordinate.
Instances For
Membership in the integral-unramified subgroup means exactly that the unramified coordinate
belongs to the image of ℤ → ℤ̂.
The image of the abelianized Weil group. A class in G_K^{ab} comes from W_K^{ab}
exactly when its unramified coordinate is an integer.
The map from W_K^{ab} to its image in G_K^{ab}, with codomain restricted to the classes
having integral unramified coordinate.
Instances For
The range-restricted abelianized inclusion has the same underlying value in G_K^{ab}.
The abelianized Weil group surjects onto the subgroup of classes with integral unramified coordinate.
The abelianized Weil group injects into the abelianized absolute Galois group: the map
W_K^{ab} → G_K^{ab} induced by the inclusion W_K → G_K is injective.
The abelianized inertia sequence: the classes of degree zero in W_K^{ab} are exactly
the classes of elements of inertia. This is the abelianized form of ker_weilDegree.
The topology of W_K^{ab}: x ↦ (x, deg x) is a topological embedding of W_K^{ab}
into G_K^{ab} × ℤ, with ℤ discrete. This is the abelianized form of the characterization of
the Weil topology, isEmbedding_weilToAbsolute_prod_weilDegree.
The abelianized Weil group is isomorphic, as an abstract group, to the subgroup of G_K^{ab}
of classes with integral unramified coordinate, through the map induced by W_K → G_K.
Instances For
The isomorphism W_K^{ab} ≃* integralUnramifiedSubgroup K has the same underlying value in
G_K^{ab} as the induced map W_K^{ab} → G_K^{ab}.
The inverse of W_K^{ab} ≃* integralUnramifiedSubgroup K sends a class with integral
unramified coordinate to the unique Weil class mapping to it in G_K^{ab}.
The induced map W_K^{ab} → G_K^{ab} has dense image. Equivalently, classes with integral
unramified coordinate are dense in G_K^{ab}.