The abstract Artin map of a class formation #
Let V ◁ U be a finite normal layer of a class formation with coefficient module A, and let
Γ = U ⧸ V. In degree r = -2, Tate's theorem for the class formation
(ClassFormation.tateIso) is cup product with the fundamental class,
H^{-2}(Γ, ℤ) ≃ H^0(Γ, A^V). Reading both sides through the low-degree identifications
H^{-2}(Γ, ℤ) ≃ Γ^ab (NormalLayer.tateHMinusTwoEquivAbelianization) and
H^0(Γ, A^V) ≃ A^U / N_{U/V}(A^V) (NormalLayer.tateHZeroEquivNormQuotient) gives the
Nakayama map Γ^ab ≃ A^U / N_{U/V}(A^V). The Artin reciprocity isomorphism is its inverse,
and the Artin map of the layer is the composite A^U → A^U / N_{U/V}(A^V) ≃ Γ^ab.
All three are ordinary definitions with bodies: Artin reciprocity artinEquiv is by definition
the inverse of cup product with the fundamental class read through the two low-degree
identifications, not an arbitrary isomorphism between two groups of the same order, and the Artin
map artinMap is its composite with NormalLayer.normQuotientMk; this fixes the direction once
for every downstream use. The characterizing property
(ClassFormation.cupFundamentalClass_artinMap, ClassFormation.artinMap_eq_iff) is that the
Artin symbol σ = artinMap a of a ∈ A^U is the unique σ ∈ Γ^ab whose degree -2 class cups
with the fundamental class to the zero-dimensional Tate class of a.
The consequences recorded here are the ones that need nothing beyond the isomorphism: the kernel
of the Artin map is exactly the norm subgroup (ClassFormation.ker_artinMap), the Artin map is
surjective (ClassFormation.surjective_artinMap), the norm quotient has as many elements as Γ^ab
(ClassFormation.natCard_normQuotient_eq_natCard_abelianization), and the Artin symbol of a
generates Γ^ab exactly when the class of a generates the norm quotient
(ClassFormation.isGenerator_artinMap_iff).
Conjugation by an element g of the ambient group carries the layer to gVg⁻¹ ◁ gUg⁻¹. Since
Tate's isomorphism commutes with conjugation (ClassFormation.tateIso_conj), so do the Nakayama
map and Artin reciprocity, and the Artin map is equivariant: the Artin symbol of g · a for the
conjugate layer is the conjugate of the Artin symbol of a (ClassFormation.artinMap_conj).
Raising the ground field from F to an intermediate field E of a layer K/F includes the ground
level A^U into A^{U'} and restricts Tate cohomology. Since Tate's isomorphism commutes with
restriction (ClassFormation.tateIso_res), restriction in degree -2 is the transfer
Gal(K/F)^ab → Gal(K/E)^ab and restriction in degree 0 is the ground-level inclusion, the Artin
symbol of a ∈ A^U over E is the transfer of its Artin symbol over F
(ClassFormation.artinMap_groundInclusion). Dually, since Tate's isomorphism commutes with
corestriction (ClassFormation.tateIso_cor), corestriction in degree -2 is induced by the
inclusion Gal(K/E) → Gal(K/F) and corestriction in degree 0 is the ground-level norm
N_{E/F}, the Artin symbol over F of the norm of b ∈ A^{U'} is the image of its Artin symbol
over E (ClassFormation.artinMap_groundNorm).
For a character χ : Γ^ab → ℚ/ℤ with connecting class δχ ∈ H^2(Γ, ℤ), the Artin map satisfies
the character formula χ(artinMap a) = inv(a₀ ∪ δχ) (ClassFormation.character_artinMap), and
since characters separate the points of Γ^ab it is the only homomorphism that does
(ClassFormation.eq_artinMap_of_character). The formula fixes the sign of the degree -2
identification: with the opposite sign the same construction would give the inverse of the
classical reciprocity map.
Refining the top field from K to L gives a quotient
Gal(L/F)^ab → Gal(K/F)^ab. Compatibility of the character cup pairing with inflation, together
with the character formula, shows that the Artin symbol for K/F is the image of the
Artin symbol for L/F (ClassFormation.artinMap_quotient).
Main definitions #
TauCeti.ClassFieldTheory.ClassFormation.nakayamaNegTwo: the Nakayama mapΓ^ab ≃ A^U / N_{U/V}(A^V).TauCeti.ClassFieldTheory.ClassFormation.artinEquiv: Artin reciprocityA^U / N_{U/V}(A^V) ≃ Γ^ab, the inverse of the Nakayama map.TauCeti.ClassFieldTheory.ClassFormation.artinMap: the Artin mapA^U → Γ^abof a layer.
Main statements #
TauCeti.ClassFieldTheory.ClassFormation.cupFundamentalClass_artinMap,TauCeti.ClassFieldTheory.ClassFormation.artinMap_eq_iff: the Artin symbol ofais the unique element ofΓ^abwhich cups with the fundamental class to the zero-dimensional class ofa.TauCeti.ClassFieldTheory.ClassFormation.ker_artinMap,TauCeti.ClassFieldTheory.ClassFormation.artinMap_eq_zero_iff: the kernel of the Artin map is the norm subgroup.TauCeti.ClassFieldTheory.ClassFormation.surjective_artinMap: the Artin map is surjective.TauCeti.ClassFieldTheory.ClassFormation.artinMap_conj: the Artin map commutes with conjugation.TauCeti.ClassFieldTheory.ClassFormation.artinMap_groundInclusion: inclusion of ground levels corresponds to the transfer of abelianized Galois groups.TauCeti.ClassFieldTheory.ClassFormation.artinMap_groundNorm: the norm between ground levels corresponds to the map of abelianized Galois groups induced by inclusion.TauCeti.ClassFieldTheory.ClassFormation.character_artinMap: the character formulaχ (artinMap a) = inv (a₀ ∪ δχ)for the Artin map.TauCeti.ClassFieldTheory.ClassFormation.eq_artinMap_of_character: the Artin map is the only homomorphism satisfying the character formula.TauCeti.ClassFieldTheory.ClassFormation.artinMap_quotient: refinement of the top field corresponds to the quotient map of abelianized Galois groups.
References #
- E. Artin and J. Tate, Class Field Theory, Chapter XIV, §§4–5.
- J.-P. Serre, Local Fields, Chapter XI, §3.
- J. Neukirch, Class Field Theory, Chapter III, §5.
The Nakayama map Γ^ab ≃ A^U / N_{U/V}(A^V) of a finite normal layer of a class
formation: Tate's theorem in degree -2, cup product with the fundamental class, read through the
layer's sign-normalized identification of H^{-2}(Γ, ℤ) with Γ^ab (the negative of the generic
one, so that the Artin map satisfies the character formula) and the canonical identification of
H^0(Γ, A^V) with the norm quotient. The codomain L.TateH F (-2 + 2) of cf.tateIso L (-2)
is L.TateH F 0 because -2 + 2 reduces to 0.
Instances For
The Nakayama map sends σ ∈ Γ^ab to the class in the norm quotient of the cup product of the
degree -2 class of σ with the fundamental class.
Artin reciprocity for a finite normal layer of a class formation,
A^U / N_{U/V}(A^V) ≃ Γ^ab. It is, by definition, the inverse of the Nakayama map.
Instances For
Artin reciprocity is the inverse of the cup product with the fundamental class in degree
-2, read through the two low-degree identifications.
The inverse of Artin reciprocity is the Nakayama map.
Artin reciprocity inverts the Nakayama map.
The Nakayama map inverts Artin reciprocity.
The Artin map A^U → Γ^ab of a finite normal layer of a class formation: the class of an
element of the ground level in the norm quotient, followed by Artin reciprocity.
Instances For
The Artin map is Artin reciprocity applied to the class modulo norms.
The Nakayama map sends the Artin symbol of a to the class of a modulo norms.
The defining property of the Artin symbol: the degree -2 class of artinMap a, cupped
with the fundamental class, is the zero-dimensional Tate class of a.
Characterization of the Artin symbol: artinMap a = σ exactly when the degree -2 class
of σ, cupped with the fundamental class, is the zero-dimensional Tate class of a.
The Artin symbol of a vanishes exactly when a is a norm.
The kernel of the Artin map is the norm subgroup N_{U/V}(A^V).
The Artin map is surjective onto the abelianized Galois group of the layer.
Norm index: the norm quotient A^U / N_{U/V}(A^V) of a finite normal layer of a class
formation has as many elements as the abelianized Galois group of the layer.
The Artin symbol of a generates the abelianized Galois group exactly when the class of a
generates the norm quotient.
Conjugation #
The Nakayama map commutes with conjugation: conjugating the abelianized Galois group by
g and conjugating the norm quotient by g are matched by the Nakayama maps of a layer and its
conjugate.
Artin reciprocity commutes with conjugation: Artin reciprocity of the conjugate layer, applied to the conjugate of a class modulo norms, is the conjugate of its Artin symbol.
The Artin map commutes with conjugation, one of the four Artin–Tate functoriality
diagrams: the Artin symbol of g · a for the conjugate layer gVg⁻¹ ◁ gUg⁻¹ is the conjugate by
g of the Artin symbol of a.
Restriction to an intermediate ground field #
Inclusion of ground levels corresponds to transfer, one of the four Artin–Tate
functoriality diagrams: for an intermediate field F ⊆ E ⊆ K of a layer K/F, the Artin symbol
over E of an element a of the ground level A^U of K/F is the transfer
Gal(K/F)^ab → Gal(K/E)^ab of its Artin symbol over F.
Corestriction from an intermediate ground field #
The norm corresponds to inclusion of Galois groups, one of the four Artin–Tate
functoriality diagrams: for an intermediate field F ⊆ E ⊆ K of a layer K/F, the Artin symbol
over F of the norm N_{E/F}(b) of an element b of the ground level A^{U'} of K/E is the
image of its Artin symbol over E under the map Gal(K/E)^ab → Gal(K/F)^ab induced by
inclusion.
The character formula #
The character formula for the Artin map: for every character χ of the abelianized Galois
group, χ (artinMap a) = inv (a₀ ∪ δχ), where a₀ is the zero-dimensional Tate class of a and
δχ is the connecting class of χ.
Uniqueness of the Artin map: a homomorphism φ from the ground level to the abelianized
Galois group which satisfies the character formula χ (φ a) = inv (a₀ ∪ δχ) for every a and
every character χ is the Artin map.
Refinement of the top field #
Passage to a quotient extension corresponds to the quotient map on Galois groups, the
fourth Artin–Tate functoriality diagram. Under a refinement of the top field from K to L, the
Artin symbol for K/F is the image of the Artin symbol for L/F under
Gal(L/F)^ab → Gal(K/F)^ab.