Idele classes with prescribed local invariants #
Let K be a number field. This file proves the cyclotomic input to the global invariant map: every
finite Galois layer L/F of the idele formation of K has a refinement L'/F carrying a class of
H²(Gal(L'/F), I_{L'}) whose local invariants sum to 1 / [L : F]
(exists_refinement_ideleSumLocalInv_eq).
The class is first built over G_K (exists_sumLocalInv_ideleLocalization_eq). Fix a finite place
v of K whose residue field has q elements, and n ≥ 1. The class of q in
(ℤ / (q ^ n - 1))ˣ has order n, so some character ψ of this group sends it to 1 / n. Let χ
be ψ of the action of G_K on a primitive (q ^ n - 1)-th root of unity ζ: a character of
the cyclotomic extension K(ζ)/K. Let a be the idele with a uniformizer π of K_v at v and
1 elsewhere. The carry class a ∪ δχ of χ and a localizes at every place to the carry class
of the restricted character and of the component of a there
(ideleBrLocalization_characterCarryCocycle). Away from v that component is 1, so the
localization vanishes. At v, the root of unity ζ generates the unramified extension of degree
n of K_v, whose arithmetic Frobenius raises ζ to the q-th power, so the local invariant is
v(π) · χ(Frob_v) = 1 / n (invMap_ideleBrLocalization_characterCarryCocycle).
The integer multiples of these classes realize every family of local Brauer classes supported at
one finite place. At a real place w, the carry class of the idele with component -1 at w and
1 elsewhere, and of a character ψ of the action of G_K on a primitive fourth root of unity
with ψ (-1) = 1 / 2, has invariant 1 / 2 at w and vanishes elsewhere: the archimedean Artin
symbol of -1 is complex conjugation (infiniteArtinAt_rootsOfUnity_of_isReal), which inverts the
root of unity. Every family of local Brauer classes is therefore the family of localizations of a
class of H²(G_K, I_{Kˢ}) (ideleLocalization_surjective).
To reach a layer V ◁ U of G_K, take n = [L : F] · [G_K : U]. The restriction of the class to
U is inflated from a refinement V' ◁ U of the layer (NormalLayer.exists_explicitInfl2_eq).
Corestricting back to G_K multiplies by the index
(exists_refinement_ideleLayerCorInfl_eq_index_nsmul), which turns the sum 1 / n into
1 / [L : F].
Over G_K itself, every value of a character on a decomposition group is the sum of the local
invariants of a carry class. Let χ be a character of G_K with open kernel, v a finite place
and g ∈ G_{K_v}, read in G_K along an embedding τ : Kˢ → K_vˢ. The Artin symbols are dense
in G_{K_v}^ab, so some b ∈ K_vˣ has χ(Art_v b) = χ(τ_* g), and the carry class of χ and of
the idele concentrated at v with component b has sum of local invariants χ(τ_* g)
(exists_sumLocalInv_ideleLocalization_characterCarryCocycle_eq).
On every layer V ◁ U the class can be found on the layer itself, not only on a refinement. Let
F be the fixed field of U, χ a character of U ⧸ V, and g ∈ G_{K_v} with τ_* g ∈ U. On
F, the embedding τ extends an embedding τ' of the completion F_w at a place w above v
(TauCeti.exists_algHom_adicCompletion_eq). The subgroup H = τ_*⁻¹(U) of G_{K_v} contains the
subgroup fixing τ'(F_w), so by local class field theory for the open subgroup H
(exists_subgroupInvMap_characterCarryCocycle_eq) some b ∈ F_wˣ has
inv_H (τ' b ∪ δχ) = χ(τ_* g). Let a be the idele of F concentrated at w with component
b, an idele of Kˢ fixed by U. The class a ∪ δχ of the layer is inflated to the carry class
of χ and a over U (NormalLayer.explicitInfl2_artinCharacterCup), and its local invariants
are computed by the Mackey formula for the corestriction from U to G_K
(invMap_ideleBrLocalization_explicitCor2): every place other than v sees only coordinates of
a equal to 1, and at v the conjugates τ ∘ s of τ outside the double coset
τ_*(G_{K_v}) · 1 · U read places of F other than w. The only surviving term is the
localization along τ, the local carry class τ' b ∪ δχ, so the sum of the local invariants of
a ∪ δχ is χ(τ_* g) (exists_ideleSumLocalInv_eq_apply_absoluteGaloisGroupMap). If the Galois
group U ⧸ V is cyclic of order n and the decomposition subgroup H maps onto it, a character
sending the image of a generator to 1 / n gives a class of the layer with sum of local
invariants 1 / n (exists_ideleSumLocalInv_eq_one_div_degree_of_isCyclic). This is the class of
invariant 1 / n that NormalLayer.map_eq_zero_of_natCard_H2_dvd_degree takes as input. Every
layer of prime degree has such a place
(NormalLayer.exists_surjective_absoluteGaloisGroupMap_of_degree_prime), so it carries a class of
invariant 1 / [U : V] (exists_ideleSumLocalInv_eq_one_div_degree_of_prime).
Main results #
TauCeti.ClassFieldTheory.exists_sumLocalInv_ideleLocalization_eq: for everyn ≥ 1there is a class ofH²(G_K, I_{Kˢ})whose local invariants sum to1 / n.TauCeti.ClassFieldTheory.ideleLocalization_surjective: every family of local Brauer classes is the family of localizations of a class ofH²(G_K, I_{Kˢ}).TauCeti.ClassFieldTheory.exists_refinement_ideleLayerCorInfl_eq_index_nsmul:[G_K : U]times every class ofH²(G_K, I_{Kˢ})is the class overG_Kof an idele-layer class of a refinement of a given layer overU.TauCeti.ClassFieldTheory.exists_refinement_ideleSumLocalInv_eq: every layer has a refinement carrying an idele-layer class whose local invariants sum to1 / [L : F].TauCeti.ClassFieldTheory.exists_sumLocalInv_ideleLocalization_characterCarryCocycle_eq: every value of a character ofG_Kon a decomposition group is the sum of the local invariants of one of its carry classes.TauCeti.ClassFieldTheory.exists_ideleSumLocalInv_eq_apply_absoluteGaloisGroupMap: on every layerV ◁ U, every value of a character ofU ⧸ Von the image of a decomposition subgroup is the sum of the local invariants of a class of the layer.TauCeti.ClassFieldTheory.exists_ideleSumLocalInv_eq_one_div_degree_of_isCyclic: a cyclic layerV ◁ Uonto whose Galois group some decomposition subgroup maps carries a class whose local invariants sum to1 / [U : V].TauCeti.ClassFieldTheory.exists_ideleSumLocalInv_eq_one_div_degree_of_prime: every layerV ◁ Uof prime degree carries such a class.
References #
- J. S. Milne, Class Field Theory, Chapter VII, Lemma 7.3, and Chapter VIII, §4.
- J. Tate, Global class field theory, in J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Chapter VII, §11.
A class of H²(G_K, I_{Kˢ}) with sum of local invariants 1 / n.
Classes over G_K come from refinements of a layer, up to the index of its ground. For a
layer V ◁ U of the idele formation and a class x ∈ H²(G_K, I_{Kˢ}), the restriction of x to
U is inflated from a refinement V' ◁ U of the layer, and corestricting it back to G_K gives
[G_K : U] • x.
The cyclotomic input: every layer L/F of the idele formation has a refinement carrying an
idele-layer class whose local invariants sum to 1 / [L : F].
Invariants prescribed on a decomposition group #
Every value of a character on a decomposition group is the sum of the local invariants of
one of its carry classes. Let χ be a character of G_K with open kernel, v a finite place
of K, τ : Kˢ → K_vˢ an embedding of separable closures and g ∈ G_{K_v}. Then for some
invariant idele a, concentrated at v, the local invariants of the carry class a ∪ δχ sum to
χ(τ_* g).
Ideles concentrated at one place of a finite subextension #
Local invariants of corestricted carry classes #
Invariants prescribed on a decomposition group #
Every layer carries a class whose invariant is any value of a character on a decomposition
group. Let χ' be a character of the abelianized Galois group U ⧸ V of a layer V ◁ U of the
idele formation, v a finite place of K, τ : Kˢ → K_vˢ an embedding of separable closures and
g ∈ G_{K_v} whose image φ_τ(g) in G_K lies in U. Then some class of the layer has local
invariants summing to χ' of the image of φ_τ(g) in U ⧸ V.
A cyclic layer with a place of full local degree carries a class of invariant
1 / [L : F]. If the Galois group U ⧸ V of a layer V ◁ U of the idele formation is cyclic
and the decomposition subgroup φ_τ⁻¹(U) at a finite place v, read along an embedding
τ : Kˢ → K_vˢ, maps onto U ⧸ V, then some class of the layer has local invariants summing to
1 / [U : V].
A prime-degree layer carries a class of invariant 1 / [U : V]. A layer V ◁ U of prime
degree has a finite place whose decomposition map onto U ⧸ V is surjective, and its Galois group
is cyclic, so the prescribed-invariant construction applies at that place.
Classes with prescribed localizations #
Every family of local Brauer classes is the family of localizations of a class of
H²(G_K, I_{Kˢ}). By additivity it suffices to treat a family supported at one place. At a finite
place this is the class of invariant 1 / n built above, and its integer multiples. At a real
place w, take the carry class of the idele concentrated at w with component -1 and of a
character ψ of the action of G_K on a primitive fourth root of unity ζ with ψ (-1) = 1 / 2.
The Artin symbol of -1 ∈ K_w = ℝ is complex conjugation, which inverts ζ, so this class has
invariant ψ (-1) = 1 / 2 at w and vanishes at every other place.
Every family of local Brauer classes is the localization of an idele class: the
localization map H²(G_K, I_{Kˢ}) → ⨁_v Br K_v is surjective.