Localization of corestricted idele classes #
Let K be a number field, U an open subgroup of G_K with fixed field F, and v a place of
K. This file computes the localization at v of the corestriction to G_K of a class
x ∈ H²(U, I_{Kˢ}). Classically, the invariant at v of a corestricted class is the sum of the
invariants of x at the places w of F above v:
inv_v (cor_U^{G_K} x) = ∑_{w ∣ v} inv_w x.
This file proves the group-theoretic form of that formula, in which the places above v are
replaced by double cosets. Fix an embedding τ : Kˢ → K_vˢ of separable closures, with
decomposition map φ_τ : G_{K_v} → G_K. For a representative s of a double coset
φ_τ(G_{K_v}) s U, the conjugate embedding τ ∘ s gives the decomposition map
φ_{τ ∘ s} = s⁻¹ φ_τ s, which carries the open subgroup φ_{τ ∘ s}⁻¹(U) of G_{K_v} into U.
Pulling x back along it and along the coordinate ideleCoeffComponent (τ ∘ s) at v is the
localization of x along τ ∘ s; classically, the double cosets index the places w of F
above v, φ_{τ ∘ s}⁻¹(U) is the absolute Galois group of the completion F_w, and this is
the localization of x at w, but that identification is not made here. The Mackey formula
along the decomposition map (TauCeti.ContCohomology.explicitMap2_explicitCor2_eq_sum) gives
loc_v (cor_U^{G_K} x) = ∑_{φ_τ(G_{K_v}) s U} cor (loc_{τ ∘ s} x),
the corestriction being from φ_{τ ∘ s}⁻¹(U) to G_{K_v}
(ideleBrLocalization_explicitCor2, and ideleInfiniteBrLocalization_explicitCor2 at an infinite
place). The local invariant of an open subgroup of G_{K_v} is the invariant of its corestriction
(TauCeti.ClassFieldTheory.subgroupInvMap), so the invariant at v is the sum of the invariants
of the localizations (invMap_ideleBrLocalization_explicitCor2). For an idele layer L/F this
computes the local invariant ideleLocalInvAt K L v of its classes
(ideleLocalInvAt_eq_sum_subgroupInvMap).
Main results #
TauCeti.ClassFieldTheory.ideleBrLocalization_explicitCor2,TauCeti.ClassFieldTheory.ideleInfiniteBrLocalization_explicitCor2: the localization of a corestricted class is the sum over the double cosets of the corestricted localizations.TauCeti.ClassFieldTheory.invMap_ideleBrLocalization_explicitCor2: the invariant at a finite place of a corestricted class is the sum of the invariants of its localizations.TauCeti.ClassFieldTheory.ideleLocalInvAt_eq_sum_subgroupInvMap: the local invariant atvof an idele-layer class is the sum of the invariants of its localizations along the conjugates ofτ.
References #
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (1.5.6) and (8.1.17).
- J. Tate, Global class field theory, in J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Chapter VII, §11.
Finite places #
The localization at a finite place of a corestricted idele class. For an open subgroup U
of G_K, an embedding τ : Kˢ → K_vˢ and representatives r D of the double cosets
φ_τ(G_{K_v}) \ G_K / U, the localization at v of the corestriction of x ∈ H²(U, I_{Kˢ}) is
the sum over the double cosets of the corestrictions, from φ_{τ ∘ r D}⁻¹(U) to G_{K_v}, of the
localizations of x along τ ∘ r D.
The invariant at a finite place of a corestricted idele class is the sum, over the double
cosets φ_τ(G_{K_v}) \ G_K / U, of the invariants of the localizations of the class along the
conjugates τ ∘ r D of τ, each read by the local invariant of the open subgroup
φ_{τ ∘ r D}⁻¹(U) of G_{K_v}: classically, inv_v (cor x) = ∑_{w ∣ v} inv_w x.
The local invariant of an idele-layer class: for a layer L/F of the idele formation of
K, cut out by V ◁ U, the invariant at the finite place v of a class x is the sum, over the
double cosets φ_τ(G_{K_v}) \ G_K / U, of the invariants of the localizations along the conjugates
τ ∘ r D of τ of its inflation to U.
Infinite places #
The localization at an infinite place of a corestricted idele class: the archimedean
counterpart of TauCeti.ClassFieldTheory.ideleBrLocalization_explicitCor2.