Localization of global character carry classes and the idelic character formula #
Let K be a number field and let χ : G_K → ℚ / ℤ be a character with open kernel. For an
invariant idele a ∈ I_{Kˢ}^{G_K}, the carry cocycle of χ and a represents the class usually
written a ∪ δχ in H²(G_K, I_{Kˢ}); for a ∈ Kˣ it represents the Brauer class a ∪ δχ in
Br K.
This file computes the localizations of these classes. At a finite place v, an embedding of
separable closures τ : Kˢ → K_vˢ restricts χ along the decomposition map and reads the
v-component of a. The localized idele class is the carry class of precisely those two local
objects (ideleBrLocalization_characterCarryCocycle). The same statement holds at an infinite
place (ideleInfiniteBrLocalization_characterCarryCocycle). In particular, the localization
vanishes where the corresponding component of a is 1. Together with the local invariant
formula for carry classes, these identities let a global character and an idele concentrated at
one place produce an idele-cohomology class with a prescribed sum of local invariants.
The local character formula evaluates these localizations for an arbitrary idele: if the
component of a at v is a_v ∈ K_vˣ, then the localization of a ∪ δχ at v has invariant
χ (Art_v a_v), the value of χ on the local Artin symbol of a_v read in G_K through the
decomposition map (invMap_ideleBrLocalization_characterCarryCocycle_of_mk_eq_artinMap), with no
hypothesis on the ramification of χ, and likewise at an infinite place
(infiniteInvMap_ideleInfiniteBrLocalization_characterCarryCocycle_of_mk_eq_infiniteArtinAt).
Hence the sum of the local invariants of the idele carry class is the sum of the values of χ on
the local Artin symbols of the components of a
(sumLocalInv_ideleLocalization_characterCarryCocycle,
sumLocalInv_ideleLocalization_characterCarryCocycle_eq_finsum).
These classes are the classes a ∪ δχ of the character formula on the layers of the idele
formation over K: for a character χ' of the Galois group of the layer V ◁ G_K, read on G_K
as χ, the class over G_K of a ∪ δχ' is the idele carry class of χ and a
(ideleLayerCorInfl_artinCharacterCup). So the sum of the local invariants of an idele layer is
computed on these classes by the local Artin maps (ideleSumLocalInv_artinCharacterCup):
Σ inv (a ∪ δχ') = ∑ᶠ v, χ (Art_v a_v) + ∑_w χ (Art_w a_w).
For a class formation on the idele-class formation whose invariant on the layers over K is the
sum of the local invariants of a lift to the ideles, the character formula of the abstract Artin
map turns this into the local–global formula for the Artin map on idele classes
(ClassFormation.apply_artinMap_eq_finsum_of_inv_eq_ideleSumLocalInv): a character of the Galois
group of the layer takes on the Artin symbol of an idele class the sum of its values on the local
Artin symbols of the components.
For a ∈ Kˣ, the localization of the Brauer class a ∪ δχ at a place is the carry class of the
restricted character and of the image of a
(TauCeti.ClassFieldTheory.brBaseChange_characterCarryCocycle), so the local character formula
(TauCeti.ClassFieldTheory.invMap_characterCarryCocycle_of_mk_eq_artinMap) evaluates its local
invariants:
inv_v (a ∪ δχ) = χ (Art_v a),
where Art_v a is the local Artin symbol of a ∈ K_vˣ, read in G_K through the decomposition
map (finiteInvAt_characterCarryCocycle_of_mk_eq_artinMap,
infiniteInvAt_characterCarryCocycle_of_mk_eq_infiniteArtinAt). Since a global Brauer class has
nonzero local invariant at only finitely many places, χ (Art_v a) = 0 for all but finitely
many finite places v (finite_setOf_apply_absoluteGaloisGroupMap_ne_zero), and the sum of the
local invariants of a ∪ δχ is the sum of the values of χ on the local Artin symbols of a
(sumLocalInv_brLocalization_characterCarryCocycle,
sumLocalInv_brLocalization_characterCarryCocycle_eq_finsum).
If E/K is cyclic and χ sends a generator of Gal(E/K) to 1 / [E : K], every Brauer class
split by E is such a carry class (exists_characterCarryCocycle_eq_relBrInfl). So the sum of the
local invariants vanishes on all classes split by E exactly when the local Artin symbols of every
a ∈ Kˣ satisfy the product formula ∑_v χ (Art_v a) = 0
(forall_sumLocalInv_brLocalization_relBrInfl_eq_zero_iff).
Main results #
TauCeti.ClassFieldTheory.ideleBrLocalization_characterCarryCocycle,TauCeti.ClassFieldTheory.ideleInfiniteBrLocalization_characterCarryCocycle: the localizations of an idele carry class are the local carry classes.TauCeti.ClassFieldTheory.invMap_ideleBrLocalization_characterCarryCocycle_of_mk_eq_artinMap,infiniteInvMap_ideleInfiniteBrLocalization_characterCarryCocycle_of_mk_eq_infiniteArtinAt: the local invariant of an idele carry classa ∪ δχat a place isχ (Art_v a_v).TauCeti.ClassFieldTheory.sumLocalInv_ideleLocalization_characterCarryCocycle,TauCeti.ClassFieldTheory.sumLocalInv_ideleLocalization_characterCarryCocycle_eq_finsum: the sum of the local invariants of an idele carry class is∑_v χ (Art_v a_v).TauCeti.ClassFieldTheory.ideleLayerCorInfl_artinCharacterCup,TauCeti.ClassFieldTheory.ideleSumLocalInv_artinCharacterCup: on the layerV ◁ G_Kof the idele formation, the classa ∪ δχ'of the character formula is the idele carry class, and its sum of local invariants is∑_v χ (Art_v a_v).TauCeti.ClassFieldTheory.ClassFormation.apply_artinMap_eq_finsum_of_inv_eq_ideleSumLocalInv: for a class formation on idele classes whose invariant is the sum of the local invariants,χ (Art [a]) = ∑_v χ (Art_v a_v).TauCeti.ClassFieldTheory.finiteInvAt_characterCarryCocycle_of_mk_eq_artinMap,TauCeti.ClassFieldTheory.infiniteInvAt_characterCarryCocycle_of_mk_eq_infiniteArtinAt: the local invariant ofa ∪ δχat a place isχ (Art_v a).TauCeti.ClassFieldTheory.finite_setOf_apply_absoluteGaloisGroupMap_ne_zero:χ (Art_v a)vanishes at all but finitely many finite places.TauCeti.ClassFieldTheory.sumLocalInv_brLocalization_characterCarryCocycle,TauCeti.ClassFieldTheory.sumLocalInv_brLocalization_characterCarryCocycle_eq_finsum: the sum of the local invariants ofa ∪ δχis∑_v χ (Art_v a).TauCeti.ClassFieldTheory.forall_sumLocalInv_brLocalization_relBrInfl_eq_zero_iff: for a cyclic extensionE/K, the sum of the local invariants vanishes on the classes split byEexactly when the local Artin symbols satisfy the product formula.
References #
- J. S. Milne, Class Field Theory, Chapter VII, Lemma 7.3 and Lemma 8.5.
- J.-P. Serre, Local Fields, Chapter XIV, §1.
- J. Tate, Global class field theory, in J. W. S. Cassels and A. Fröhlich (eds.), Algebraic Number Theory, Chapter VII, §11.
- J. Neukirch, Algebraic Number Theory, Chapter VI, §5.
Finite places #
A global idele carry class localizes to the corresponding local carry class. The local character is obtained by restricting along the decomposition map, and its invariant coefficient is the component of the global idele along the chosen embedding of separable closures.
A global idele carry class has zero localization at a finite place where its idele coefficient
has component 1 (written 0 in the additive coefficient module).
The local invariant of a global idele carry class. Suppose the restricted global
character is the character of a finite unramified extension E / K_v, and the selected component
of the invariant idele is the image of b ∈ K_vˣ. Then the localized class has invariant
v(b) · χ_v(Frob).
The local character formula for a global idele carry class. Let χ : G_K → ℚ/ℤ be a
character with open kernel and a an invariant idele of Kˢ whose component at v along τ is
b ∈ K_vˣ. If σ represents the absolute local Artin symbol of b, then the localization at v
of the carry class a ∪ δχ has invariant χ of the image of σ under the decomposition map of
τ. No hypothesis on the ramification of χ at v is needed.
Infinite places #
A global idele carry class localizes at an infinite place to the corresponding local carry class. The local character is the restriction along the archimedean decomposition map and the coefficient is the selected infinite component of the idele.
A global idele carry class has zero localization at an infinite place where its idele
coefficient has component 1 (written 0 in the additive coefficient module).
The archimedean character formula for a global idele carry class. Let
χ : G_K → ℚ/ℤ be a character with open kernel and a an invariant idele of Kˢ whose component
at the infinite place w along τ is b ∈ K_wˣ. If σ represents the archimedean Artin symbol
of b, then the localization at w of the carry class a ∪ δχ has invariant χ of the image of
σ under the decomposition map of τ.
The sum of the local invariants of an idele carry class #
The sum of the local invariants of a global idele carry class. Let χ : G_K → ℚ/ℤ be a
character with open kernel and a an invariant idele of Kˢ with components b v ∈ K_vˣ and
b' w ∈ K_wˣ along chosen embeddings of separable closures. Choose at every place a
representative of the local Artin symbol of the component. Then the sum of the local invariants of
a ∪ δχ is the sum of the values of χ on the images of these representatives in G_K, computed
over any finite set S of finite places outside which those values vanish.
A character vanishes on almost all local Artin symbols of the components of an idele. In
the setting of sumLocalInv_ideleLocalization_characterCarryCocycle, χ vanishes on the image
in G_K of the local Artin symbol of the component of a at all but finitely many finite places:
these values are the local invariants of the idele carry class a ∪ δχ.
The sum of the local invariants of a global idele carry class, as a finite sum over all
finite places: Σ inv (a ∪ δχ) = ∑ᶠ v, χ (Art_v a_v) + ∑_w χ (Art_w a_w), the local Artin
symbols of the components of a being read in G_K through chosen representatives and
decomposition maps.
Carry classes on the layers over K #
The class over G_K of the Artin character cup of an idele is its carry class. Let χ' be
a character of the abelianized Galois group of the layer V ◁ G_K of the idele formation, read on
G_K as χ, and let a' be an element of the ground level of the layer, an invariant idele, which
is a as an element of H⁰(G_K, I_{Kˢ}). Then the class over G_K of the class a' ∪ δχ' of the
character formula of the layer is the idele carry class of χ and a.
The idelic character formula on the layers over K. Let χ' be a character of the
abelianized Galois group of the layer V ◁ G_K of the idele formation, read on G_K as χ, and
let a' be an element of the ground level of the layer, an invariant idele, with components b v
and b' w along chosen embeddings of separable closures. Choose at every place a representative
of the local Artin symbol of the component. Then the sum of the local invariants of the class
a' ∪ δχ' of the character formula of the layer is the sum over all places of the values of χ
on the images of these representatives in G_K:
Σ inv (a' ∪ δχ') = ∑ᶠ v, χ (Art_v a_v) + ∑_w χ (Art_w a_w).
The local–global character formula for the Artin map on idele classes. Let cf be a class
formation on the idele-class formation of K whose invariant on the layer V ◁ G_K of an open
normal subgroup V is the sum of the local invariants of a lift to the ideles. Let χ' be a
character of the abelianized Galois group of the layer, read on G_K as χ, and let a' be an
invariant idele, with components b v and b' w along chosen embeddings of separable closures.
Then the character χ' of the Artin symbol of the idele class of a' is the sum over all places
of the values of χ on the local Artin symbols of the components of a':
χ' (Art [a']) = ∑ᶠ v, χ (Art_v a_v) + ∑_w χ (Art_w a_w).
Brauer classes of global elements #
The archimedean invariant of a global carry class. Let w be an infinite place of a
field K, χ : G_K → ℚ/ℤ a character with open kernel and a ∈ Kˣ. If σ represents the
archimedean Artin symbol of a ∈ K_wˣ, then the local invariant at w of the carry class
a ∪ δχ is χ of the image of σ under the decomposition map of an embedding Kˢ → K_wˢ.
The local invariant of a global carry class at a finite place. Let v be a finite place
of a number field K, χ : G_K → ℚ/ℤ a character with open kernel and a ∈ Kˣ. If σ
represents the absolute local Artin symbol of a ∈ K_vˣ, then the local invariant at v of the
carry class a ∪ δχ is χ of the image of σ under the decomposition map of an embedding
Kˢ → K_vˢ.
A character vanishes on almost all local Artin symbols of a global element. For a
character χ : G_K → ℚ/ℤ with open kernel and a ∈ Kˣ, choose at every finite place v a
representative σ v of the absolute local Artin symbol of a ∈ K_vˣ. Then χ vanishes on the
image of σ v in G_K at all but finitely many v: these values are the local invariants of
the global carry class a ∪ δχ.
The sum of the local invariants of a global carry class. For a character
χ : G_K → ℚ/ℤ with open kernel and a ∈ Kˣ, choose at every place a representative of the
local Artin symbol of a. Then the sum of the local invariants of the carry class a ∪ δχ is the
sum of the values of χ on the images of these representatives in G_K, computed over any finite
set S of finite places outside which those values vanish.
The sum of the local invariants of a global carry class, as a finite sum over all finite
places: for a character χ : G_K → ℚ/ℤ with open kernel and a ∈ Kˣ, the sum of the local
invariants of a ∪ δχ is ∑ᶠ v, χ (Art_v a) + ∑_w χ (Art_w a), the local Artin symbols being
read in G_K through chosen representatives and decomposition maps.
Classes split by a cyclic extension #
The sum of the local invariants on the classes split by a cyclic extension. Let E/K be a
cyclic Galois extension of a number field inside Kˢ, with generator g, and χ the character of
Gal(E/K) with χ(g) = 1 / [E : K]. For every a ∈ Kˣ, choose at every place a representative
of the local Artin symbol of a and read it in Gal(E/K) through a decomposition map. Then the
sum of the local invariants vanishes on every Brauer class split by E exactly when the product
formula ∑_v χ (Art_v a) = 0 holds for every a ∈ Kˣ.