The cyclotomic character of local Artin symbols along the norm #
Let L/K be an extension of number fields, p a prime and x ∈ Lˣ. At every place w of L
the local Artin symbol Art_w(x) of the image of x in the completion L_w acts on the roots of
unity of p-power order through the p-adic cyclotomic character
χ_p = TauCeti.abelianizedLocalCyclotomicCharacter p L_w. This file proves that these values
are carried to K by the norm, place by place and then globally:
∏_{w ∣ v} χ_p(Art_w(x)) = χ_p(Art_v(N_{L/K} x)) for every place v of K,
∏_w χ_p(Art_w(x)) = ∏_v χ_p(Art_v(N_{L/K} x)),
the second over the finite and over the infinite places separately. Since the action of a Galois
automorphism on the m-th roots of unity is read off from its p-adic cyclotomic characters
for the primes p ∣ m, this transfers the product formula ∏_v Art_v(a) = 1 for the local
Artin symbols on the cyclotomic extensions K(μ_m)/K from N_{L/K} x to x. Taking K = ℚ,
this reduces the product formula for the cyclotomic extensions of every number field to the one
over ℚ, where the local symbols are explicit; that product formula is the input from which the
sum of the local invariants of a global Brauer class is shown to vanish.
At a finite place v the identity is the local norm functoriality
TauCeti.ClassFieldTheory.abelianizedLocalCyclotomicCharacter_artinMap_norm at each
w ∣ v, combined with the factorization N_{L/K}(x) = ∏_{w ∣ v} N_{L_w/K_v}(x) in K_v
(TauCeti.algebraMap_norm_eq_prod_norm); at an infinite place it is the archimedean norm
functoriality TauCeti.ClassFieldTheory.infiniteArtinAt_norm with the archimedean
factorization TauCeti.GlobalNumberFields.algebraMap_norm_eq_prod_norm_infiniteCompletion.
The global finite identity regroups the places of L by the place below. This needs the
finiteness of the support of w ↦ χ_p(Art_w(x)): at a place not above p where x is a unit,
the unramified formula χ_p(Art_w(x)) = q_w ^ {v_w(x)}
(TauCeti.ClassFieldTheory.abelianizedLocalCyclotomicCharacter_artinMap) gives 1.
The same functoriality controls the Artin symbols of local norms. If L contains a primitive
pⁿ-th root of unity, every completion L_w does, so χ_p(Art_w(u)) ≡ 1 mod pⁿ for every
u ∈ L_wˣ; hence χ_p(Art_v(a)) ≡ 1 mod pⁿ for every local norm a from K_v ⊗[K] L, at finite
and at infinite places alike.
Main results #
TauCeti.ClassFieldTheory.abelianizedLocalCyclotomicCharacter_artinMap_normUnits_eq_finprodandabelianizedLocalCyclotomicCharacter_infiniteArtinAt_normUnits_eq_finprod: the value at a placevofKon the norm is the product of the values at the placesw ∣ v.TauCeti.ClassFieldTheory.abelianizedLocalCyclotomicCharacter_infiniteArtinAt_norm: norm functoriality of the character of archimedean Artin symbols for one placew ∣ v.TauCeti.ClassFieldTheory.abelianizedLocalCyclotomicCharacter_artinMap_mem_unitsPrincipalandabelianizedLocalCyclotomicCharacter_infiniteArtinAt_mem_unitsPrincipal: ifLcontains a primitivepⁿ-th root of unity, the character of the Artin symbol of a local norm fromK_v ⊗[K] Llies inU^(n) = 1 + pⁿ ℤ_p.TauCeti.ClassFieldTheory.hasFiniteMulSupport_abelianizedLocalCyclotomicCharacter_artinMap: forx ∈ Kˣ,χ_p(Art_v(x)) = 1at all but finitely many finite placesv.TauCeti.ClassFieldTheory.finprod_abelianizedLocalCyclotomicCharacter_artinMap_normUnitsandprod_abelianizedLocalCyclotomicCharacter_infiniteArtinAt_normUnits: the products over all finite, respectively all infinite, places agree.
References #
- J. S. Milne, Class Field Theory, Chapter VII, Lemma 8.4.
Place by place #
Norm functoriality at a finite place. For x ∈ Lˣ and a finite place v of K, the
p-adic cyclotomic character of the local Artin symbol of N_{L/K} x at v is the product of
those of the local Artin symbols of x at the places of L above v.
The cyclotomic character of archimedean Artin symbols along a local norm, the
archimedean analogue of abelianizedLocalCyclotomicCharacter_artinMap_norm. For a place w of
L over the infinite place v of K and x ∈ L_wˣ, the p-adic cyclotomic character of
Art_w(x) is that of Art_v(N_{L_w/K_v} x).
Norm functoriality at an infinite place. For x ∈ Lˣ and an infinite place v of K,
the p-adic cyclotomic character of the archimedean Artin symbol of N_{L/K} x at v is the
product of those of the archimedean Artin symbols of x at the places of L above v.
Local norms from an extension containing roots of unity #
Local norms from an extension containing μ_{pⁿ}, at a finite place. If L contains a
primitive pⁿ-th root of unity, then for every local norm a from K_v ⊗[K] L at a finite place
v of K, the p-adic cyclotomic character of Art_v(a) is ≡ 1 mod pⁿ.
Local norms from an extension containing μ_{pⁿ}, at an infinite place. If L contains a
primitive pⁿ-th root of unity, then for every local norm a from K_v ⊗[K] L at an infinite
place v of K, the p-adic cyclotomic character of the archimedean Artin symbol Art_v(a) is
≡ 1 mod pⁿ.
Over all places #
For x ∈ Kˣ, the p-adic cyclotomic character of the local Artin symbol of x is 1 at
all but finitely many finite places of K: at a place not above p where x is a unit, it is
q ^ v(x) = 1 for the residue cardinality q.
Norm functoriality over the finite places. For x ∈ Lˣ, the product over the finite
places v of K of the p-adic cyclotomic characters of the local Artin symbols of
N_{L/K} x is the corresponding product for x over the finite places of L.
Norm functoriality over the infinite places. For x ∈ Lˣ, the product over the infinite
places v of K of the p-adic cyclotomic characters of the archimedean Artin symbols of
N_{L/K} x is the corresponding product for x over the infinite places of L.