The cyclotomic character of the local Artin symbols of ℚ_p #
The absolute local Artin map of ℚ_p acts on the roots of unity of p-power order through the
inverse of the unit part:
χ_cyc (Art_{ℚ_p} (pᵏ u)) = u⁻¹ (u ∈ ℤ_pˣ),
and for a finite extension F/ℚ_p the Artin symbol of a unit u of the valuation ring of F
(v(u) = 1) has cyclotomic character N_{F/ℚ_p}(u)⁻¹. This is the cyclotomic normalization of
local class field theory, with the arithmetic Frobenius convention. On the roots of unity of
p-power order it identifies the Artin map with the explicit local symbols cyclotomicSymbol,
which act there through u⁻¹ as well (TauCeti.unitsMap_cyclotomicSymbol_primePow), so the
product formula for those explicit symbols becomes a statement about Artin symbols.
The proof uses no Lubin–Tate theory. It compares, at each finite level pⁿ, a representative of
the Artin symbol of a uniformizer p u with an automorphism acting through the explicit symbol
cyclotomicSymbol, inside a cyclotomic extension of ℚ_p on which local reciprocity forces the
two to agree on the roots of unity of order pⁿ. Norm functoriality of the Artin map then passes
from ℚ_p to its finite extensions.
At a finite level, a representative in Gal(L/ℚ_p) of the Artin symbol of pᵏ u raises every
root of unity of order dividing pⁿ in L to the power u⁻¹ mod pⁿ
(localArtinMap_cyclotomic_padic_primePow). Two consequences test the normalization on explicit
units. The Artin symbol of −1 inverts these roots of unity, so −1 is not a norm from any finite
Galois extension of ℚ_p containing a root of unity of p-power order other than ±1, for
instance from ℚ₂(i). Over ℚ₂, the Artin symbol of −3 acts on the sixteenth roots of unity
by ζ ↦ ζ ^ 5, whereas the geometric normalization would give ζ ↦ ζ ^ 13.
Since Kˣ is generated by a uniformizer ϖ and the units of its valuation ring, and the image of
the cyclotomic character is the closure of its values on Artin symbols
(TauCeti.ClassFieldTheory.range_localCyclotomicCharacter), these evaluations compute the image:
it is topologically generated by the unit norms N_{K/ℚ_p}(𝒪[K]ˣ) and the one further value
p ^ f · N_{K/ℚ_p}(ϖ)⁻¹. The uniformizer value cannot be dropped, because K(μ_{pⁿ})/K need not
be totally ramified: over ℚ_3(√3), where ℚ_3(√3, μ_3) is unramified over ℚ_3(√3), the unit
norms a² - 3b² fill only 1 + 3ℤ_3, while the image is all of ℤ_3ˣ.
Main results #
TauCeti.ClassFieldTheory.abelianizedLocalCyclotomicCharacter_artinMap_padic: the cyclotomic character of the Artin symbol ofpᵏ uisu⁻¹.TauCeti.ClassFieldTheory.localCyclotomicCharacter_artinMap_padic: a representative of the Artin symbol of a unitu ∈ ℤ_pˣhas cyclotomic characteru⁻¹.TauCeti.ClassFieldTheory.localCyclotomicCharacter_artinMap_unit: forF/ℚ_pfinite, a representative of the Artin symbol of a unituof the valuation ring ofFhas cyclotomic characterN_{F/ℚ_p}(u)⁻¹.TauCeti.ClassFieldTheory.localArtinMap_cyclotomic_padic_primePow: overℚ_p, a representative of the finite Artin symbol ofpᵏ uacts on the roots of unity of order dividingpⁿby the poweru⁻¹ mod pⁿ.TauCeti.ClassFieldTheory.localArtinMap_cyclotomic_padic_neg_oneandTauCeti.ClassFieldTheory.neg_one_notMem_normGroup_padic: the Artin symbol of−1inverts the roots of unity ofp-power order, so−1is not a norm from a field containing one other than±1.TauCeti.ClassFieldTheory.localArtinMap_Q2_neg_three: overℚ₂, the Artin symbol of−3sends a sixteenth root of unityζtoζ ^ 5.TauCeti.ClassFieldTheory.localCyclotomicCharacter_artinMap_uniformizer: for a uniformizerϖof a finite extensionK/ℚ_p, its Artin symbolσsatisfiesχ_cyc(σ) · N_{K/ℚ_p}(ϖ) = p ^ f(K/ℚ_p).TauCeti.ClassFieldTheory.map_unitFiltration_zero_artinMap_eq_comap_map_normUnits: the cyclotomic characters of the Artin symbols of the units of the valuation ring ofKare the norms of those units.TauCeti.ClassFieldTheory.range_localCyclotomicCharacter_eq_topologicalClosure_zpowers_sup: the image of the cyclotomic character ofG_Kis the closed subgroup ofℤ_pˣgenerated by the norms of the units ofKand the valuep ^ f · N_{K/ℚ_p}(ϖ)⁻¹at a uniformizerϖ.
References #
- J. S. Milne, Class Field Theory, Chapter VII, Example 8.2, for the explicit symbols.
- J. Neukirch, Algebraic Number Theory, Chapter V, §2, for the norm residue symbol over
ℚ_p.
The cyclotomic character of the Artin symbols of ℚ_p. If x = pᵏ u with u ∈ ℤ_pˣ, the
p-adic cyclotomic character of the absolute Artin symbol of x is u⁻¹:
χ_cyc (Art_{ℚ_p} (pᵏ u)) = u⁻¹.
So the Artin symbol of x acts on the roots of unity of p-power order through the inverse of the
unit part of x, exactly as the explicit symbol cyclotomicSymbol prescribes.
The cyclotomic normalization of the local Artin map of ℚ_p. If σ ∈ G_{ℚ_p} represents
the absolute Artin symbol of a unit u ∈ ℤ_pˣ, then the p-adic cyclotomic character of σ is
u⁻¹: σ raises every root of unity of order pⁿ to the power u⁻¹ mod pⁿ.
The cyclotomic normalization of the local Artin map of a finite extension of ℚ_p. Let
F/ℚ_p be finite. If σ ∈ G_F represents the absolute Artin symbol of a unit u of the valuation
ring of F (v(u) = 1), then the p-adic cyclotomic character of σ is the inverse of the norm
of u:
χ_cyc (Art_F u) = N_{F/ℚ_p}(u)⁻¹.
Finite-level Artin symbols on roots of unity of p-power order #
The local Artin symbols of ℚ_p on the roots of unity of p-power order. Let L/ℚ_p be a
finite Galois extension and let σ ∈ Gal(L/ℚ_p) represent the local Artin symbol of x = pᵏ u,
u ∈ ℤ_pˣ. Then σ raises every ζ ∈ L with ζ ^ pⁿ = 1 to the power u⁻¹ mod pⁿ. This is
the finite-level form of abelianizedLocalCyclotomicCharacter_artinMap_padic; on the roots of
unity of order prime to p, the Artin symbol of p acts by the p-th power instead
(localArtinMap_cyclotomic_padic).
The Artin symbol of −1 over ℚ_p inverts the roots of unity of p-power order. If
σ ∈ Gal(L/ℚ_p) represents the local Artin symbol of −1, then σ ζ = ζ⁻¹ for every ζ ∈ L
with ζ ^ pⁿ = 1.
−1 is not a norm from a field containing a root of unity of p-power order other than
±1. If a finite Galois extension L/ℚ_p contains ζ with ζ ^ pⁿ = 1 and ζ ^ 2 ≠ 1, then
−1 is not a norm from L: its Artin symbol inverts ζ
(localArtinMap_cyclotomic_padic_neg_one), so it is not trivial. For p = 2 and ζ = i this
says that −1 is not a norm from ℚ₂(i).
The ℚ₂ test at −3. Over ℚ_[2], the Artin symbol of the unit −3 sends every ζ with
ζ ^ 16 = 1 to ζ ^ 5, since (−3)⁻¹ ≡ 5 modulo 16. With the geometric normalization the
exponent would be −3 ≡ 13.
If σ represents the arithmetic Artin symbol of a uniformizer ϖ of K, then
χ_cyc(σ) · N_{K/ℚ_p}(ϖ) = p ^ f(K/ℚ_p) in ℚ_pˣ.
The cyclotomic characters of the Artin symbols of units are the unit norms. For a finite
extension K/ℚ_p, the values χ_cyc(Art_K u) at the units u of the valuation ring of K form
the subgroup of ℤ_pˣ of norms N_{K/ℚ_p}(u) of such units, since
χ_cyc(Art_K u) = N_{K/ℚ_p}(u)⁻¹ (localCyclotomicCharacter_artinMap_unit).
The image of the cyclotomic character, from unit norms and a uniformizer. Let K/ℚ_p be
finite with inertia degree f, let ϖ be a uniformizer of K, and let w ∈ ℤ_pˣ be the unit
with w · N_{K/ℚ_p}(ϖ) = p ^ f; it is the cyclotomic character of the Artin symbol of ϖ
(localCyclotomicCharacter_artinMap_uniformizer). Then the image of the p-adic cyclotomic
character of G_K is the closed subgroup of ℤ_pˣ generated by w and the norms
N_{K/ℚ_p}(u) of the units u of the valuation ring of K.