Marked presentations of local maximal pro-p Galois groups #
Let K be a nonarchimedean local field that is a finite extension of ℚ_[p] and contains a
primitive pth root of unity. Then G_K(p) is a Demushkin group of rank N + 2, where
N = [K : ℚ_[p]] (TauCeti.demushkinRank_absoluteGaloisGroupProP), its q-invariant is the
number q(K) of p-power roots of unity of K (TauCeti.demushkinQ_absoluteGaloisGroupProP),
and its canonical character is the cyclotomic orientation
(TauCeti.demushkinCharacter_absoluteGaloisGroupProP). Labute's marked classification of
Demushkin groups (TauCeti.isDemushkin_marked_of_q_ne_two,
TauCeti.isDemushkin_marked_of_q_two_odd, TauCeti.isDemushkin_marked_of_q_two_even_unitsPlusMinus
and TauCeti.isDemushkin_marked_of_q_two_even_twisted) therefore presents G_K(p) by a single
relator on N + 2 generators, and records the values of the cyclotomic character on the
generators.
If q(K) ≠ 2, which is the case TauCeti.IsQNeTwoCase,
G_K(p) ≃ ⟨x₁, …, x_{N+2} ∣ x₁^{q(K)} (x₁, x₂) (x₃, x₄) ⋯ (x_{N+1}, x_{N+2})⟩,
with χ(x₂) = (1 - q(K))⁻¹ and χ(xᵢ) = 1 for the other generators.
If p = 2 and N is odd, the cyclotomic character of K is onto ℤ₂ˣ
(TauCeti.range_localCyclotomicCharacter_of_odd_finrank), which is the image {±1} × U^(2), so
the level of Labute's odd-rank relator is f = 2 and
G_K(2) ≃ ⟨x₁, …, x_{N+2} ∣ x₁² x₂⁴ (x₂, x₃) (x₄, x₅) ⋯ (x_{N+1}, x_{N+2})⟩,
with χ(x₁) = -1, χ(x₃) = (1 - 4)⁻¹ = -1/3 and χ(xᵢ) = 1 for the other generators. For
K = ℚ₂ this is the standard dyadic group ⟨x₁, x₂, x₃ ∣ x₁² x₂⁴ (x₂, x₃)⟩. The only hypothesis
on K is odd degree, which already forces K to contain no primitive fourth root of unity; so it
applies to every field satisfying TauCeti.IsDyadicOddCase, through
TauCeti.IsDyadicOddCase.odd_finrank.
If p = 2 and N is even, the relator is
x₁^{2 + α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (x_{N+1}, x_{N+2}),
with χ(x₂) = -(1 + α)⁻¹, χ(x₄) = (1 - 2^f)⁻¹ and χ(xᵢ) = 1 for the other generators. The
image of the cyclotomic character is a hypothesis of each even marked theorem, and it constrains
the parameters: it determines the level f in the first case below and the index k in the
second, while the exponent α, and in the second case the level f, may be any values satisfying
the stated conditions. If the image contains -1 it is {±1} × U^(f) for some f ≥ 2
(TauCeti.range_localCyclotomicCharacter_of_degree_even_plusMinus), and that level f is the
level of the relator, with any exponent α divisible by 2^f, for instance α = 0. Otherwise,
when K has no primitive fourth root of unity, the image is the twisted subgroup U^[k]
topologically generated by -1 + 2^k, k ≥ 2
(TauCeti.range_localCyclotomicCharacter_of_degree_even_principal), and the relator takes any
exponent α with v₂(α) = k, for instance α = 2^k, and any level f > k.
Main results #
TauCeti.absoluteGaloisGroupProP_marked_of_q_ne_two: the marked presentation ofG_K(p)whenq(K) ≠ 2.TauCeti.absoluteGaloisGroupProP_two_marked_of_degree_odd: the marked presentation ofG_K(2)for a finite extensionKofℚ₂of odd degree.TauCeti.absoluteGaloisGroupProP_two_marked_of_degree_even_plusMinus,TauCeti.absoluteGaloisGroupProP_two_marked_of_degree_even_principal: the marked presentations ofG_K(2)for a finite extensionKofℚ₂of even degree, according as the cyclotomic image is{±1} × U^(f)orU^[k].TauCeti.absoluteGaloisGroupProP_two_of_degree_even: in even degree, whenKcontains no primitive fourth root of unity, some even-rank dyadic relator presentsG_K(2).
References #
- J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, Theorems 7–9.
- J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., (7.5.11) and (7.5.12).
The marked presentation of G_K(p) when q(K) ≠ 2. Let K be a finite extension of
ℚ_[p] of degree N containing a primitive pth root of unity, whose number q(K) of p-power
roots of unity is not 2. Then G_K(p) is isomorphic to
⟨x₁, …, x_{N+2} ∣ x₁^{q(K)} (x₁, x₂) (x₃, x₄) ⋯ (x_{N+1}, x_{N+2})⟩, and under this isomorphism
the cyclotomic orientation takes the value (1 - q(K))⁻¹ on the second marked generator and is
trivial on every other marked generator.
The marked presentation of G_K(2) in odd degree. Let K be a finite extension of ℚ₂
of odd degree N. Then G_K(2) is isomorphic to
⟨x₁, …, x_{N+2} ∣ x₁² x₂⁴ (x₂, x₃) (x₄, x₅) ⋯ (x_{N+1}, x_{N+2})⟩, and under this isomorphism
the cyclotomic orientation takes the value -1 on the first marked generator, the value
(1 - 4)⁻¹ on the third, and is trivial on every other marked generator.
The marked presentation of G_K(2) in even degree, image {±1} × U^(f). Let K be a
finite extension of ℚ₂ of even degree N whose cyclotomic character has image
{±1} × U^(f), with f ≥ 2. Then for every natural exponent α divisible by 2^f, for
instance α = 0, G_K(2) is isomorphic to
⟨x₁, …, x_{N+2} ∣ x₁^{2 + α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (x_{N+1}, x_{N+2})⟩, and under this
isomorphism the cyclotomic orientation takes the value -(1 + α)⁻¹ on the second marked
generator, the value (1 - 2^f)⁻¹ on the fourth, and is trivial on every other marked
generator.
The marked presentation of G_K(2) in even degree, image U^[k]. Let K be a finite
extension of ℚ₂ of degree N whose cyclotomic character has image the twisted subgroup
U^[k], topologically generated by -1 + 2^k, with k ≥ 2. Since -1 does not lie in U^[k],
while the image is everything in odd degree, N is even. Then for every natural exponent α
with v₂(α) = k, for instance α = 2^k, and every level f > k, G_K(2) is isomorphic to
⟨x₁, …, x_{N+2} ∣ x₁^{2 + α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (x_{N+1}, x_{N+2})⟩, and under this
isomorphism the cyclotomic orientation takes the value -(1 + α)⁻¹ on the second marked
generator, the value (1 - 2^f)⁻¹ on the fourth, and is trivial on every other marked
generator.
G_K(2) in even degree is an even-rank dyadic Demushkin group. Let K be a finite
extension of ℚ₂ of even degree N that contains no primitive fourth root of unity, that is,
whose number of 2-power roots of unity is 2. Then for some exponent α divisible by 4 and
some level f ≥ 2, G_K(2) is isomorphic to
⟨x₁, …, x_{N+2} ∣ x₁^{2 + α} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (x_{N+1}, x_{N+2})⟩.