Labute's normal forms modulo λ_2 for the relator of a Demushkin group #
Let F = freeProP p (Fin n) be the free pro-p group on n generators, let r ∈ Φ(F), and let
G ≅ ⟨x₁, …, x_n ∣ r⟩ be a Demushkin group. By Labute's criterion
(TauCeti.IsDemushkin.nondegenerate_degreeOneForm), the degree-one form of the class of r in
gr_1(F) is nondegenerate, and it is alternating exactly when every cup square a ⌣ a on
H¹(G, 𝔽_p) vanishes (TauCeti.freeProP.isAlt_degreeOneForm_iff_forall_cupFp_self_eq_zero).
Feeding the relator through the normal forms of
TauCeti.Topology.Algebra.Group.Profinite.Demushkin.NormalForm.DegreeOneForm gives Labute's normal
forms modulo λ_2(F): when every cup square on H¹(G, 𝔽_p) vanishes, a change of basis of F
brings r to x₁^q (x₁, x₂) ⋯ (x_{n-1}, x_n) modulo λ_2(F) with q ∈ {0, p}, and n is even;
when some cup square does not vanish, which forces p = 2, it brings r to
x₁² x₂^{2^f} (x₂, x₃) ⋯ (x_{n-1}, x_n) for any f ≥ 2, equivalently to
x₁² (x₂, x₃) ⋯ (x_{n-1}, x_n) (for f ≥ 2 the factor x₂^{2^f} is a fourth power, so the two
words have the same class in gr_1(F)), for odd n and to
x₁^{2+a} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (x_{n-1}, x_n) for even n.
Main results #
TauCeti.IsDemushkin.exists_continuousMulEquiv_gradedMap_eq_gradedMk_demushkinWordNeTwo,TauCeti.IsDemushkin.exists_continuousMulEquiv_gradedMap_eq_gradedMk_demushkinWordTwoOdd,TauCeti.IsDemushkin.exists_continuousMulEquiv_gradedMap_eq_gradedMk_demushkinWordTwoOddTop,TauCeti.IsDemushkin.exists_continuousMulEquiv_gradedMap_eq_gradedMk_demushkinWordTwoEven: Labute's normal forms moduloλ_2(F)for the relator of a Demushkin group, according to whether the cup form onH¹(G, 𝔽_p)is alternating or not.
References #
- J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §3, Propositions 3 and 4.
- J.-P. Serre, Galois Cohomology, Chapter I, §4.5.
- J. Neukirch, A. Schmidt and K. Wingberg, Cohomology of Number Fields, 2nd ed., Chapter III, §9.
Labute's normal form modulo λ_2 for a Demushkin relator, the alternating case. Let
G ≅ ⟨x₁, …, x_n ∣ r⟩ with r ∈ Φ(F) be a Demushkin group on which every cup square
a ⌣ a vanishes, which for odd p is automatic. Then n is even, and a continuous automorphism
of F carries the class of r in gr_1(F) to the class of (x₁, x₂)(x₃, x₄) ⋯ (x_{n-1}, x_n) or
to the class of x₁^p (x₁, x₂)(x₃, x₄) ⋯ (x_{n-1}, x_n): after a change of basis,
r ≡ x₁^q (x₁, x₂) ⋯ (x_{n-1}, x_n) mod λ_2(F) with q = 0 or q = p.
Labute's normal form modulo λ_2 for a Demushkin relator, the nonalternating case of odd
rank. Let G ≅ ⟨x₁, …, x_n ∣ r⟩ with r ∈ Φ(F) be a Demushkin group at p = 2 on which some
cup square a ⌣ a does not vanish, with n odd. Then for every f ≥ 2 a continuous automorphism
of F carries the class of r in gr_1(F) to the class of
x₁² x₂^{2^f} (x₂, x₃) ⋯ (x_{n-1}, x_n).
Labute's normal form modulo λ_2 for a Demushkin relator, the nonalternating case of odd
rank, at level f = ∞. Let G ≅ ⟨x₁, …, x_n ∣ r⟩ with r ∈ Φ(F) be a Demushkin group at
p = 2 on which some cup square a ⌣ a does not vanish, with n odd. Then a continuous
automorphism of F carries the class of r in gr_1(F) to the class of
x₁² (x₂, x₃) ⋯ (x_{n-1}, x_n).
Labute's normal form modulo λ_2 for a Demushkin relator, the nonalternating case of even
rank. Let G ≅ ⟨x₁, …, x_n ∣ r⟩ with r ∈ Φ(F) be a Demushkin group at p = 2 on which some
cup square a ⌣ a does not vanish, with n even. Then for every a divisible by 4 and every
f ≥ 2 a continuous automorphism of F carries the class of r in gr_1(F) to the class of
x₁^{2+a} (x₁, x₂) x₃^{2^f} (x₃, x₄) ⋯ (x_{n-1}, x_n).