The cup squares of a Demushkin group at p = 2 vanish exactly when q ≠ 2

Let G be a Demushkin group at p = 2. The cup square of a class of H¹(G, 𝔽₂) vanishes exactly when the corresponding character G → 𝔽₂ lifts to a continuous character G → ℤ/4 (TauCeti.cupFp_self_eq_zero_iff_exists_zmodFourReductionClass_eq, the Bockstein description of the cup square). Characters with values in 𝔽₂ or ℤ/4 factor through the topological abelianization G^{ab} ≅ ℤ_2^{n-1} × ℤ_2 ⧸ (q), where q = q(G) is Labute's invariant, so the question becomes one about this abelian pro-2 group. If q ≠ 2, then 4 ∣ q, including q = 0, and every character lifts, so every cup square vanishes: the cup form is alternating. If q = 2, the projection onto the torsion factor ℤ_2 ⧸ (2) = 𝔽₂ does not lift, because an element of order two cannot map to an odd element of ℤ/4, so some cup square is nonzero: the cup form is symmetric but not alternating.

This is the invariant-theoretic content of the trichotomy in Labute's classification: the cup form of a Demushkin group is alternating exactly when q ≠ 2, at every prime, since for odd p every cup square vanishes by graded commutativity. It decides which of Labute's normal forms the relator of G can be brought to: the alternating form x₁^q (x₁, x₂) ⋯ (x_{n-1}, x_n) when q ≠ 2, and the dyadic forms with a square x₁² when q = 2. In particular a Demushkin group with q ≠ 2 has even rank, at every prime, and one of odd rank has q = 2.

Main results

• TauCeti.IsDemushkin.forall_exists_zmodFourReduction_eq_iff_demushkinQ_ne_two: every continuous character G → 𝔽₂ of a Demushkin group at p = 2 lifts to ℤ/4 exactly when q(G) ≠ 2.
• TauCeti.IsDemushkin.forall_cupFp_self_eq_zero_iff_demushkinQ_ne_two: every cup square on H¹(G, 𝔽₂) vanishes exactly when q(G) ≠ 2.
• TauCeti.IsDemushkin.exists_cupFp_self_ne_zero_iff_demushkinQ_eq_two: some cup square is nonzero exactly when q(G) = 2.
• TauCeti.IsDemushkin.even_demushkinRank_of_demushkinQ_ne_two, TauCeti.IsDemushkin.demushkinQ_eq_two_of_odd_demushkinRank: at every prime, q(G) ≠ 2 forces the rank to be even, and an odd rank forces q(G) = 2.

References

• J. P. Labute, Classification of Demushkin groups, Canad. J. Math. 19 (1967), 106–132, §1 and §3.
• J.-P. Serre, Galois Cohomology, Springer (1997), Chapter I, §4.5.
• J. Neukirch, A. Schmidt, K. Wingberg, Cohomology of Number Fields, 2nd ed., Springer (2008), Chapter III, §9.

Demushkin groups at p = 2

theorem TauCeti.IsDemushkin.forall_exists_zmodFourReduction_eq_iff_demushkinQ_ne_two {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin 2 G) : (∀ (χ : continuousZModDual 2 G), ∃ (φ : G →ₜ* Multiplicative (ZMod 4)), φ.zmodFourReduction = χ) ↔ demushkinQ hG ≠ 2

Every continuous character G → 𝔽₂ of a Demushkin group at p = 2 lifts to ℤ/4 exactly when q(G) ≠ 2. Through G^{ab} ≅ ℤ_2^{n-1} × ℤ_2 ⧸ (q): if q ≠ 2 then 4 ∣ q and every character lifts coordinatewise, while if q = 2 the projection onto the torsion factor ℤ_2 ⧸ (2) = 𝔽₂ does not lift.

theorem TauCeti.IsDemushkin.forall_cupFp_self_eq_zero_iff_demushkinQ_ne_two {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin 2 G) : (∀ (a : ↑(cohomFp 2 G 1).toModuleCat), ((cupFp 2 G) a) a = 0) ↔ demushkinQ hG ≠ 2

The cup form of a Demushkin group at p = 2 is alternating exactly when q(G) ≠ 2: every cup square on H¹(G, 𝔽₂) vanishes if and only if q(G) ≠ 2.

theorem TauCeti.IsDemushkin.exists_cupFp_self_ne_zero_iff_demushkinQ_eq_two {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] (hG : IsDemushkin 2 G) : (∃ (a : ↑(cohomFp 2 G 1).toModuleCat), ((cupFp 2 G) a) a ≠ 0) ↔ demushkinQ hG = 2

The cup form of a Demushkin group at p = 2 is not alternating exactly when q(G) = 2: some cup square on H¹(G, 𝔽₂) is nonzero if and only if q(G) = 2.

Every prime

theorem TauCeti.IsDemushkin.even_demushkinRank_of_demushkinQ_ne_two {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {p : ℕ} [Fact (Nat.Prime p)] (hG : IsDemushkin p G) (hq : demushkinQ hG ≠ 2) : Even (demushkinRank hG)

A Demushkin group with q(G) ≠ 2 has even rank, at every prime p: its cup form is then a nondegenerate alternating form on H¹(G, 𝔽_p). At an odd prime every Demushkin group has even rank (TauCeti.IsDemushkin.even_demushkinRank_of_ne_two); at p = 2 the cup squares vanish exactly when q(G) ≠ 2.

theorem TauCeti.IsDemushkin.demushkinQ_eq_two_of_odd_demushkinRank {G : Type u} [Group G] [TopologicalSpace G] [IsTopologicalGroup G] [CompactSpace G] [TotallyDisconnectedSpace G] {p : ℕ} [Fact (Nat.Prime p)] (hG : IsDemushkin p G) (hn : Odd (demushkinRank hG)) : demushkinQ hG = 2

A Demushkin group of odd rank has q(G) = 2, at every prime p; at an odd prime there is no Demushkin group of odd rank.