Rank-two dyadic blocks are u^{(2)}(2^k) or v^{(2)}(2^k)

Let A be a finite quadratic module and K ≥ 1. Suppose x, y ∈ A are killed by 2^K, their quadratic values q(x) and q(y) are killed by 2^K, and their pairing b(x, y) has exact order 2^K. Then the subgroup ⟨x, y⟩ they generate, with the restricted form, is isometric to one of Nikulin's rank-two dyadic generators u^{(2)}(2^K) or v^{(2)}(2^K). In particular it is nondegenerate, so it splits off orthogonally from A. This is the rank-two summand in Nikulin's decomposition of a nondegenerate 2-primary finite quadratic module: an element x of maximal order 2^K either generates a nondegenerate cyclic summand q_θ^{(2)}(2^K), or 2^{K-1} b(x, x) vanishes, and then an element y of 2-power order with 2^{K-1} b(x, y) ≠ 0 either generates a cyclic summand or, together with x, satisfies the hypotheses above.

Write q(x) = a / 2^K, q(y) = b / 2^K and b(x, y) = u / 2^K with a, b, u ∈ ℤ/2^K and u odd, so that

q(m x + n y) = (a m² + u m n + b n²) / 2^K.

The classification is then the normal form of the binary form a m² + u m n + b n² over ℤ/2^K, and it is decided by the parities of a and b.

• If a is even, the form has an isotropic vector (m₀, 1), and completing it to a hyperbolic pair gives u^{(2)}(2^K), where q(m, n) = mn / 2^K. The case of even b is symmetric.
• If a and b are both odd, the form represents 1 at a vector (2j, 1), and a second vector can be adjusted to value 1 and pairing 1 with it, giving v^{(2)}(2^K), where q(m, n) = (m² + mn + n²) / 2^K.

Each equation to be solved is reduced to a value of s ↦ 2cs² + us with u odd. That map is a bijection of ℤ/2^K, because (2cs² + us) - (2ct² + ut) = (s - t)(2c(s + t) + u) and 2c(s + t) + u is a unit, 2 being nilpotent in ℤ/2^K; this replaces Hensel's lemma.

For K = 1 the hypotheses say that x and y are 2-torsion with b(x, x) = b(y, y) = 0 and b(x, y) = 1/2, and the conclusion recovers, for the subgroup ⟨x, y⟩, the classification of nondegenerate alternating forms on a Klein four-group in TauCeti.LinearAlgebra.FiniteBilinearModule.Dyadic.KleinFour.

Main results

• TauCeti.FiniteQuadraticModule.nonempty_isometry_dyadicU_or_dyadicV_restrict_zmultiples_sup: the subgroup generated by such x and y is isometric to u^{(2)}(2^K) or v^{(2)}(2^K).
• TauCeti.FiniteQuadraticModule.isNondegenerate_restrict_zmultiples_sup: that subgroup is nondegenerate.

References

• V. V. Nikulin, Integral symmetric bilinear forms and some of their applications, Proposition 1.8.1.
• C. T. C. Wall, Quadratic forms on finite groups, and related topics, Topology 2 (1963), 281–298.

Transport to a finite quadratic module

The classification

theorem TauCeti.FiniteQuadraticModule.nonempty_isometry_dyadicU_or_dyadicV_restrict_zmultiples_sup {k : ℕ} {A : FiniteQuadraticModule} {x y : A.carrier} (hx : 2 ^ (k + 1) • x = 0) (hy : 2 ^ (k + 1) • y = 0) (hqx : 2 ^ (k + 1) • A.quadratic x = 0) (hqy : 2 ^ (k + 1) • A.quadratic y = 0) (hxy : 2 ^ k • (A.pairing x) y ≠ 0) : Nonempty ((dyadicU (k + 1)).Isometry (A.restrict (AddSubgroup.zmultiples x ⊔ AddSubgroup.zmultiples y))) ∨ Nonempty ((dyadicV (k + 1)).Isometry (A.restrict (AddSubgroup.zmultiples x ⊔ AddSubgroup.zmultiples y)))

Rank-two dyadic blocks. If x, y, q(x), and q(y) are all killed by 2^{k+1}, and 2^k b(x, y) ≠ 0, then the subgroup generated by x and y, with the restricted form, is isometric to u^{(2)}(2^{k+1}) or to v^{(2)}(2^{k+1}).

theorem TauCeti.FiniteQuadraticModule.isNondegenerate_restrict_zmultiples_sup {k : ℕ} {A : FiniteQuadraticModule} {x y : A.carrier} (hx : 2 ^ (k + 1) • x = 0) (hy : 2 ^ (k + 1) • y = 0) (hqx : 2 ^ (k + 1) • A.quadratic x = 0) (hqy : 2 ^ (k + 1) • A.quadratic y = 0) (hxy : 2 ^ k • (A.pairing x) y ≠ 0) : (A.restrict (AddSubgroup.zmultiples x ⊔ AddSubgroup.zmultiples y)).IsNondegenerate

Under the hypotheses of TauCeti.FiniteQuadraticModule.nonempty_isometry_dyadicU_or_dyadicV_restrict_zmultiples_sup, the subgroup generated by x and y is nondegenerate, so TauCeti.FiniteQuadraticModule.restrictProdOrthogonalComplementIsometry splits it off A.