The second Stiefel–Whitney class and the local Hasse invariant

Let K be a nonarchimedean local field in which 2 is invertible, for instance a finite extension of ℚ₂. The second Stiefel–Whitney class of a regular form over K is determined by its local Hasse invariant ∏_{i<j} (aᵢ, aⱼ)_K. So every identity between second Stiefel–Whitney classes over K is an identity between signs, which can be evaluated with the Hilbert symbol.

Over a general field w₂ corresponds to the Brauer-valued Hasse invariant, a sum of quaternion classes. Over a local field a quaternion algebra is determined by its Hilbert symbol, so this Brauer-valued invariant carries exactly the information of the local sign-valued one.

Main results

• TauCeti.sw2Class_eq_iff_localHasse_eq: two regular form classes have the same w₂ exactly when their local Hasse invariants agree.
• TauCeti.sw2Class_eq_cup_kummerClass_iff_localHasse_eq: w₂(q) = (a) ∪ (b) if and only if the local Hasse invariant of q is (a, b)_K.

References

• J.-P. Serre, Local Fields, GTM 67 (1979), XIV §2, Proposition 5.
• T. Y. Lam, Introduction to Quadratic Forms over Fields, GSM 67 (2005), Chapter V, §3 and Chapter VI, §2.

theorem TauCeti.sw2Class_eq_iff_localHasse_eq {K : Type u} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] [Invertible 2] (q r : RegularFormClass K) : sw2Class q = sw2Class r ↔ q.localHasse = r.localHasse

The second Stiefel–Whitney class over a local field is the local Hasse invariant: two regular form classes over K have the same w₂ exactly when their local Hasse invariants agree. No equality of ranks or discriminants is required.

theorem TauCeti.sw2Class_eq_zero_iff_localHasse_eq_one {K : Type u} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] [Invertible 2] (q : RegularFormClass K) : sw2Class q = 0 ↔ q.localHasse = 1

The second Stiefel–Whitney class of a regular form class over a local field vanishes exactly when its local Hasse invariant is 1.

theorem TauCeti.sw2Class_eq_cup_kummerClass_iff_localHasse_eq {K : Type u} [Field K] [ValuativeRel K] [TopologicalSpace K] [IsNonarchimedeanLocalField K] [Invertible 2] (q : RegularFormClass K) (a b : Kˣ) : sw2Class q = (((trivialF2TopPairing (AbsoluteGaloisGroup K)).cup 1 1) (kummerClass a)) (kummerClass b) ↔ q.localHasse = hilbertSymbol a b

The second Stiefel–Whitney class of a regular form class over a local field is the Kummer cup (a) ∪ (b) exactly when its local Hasse invariant is the Hilbert symbol (a, b)_K.