The Stiefel–Whitney classes of the trace forms of a quadratic extension #
Let K be a field in which 2 is invertible and let L = K(x) be a quadratic extension with
x ∉ K and x² = d ∈ Kˣ. For a ∈ Lˣ write Tr_*⟨a⟩ for the twisted trace form
y ↦ Tr_{L/K}(a y²) on L regarded as a K-space, and (c) ∈ H¹(G_K, 𝔽₂) for the Kummer class
of c ∈ Kˣ. This file evaluates the Stiefel–Whitney classes of these forms in degrees one and
two:
w₁(Tr_*⟨1⟩) = (d), w₂(Tr_*⟨1⟩) = (2) ∪ (d),
w₂(Tr_*⟨a⟩) = (Tr a) ∪ (−d · N a) if Tr a ≠ 0,
w₂(Tr_*⟨a⟩) = 0 if Tr a = 0.
The trace form is ⟨2, 2d⟩ (TauCeti.equivalent_traceTransfer_sq_weightedSumSquares_of_sq), and
(2) ∪ (2) = 0 because 2 = 2² − 2 · 1². If Tr a ≠ 0, Kahn's basis (1, x / a) gives
Tr_*⟨a⟩ ≅ ⟨Tr a, d · Tr a / N a⟩
(TauCeti.equivalent_traceTransfer_smul_sq_weightedSumSquares_of_sq), and
(t) ∪ (d t / N) = (t) ∪ (−d N) + (t) ∪ (−t) = (t) ∪ (−d N), because d t / N and
(−d N)(−t) differ by the square N² and (t) ∪ (−t) = 0. If Tr a = 0, the form is the
hyperbolic plane (TauCeti.equivalent_traceTransfer_smul_sq_hyperbolicPlane).
Finally, for a K-embedding σ of L into a separable closure of K, the cross term
w₁(Tr_*⟨1⟩) ∪ cor (a) vanishes: it is (d) ∪ (N a), and N a = u² − d v².
Together with the value of the Evens norm N^{Ev}((a)) on the Kummer class of a, which is
(Tr a) ∪ (−d · N a) + (2) ∪ (d), and (2) ∪ (d) at trace zero
(TauCeti.galoisEvens2_kummerClass, TauCeti.galoisEvens2_kummerClass_of_trace_eq_zero), these
give Kahn's relative Stiefel–Whitney formula in degree two,
w₂(Tr_*⟨a⟩) = w₂(Tr_*⟨1⟩) + N^{Ev}((a)) + w₁(Tr_*⟨1⟩) ∪ cor (a),
on the trace forms of an arbitrary quadratic extension L/K
(TauCeti.sw2Class_formClass_traceTransfer_smul_sq). Its degree-one companion
w₁(Tr_*⟨a⟩) = w₁(Tr_*⟨1⟩) + cor (a) holds for every finite extension
(TauCeti.sw1Class_formClass_traceTransfer_smul_sq).
Read on diagonal presentations Tr_*⟨1⟩ ≅ ⟨t₁, …, tₘ⟩ and Tr_*⟨a⟩ ≅ ⟨b₁, …, bₙ⟩, the formula
is an identity between the classes of the tuples t and b
(TauCeti.sw2_eq_add_galoisEvens_add_cup_of_equivalent_traceTransfer), the shape in which it is
applied to an explicit diagonalization over a fixed base field. Its degree-one companion is
TauCeti.sw1_eq_add_galoisCor_of_equivalent_traceTransfer.
Main results #
TauCeti.sw1Class_formClass_traceTransfer_sq_of_sq:w₁(Tr_*⟨1⟩) = (d).TauCeti.sw2Class_formClass_traceTransfer_sq_of_sq:w₂(Tr_*⟨1⟩) = (2) ∪ (d).TauCeti.sw2Class_formClass_traceTransfer_smul_sq_of_sq: forTr a ≠ 0,w₂(Tr_*⟨a⟩) = (Tr a) ∪ (−d · N a).TauCeti.sw2Class_formClass_traceTransfer_smul_sq_of_trace_eq_zero: forTr a = 0,w₂(Tr_*⟨a⟩) = 0.TauCeti.cup_sw1Class_formClass_traceTransfer_sq_galoisCor_eq_zero:w₁(Tr_*⟨1⟩) ∪ cor (a) = 0.TauCeti.sw2Class_formClass_traceTransfer_smul_sq_eq_add_galoisEvens:w₂(Tr_*⟨a⟩) = w₂(Tr_*⟨1⟩) + N^{Ev}((a)).TauCeti.sw2Class_formClass_traceTransfer_smul_sq: the relative Stiefel–Whitney formula in degree two,w₂(Tr_*⟨a⟩) = w₂(Tr_*⟨1⟩) + N^{Ev}((a)) + w₁(Tr_*⟨1⟩) ∪ cor (a).TauCeti.sw2_eq_add_galoisEvens_add_cup_of_equivalent_traceTransfer: the same formula on diagonal presentations of the trace forms,w₂(b) = w₂(t) + N^{Ev}((a)) + w₁(t) ∪ cor (a).
References #
- B. Kahn, Classes de Stiefel-Whitney de formes quadratiques et de représentations galoisiennes réelles, Invent. Math. 78 (1984), 223–256, Lemme II.2.1 and Théorème 2.
- J.-P. Serre, L'invariant de Witt de la forme Tr(x²), Comment. Math. Helv. 59 (1984), 651–676.
The first Stiefel–Whitney class of the trace form of a quadratic algebra: if L is
generated over K by a square root x ∉ K of d, then w₁(Tr_*⟨1⟩) = (d).
The second Stiefel–Whitney class of the trace form of a quadratic algebra: if L is
generated over K by a square root x ∉ K of d, then w₂(Tr_*⟨1⟩) = (2) ∪ (d). Indeed
Tr_*⟨1⟩ ≅ ⟨2, 2d⟩, and (2) ∪ (2) = 0 because 2 = 2² − 2 · 1².
The second Stiefel–Whitney class of a twisted trace form, by Kahn's basis: let L be
generated over K by a square root x ∉ K of d, and let a ∈ Lˣ have nonzero trace t. Then
w₂(Tr_*⟨a⟩) = (t) ∪ (−d · N a).
The second Stiefel–Whitney class of a twisted trace form at trace zero: if L/K is a
quadratic extension and a ∈ Lˣ has trace zero, then Tr_*⟨a⟩ is the hyperbolic plane, so
w₂(Tr_*⟨a⟩) = 0.
The cross term of the relative Stiefel–Whitney formula vanishes for a quadratic
extension: if L is generated over K by a square root x ∉ K of d, as every quadratic
extension is, then w₁(Tr_*⟨1⟩) ∪ cor (a) = (d) ∪ (N a) = 0 for every a ∈ Lˣ, with
corestriction taken along any K-embedding σ of L into a separable closure of K.
The Evens norm is the change of w₂ under twisting the trace form: for a quadratic
extension L/K and a ∈ Lˣ,
w₂(Tr_*⟨a⟩) = w₂(Tr_*⟨1⟩) + N^{Ev}((a)),
where the Evens norm is taken along any K-embedding σ of L into a separable closure of K.
This is the relative formula TauCeti.sw2Class_formClass_traceTransfer_smul_sq without its
cross term w₁(Tr_*⟨1⟩) ∪ cor (a), which vanishes for a quadratic extension
(TauCeti.cup_sw1Class_formClass_traceTransfer_sq_galoisCor_eq_zero).
The relative Stiefel–Whitney formula in degree two, on the trace forms themselves (Kahn,
Théorème 2 in degree two): for a quadratic extension L/K, which is separable as 2 is
invertible, and a ∈ Lˣ with Kummer class (a) ∈ H¹(G_L, 𝔽₂),
w₂(Tr_*⟨a⟩) = w₂(Tr_*⟨1⟩) + N^{Ev}((a)) + w₁(Tr_*⟨1⟩) ∪ cor (a),
where the Evens norm N^{Ev} and the corestriction cor are taken along any K-embedding σ
of L into a separable closure of K. The two sides are the classes of the regular forms
y ↦ Tr_{L/K}(a y²) and y ↦ Tr_{L/K}(y²) on L; their regularity is a hypothesis.
The relative Stiefel–Whitney formula in degree two, on diagonal presentations of the trace
forms: for a quadratic extension L/K and a ∈ Lˣ, if Tr_*⟨1⟩ ≅ ⟨t₁, …, tₘ⟩ and
Tr_*⟨a⟩ ≅ ⟨b₁, …, bₙ⟩, then w₂(b) = w₂(t) + N^{Ev}((a)) + w₁(t) ∪ cor (a), with the Evens
norm and the corestriction taken along any K-embedding σ of L into a separable closure of
K. No regularity hypothesis is needed: isometry with a diagonal form with unit coefficients
already makes both trace forms regular.