The relative Stiefel–Whitney formula over a local field #
Let K be a nonarchimedean local field in which 2 is invertible, for instance a finite
extension of ℚ₂, 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). Over any field in
which 2 is invertible, the relative Stiefel–Whitney formula in degree two reduces, for a
quadratic extension, to
w₂(Tr_*⟨a⟩) = w₂(Tr_*⟨1⟩) + N^{Ev}((a))
(TauCeti.sw2Class_formClass_traceTransfer_smul_sq_eq_add_galoisEvens). Over K the second
Stiefel–Whitney class is determined by the local Hasse invariant
(TauCeti.sw2Class_eq_iff_localHasse_eq), so the identity becomes a statement about signs: the
Evens norm of (a) vanishes exactly when the twisted and the untwisted trace forms have the same
local Hasse invariant. These invariants are Hilbert symbols:
s(Tr_*⟨1⟩) = (2, d)_K,
s(Tr_*⟨a⟩) = (Tr a, −d · N a)_K if Tr a ≠ 0,
s(Tr_*⟨a⟩) = 1 if Tr a = 0.
Correspondingly, when Tr a ≠ 0 the Evens norm N^{Ev}((a)) = (Tr a) ∪ (−d · N a) + (2) ∪ (d)
vanishes exactly when (Tr a, −d · N a)_K = (2, d)_K. Over a finite extension of ℚ₂ this
turns both sides of the relative formula into Hilbert-symbol computations, for every a ∈ Lˣ.
The degree-one formula w₁(Tr_*⟨a⟩) = w₁(Tr_*⟨1⟩) + cor (a) is already an identity of square
classes over every field (TauCeti.sw1Class_formClass_traceTransfer_smul_sq).
Fix c, e ∈ Kˣ with (c, e)_K = -1, so that κ = (c) ∪ (e) is a nonzero class. Every Kummer
cup over K is a multiple of κ (TauCeti.cup_kummerClass_eq_hilbertSymbol_smul), so the Evens
norm is known exactly and not only up to its vanishing:
N^{Ev}((a)) = ε((Tr a, −d · N a)_K · (2, d)_K) • κ if Tr a ≠ 0,
N^{Ev}((a)) = ε((2, d)_K) • κ if Tr a = 0,
where ε : {±1} → ZMod 2 sends +1 to 0 and -1 to 1. Together with the Hilbert-symbol
values of w₂ above, both sides of the degree-two relative formula become multiples of κ with
coefficients in ZMod 2.
Main results #
TauCeti.localHasse_formClass_traceTransfer_sq_of_sq:s(Tr_*⟨1⟩) = (2, d)_K.TauCeti.localHasse_formClass_traceTransfer_smul_sq_of_sq: forTr a ≠ 0,s(Tr_*⟨a⟩) = (Tr a, −d · N a)_K.TauCeti.localHasse_formClass_traceTransfer_smul_sq_of_trace_eq_zero: forTr a = 0,s(Tr_*⟨a⟩) = 1.TauCeti.galoisEvens_kummerClass_eq_zero_iff_localHasse_eq:N^{Ev}((a)) = 0if and only ifs(Tr_*⟨a⟩) = s(Tr_*⟨1⟩).TauCeti.galoisEvens_kummerClass_eq_zero_iff_hilbertSymbol_eq: forTr a ≠ 0,N^{Ev}((a)) = 0if and only if(Tr a, −d · N a)_K = (2, d)_K.TauCeti.galoisEvens_kummerClass_eq_hilbertSymbol_smul: forTr a ≠ 0andκ = (c) ∪ (e)with(c, e)_K = -1,N^{Ev}((a)) = ε((Tr a, −d · N a)_K · (2, d)_K) • κ.TauCeti.galoisEvens_kummerClass_eq_hilbertSymbol_smul_of_trace_eq_zero: forTr a = 0andκas above,N^{Ev}((a)) = ε((2, d)_K) • κ.
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.
- J.-P. Serre, Local Fields, GTM 67 (1979), XIV §2, Proposition 5.
The local Hasse invariant of the trace form of a quadratic extension: if L is generated
over a local field K by a square root x ∉ K of d, then s(Tr_*⟨1⟩) = (2, d)_K.
The local Hasse invariant of a twisted trace form: if L is generated over a local field
K by a square root x ∉ K of d, and a ∈ Lˣ has nonzero trace t, then
s(Tr_*⟨a⟩) = (t, −d · N a)_K.
The twisted trace form Tr_*⟨a⟩ of a quadratic extension of a local field has trivial local
Hasse invariant when Tr a = 0, since it is then a hyperbolic plane.
The relative Stiefel–Whitney formula over a local field: for a quadratic extension L of a
local field K and a ∈ Lˣ, the Evens norm N^{Ev}((a)) ∈ H²(G_K, 𝔽₂), taken along any
K-embedding σ of L into a separable closure of K, vanishes exactly when the twisted trace
form Tr_*⟨a⟩ and the trace form Tr_*⟨1⟩ have the same local Hasse invariant.
The Evens norm of a Kummer class over a local field, as Hilbert symbols: if L is
generated over a local field K by a square root x ∉ K of d, and a ∈ Lˣ has nonzero trace
t, then N^{Ev}((a)) = (t) ∪ (−d · N a) + (2) ∪ (d) vanishes exactly when
(t, −d · N a)_K = (2, d)_K.
The Evens norm of a Kummer class over a local field, as an element of H²(G_K, 𝔽₂): if
L is generated over a local field K by a square root x ∉ K of d, and a ∈ Lˣ has nonzero
trace t, then N^{Ev}((a)) = ε((t, −d · N a)_K · (2, d)_K) • ((c) ∪ (e)) for all c, e with
(c, e)_K = -1, where ε sends the signs +1, -1 to 0, 1.
The Evens norm of a trace-zero Kummer class over a local field: if L is generated over a
local field K by a square root x ∉ K of d, and a ∈ Lˣ has trace zero, then
N^{Ev}((a)) = ε((2, d)_K) • ((c) ∪ (e)) for all c, e with (c, e)_K = -1, where ε sends the
signs +1, -1 to 0, 1.