The local Hasse invariant #
Let K be a nonarchimedean local field in which 2 is invertible, in any residue
characteristic. For a regular quadratic form diagonalized as ⟨a₁, …, aₙ⟩, the local Hasse
invariant is the product ∏_{i<j} (aᵢ, aⱼ)_K ∈ {±1} of norm-equation Hilbert symbols, with the
empty product in ranks 0 and 1. It is defined on TauCeti.RegularFormClass, so it depends
only on the isometry class of the form.
That the product does not depend on the diagonalization is Witt's chain theorem, through the
descent principle TauCeti.RegularFormClass.liftDiagonal: the Hilbert symbol is symmetric, takes
equal values on isometric binary forms, and is bimultiplicative over a local field, the last
including the dyadic case (TauCeti.hilbertSymbol_mul_left).
Together with the rank and the discriminant, this sign is the third invariant in the
classification of regular quadratic forms over a nonarchimedean local field. It is the
{±1}-valued counterpart of the Brauer-valued TauCeti.RegularFormClass.hasseInvariant of
TauCeti.LinearAlgebra.QuadraticForm.RegularFormClass.Hasse.Basic, whose construction and whose
orthogonal-sum and scaling formulas are adapted here to the Hilbert symbol. O'Meara's i ≤ j
Hasse symbol differs from this i < j sign by the Hilbert symbol of the discriminant with -1.
Main definitions #
TauCeti.RegularFormClass.localHasse: the local Hasse invariant of an isometry class of regular quadratic forms.
Main results #
TauCeti.RegularFormClass.localHasse_mk: its value∏_{i<j} (aᵢ, aⱼ)_Kon a diagonal presentation⟨a₁, …, aₙ⟩.TauCeti.RegularFormClass.localHasse_congr: isometric diagonal forms have the same local Hasse invariant.TauCeti.RegularFormClass.localHasse_formClass: the same value on the class of any regular form isometric to⟨a₁, …, aₙ⟩.TauCeti.RegularFormClass.localHasse_eq_one_of_rank_eq_two_of_discr_eq_neg_one: a class of rank two and discriminant[-1]has trivial local Hasse invariant.TauCeti.RegularFormClass.localHasse_add:s(q ⊥ r) = s(q) · s(r) · (d(q), d(r))_K.TauCeti.RegularFormClass.localHasse_nsmul_hyperbolicClass:mhyperbolic planes have local Hasse invariant(-1, -1)_K^{m(m-1)/2}.TauCeti.RegularFormClass.localHasse_mk_rankOne_add:s(⟨a⟩ ⊥ q) = s(q) · (a, d(q))_K.TauCeti.RegularFormClass.localHasse_mk_rankOne_mul: the formula for scaling by a unit.
References #
- T. Y. Lam, Introduction to Quadratic Forms over Fields, Chapter V, §3.
- J.-P. Serre, A Course in Arithmetic, Chapter IV, §2.1.
- O. T. O'Meara, Introduction to Quadratic Forms, §63:20.
The local Hasse invariant of an isometry class of regular quadratic forms over a
nonarchimedean local field: for a diagonal presentation ⟨a₁, …, aₙ⟩ of the class, the product
∏_{i<j} (aᵢ, aⱼ)_K of Hilbert symbols, which is 1 in ranks 0 and 1. It does not depend on
the presentation.
Instances For
The local Hasse invariant of the class of a diagonal presentation ⟨a₁, …, aₙ⟩ is
∏_{i<j} (aᵢ, aⱼ)_K.
The local Hasse invariant descends along Quotient.mk. Isometric diagonal forms
⟨a₁, …, aₘ⟩ ≅ ⟨b₁, …, bₙ⟩ have the same local Hasse invariant; by localHasse_mk this is
∏_{i<j} (aᵢ, aⱼ)_K = ∏_{i<j} (bᵢ, bⱼ)_K.
O'Meara's i ≤ j Hasse symbol of local Hilbert symbols is the local Hasse invariant times
the symbol of the discriminant with -1.
The local Hasse invariant of a regular form isometric to ⟨a₁, …, aₙ⟩ is
∏_{i<j} (aᵢ, aⱼ)_K.
The local Hasse invariant is trivial in ranks 0 and 1.
The zero class has trivial local Hasse invariant.
The unit class ⟨1⟩ has trivial local Hasse invariant.
A rank-one diagonal form ⟨a⟩ has trivial local Hasse invariant.
The local Hasse invariant of a binary diagonal form ⟨a, b⟩ is its Hilbert symbol
(a, b)_K.
The hyperbolic class has trivial local Hasse invariant (1, -1)_K = 1.
A class of rank two and discriminant [-1] has trivial local Hasse invariant: it is the
hyperbolic class.
The orthogonal-sum formula on diagonal presentations: the local Hasse invariant of
⟨a₁, …, aₘ⟩ ⊥ ⟨b₁, …, bₙ⟩ is s⟨a⟩ · s⟨b⟩ · (∏ aᵢ, ∏ bⱼ)_K.
The orthogonal-sum formula s(q ⊥ r) = s(q) · s(r) · (d(q), d(r))_K, with the cross
term the Hilbert symbol of the two discriminants.
The local Hasse invariant of m hyperbolic planes is (-1, -1)_K^{m(m-1)/2}.
Adjoining a line: s(⟨a⟩ ⊥ q) = s(q) · (a, d(q))_K, the orthogonal-sum formula with the
trivial local Hasse invariant of the rank-one summand.
The scaling formula on diagonal presentations: scaling ⟨a₁, …, aₙ⟩ by a unit λ
multiplies its local Hasse invariant by (λ, -1)_K^{n(n-1)/2} · (λ, ∏ aᵢ)_K^{n-1}.
The scaling formula for multiplication of a diagonal presentation by the rank-one class
⟨λ⟩.
The scaling formula s(λ • q) = s(q) · (λ, -1)_K^{n(n-1)/2} · (λ, d(q))_K^{n-1} for a
class q of rank n, where λ • q is the product with the rank-one class ⟨λ⟩.