Finite support of global Hasse signs #
The Hasse signs of a regular quadratic form over a number field are nontrivial at only finitely many finite places. After choosing one diagonalization, the Hasse sign at each finite place is the pairwise product of local Hilbert symbols of the diagonal coefficients, and each such symbol is trivial away from the dyadic places and the primes supporting the two coefficients.
Main result #
QuadraticForm.hasFiniteMulSupport_finiteHasse: the finite-place Hasse signs of a regular global form have finite multiplicative support.
The good-place calculation follows O'Meara, Introduction to Quadratic Forms, 66:6, using the unramified norm-equation calculation for each pair of coefficients.
theorem
QuadraticForm.hasFiniteMulSupport_finiteHasse
{K : Type u_1}
[Field K]
[NumberField K]
{V : Type u_2}
[AddCommGroup V]
[Module K V]
[FiniteDimensional K V]
(Q : QuadraticForm K V)
(hQ : QuadraticMap.Nondegenerate)
:
Almost-all triviality of the finite Hasse sign. The Hasse sign of a regular quadratic form over a number field is one at all but finitely many finite places.