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QuadraticForm: implement clifford_invariant and replace hasse_conductor with clifford_conductor #5411

Closed tornaria closed 15 years ago

tornaria commented 15 years ago

The hasse_invariant of a quadratic form doesn't match the standard invariant (brauer class) for quaternion algebras (e.g. for ternary quadratic forms, the ramification of the corresponding quaternion algebra).

The clifford_invariant can defined in terms of the clifford algebra of the quadratic form. See Lam (AMS GSM 67) p. 117 for the definition, and p. 119 for the formula relating it to the Hasse invariant.

It also has the property that hyperbolic spaces have clifford_invariant == +1 at all primes.

It also makes more sense to define a clifford_conductor instead of a hasse_conductor as the product of all the primes with clifford_invariant == -1.

Component: quadratic forms

Issue created by migration from https://trac.sagemath.org/ticket/5411

tornaria commented 15 years ago
comment:1

Attachment: patch.5411.clifford_invariant.gz

tornaria commented 15 years ago

same file with correct name

tornaria commented 15 years ago
comment:2

Attachment: trac_5411.clifford_invariant.patch.gz

85eec1a4-3d04-4b4d-b711-d4db03337c41 commented 15 years ago
comment:3

Positive review.

Cheers,

Michael

85eec1a4-3d04-4b4d-b711-d4db03337c41 commented 15 years ago
comment:4

Jon expressed some further wishes, so "needs review" again for now :-)

Cheers,

Michael

tornaria commented 15 years ago

replaces previous patch --- this one doesn't remove hasse_conductor

tornaria commented 15 years ago
comment:5

Attachment: trac_5411.clifford_invariant.2nd.patch.gz

The new patch (2nd patch) preserves hasse_conductor as requested by Jon, and it also fixes the imports in quadratic_form.py, so it is meant to be applied on top of #5403.

jonhanke commented 15 years ago
comment:6

Looks good. Positive review.

85eec1a4-3d04-4b4d-b711-d4db03337c41 commented 15 years ago
comment:7

Merged trac_5411.clifford_invariant.2nd.patch in Sage 3.4.rc0.

Cheers,

Michael