Ticket #5411 (closed enhancement: fixed)

Opened 4 years ago

Last modified 4 years ago

[with patch, positive review] QuadraticForm: implement clifford_invariant and replace hasse_conductor with clifford_conductor

Reported by: tornaria Owned by: tornaria
Priority: major Milestone: sage-3.4
Component: quadratic forms Keywords:
Cc: Work issues:
Report Upstream: Reviewers:
Authors: Merged in:
Dependencies: Stopgaps:

Description

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.

Attachments

patch.5411.clifford_invariant Download (3.7 KB) - added by tornaria 4 years ago.
trac_5411.clifford_invariant.patch Download (3.7 KB) - added by tornaria 4 years ago.
same file with correct name
trac_5411.clifford_invariant.2nd.patch Download (4.8 KB) - added by tornaria 4 years ago.
replaces previous patch --- this one doesn't remove hasse_conductor

Change History

Changed 4 years ago by tornaria

comment:1 Changed 4 years ago by tornaria

  • Summary changed from QuadraticForm: implement clifford_invariant and replace hasse_conductor with clifford_conductor to [with patch, needs review] QuadraticForm: implement clifford_invariant and replace hasse_conductor with clifford_conductor

Changed 4 years ago by tornaria

same file with correct name

comment:2 Changed 4 years ago by tornaria

  • Owner changed from gonzalo to tornaria
  • Status changed from new to assigned

comment:3 Changed 4 years ago by mabshoff

  • Summary changed from [with patch, needs review] QuadraticForm: implement clifford_invariant and replace hasse_conductor with clifford_conductor to [with patch, positive review] QuadraticForm: implement clifford_invariant and replace hasse_conductor with clifford_conductor

Positive review.

Cheers,

Michael

comment:4 Changed 4 years ago by mabshoff

  • Summary changed from [with patch, positive review] QuadraticForm: implement clifford_invariant and replace hasse_conductor with clifford_conductor to [with patch, needs review] QuadraticForm: implement clifford_invariant and replace hasse_conductor with clifford_conductor

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

Cheers,

Michael

Changed 4 years ago by tornaria

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

comment:5 Changed 4 years ago by tornaria

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.

comment:6 Changed 4 years ago by jonhanke

  • Summary changed from [with patch, needs review] QuadraticForm: implement clifford_invariant and replace hasse_conductor with clifford_conductor to [with patch, positive review] QuadraticForm: implement clifford_invariant and replace hasse_conductor with clifford_conductor

Looks good. Positive review.

comment:7 Changed 4 years ago by mabshoff

  • Status changed from assigned to closed
  • Resolution set to fixed

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

Cheers,

Michael

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