Ideal equivalence in indefinite quaternion algebras

[jtcc2]

Steps To Reproduce

Define an indefinite quaternion algebra. Define two equivalent right ideals I and J. Then I.is_equivalent(J) throws an error.

B.<i,j,k> = QuaternionAlgebra(-2, 5)
O0 = B.maximal_order()
I = O0.unit_ideal()
I.is_equivalent(I)

Expected Behavior

Should return True.

Actual Behavior

Throws error PariError: domain error in minim0: form is not positive definite

Additional Information

The is_equivalent function calls I.theta_series_vector and a theta series only makes sense in a definite quaternion algebra when the norm form of the ideal is positive definite. In turn it calls Pari's qfrep which does a positive definite check, and throws the error. An alternative method to test equivalence would be to implement Minkowski reduced basis, and look for the smallest norm element of J.conjugate() * I

@S17A05

Environment

- **OS**: Windows 10 (WSL)
- **Sage Version**: 10.3.beta4

Checklist

• I have searched the existing issues for a bug report that matches the one I want to file, without success.
• I have read the documentation and troubleshoot guide