Quadratic Forms Local Representation Returns Incorrect Output

[annahaensch]

The function is_locally_represented_number_at_place returns the wrong output in at least one case. For the binary form <4,-4> we can see that it integrally represents 44, since 44=4(36)-4(25), so it certainly represents 44 locally at 2. However,

sage: L=DiagonalQuadraticForm(ZZ,[4,-4])
sage: L.is_locally_represented_number_at_place(44,2)
False

This should return True. The same error does not seem to appear at other primes,

sage: L.is_locally_represented_number_at_place(44,5)
True
sage: L.is_locally_represented_number_at_place(44,3)
True

[fli]

Duplicate of #20180?

[annahaensch]

Yes, I had totally forgotten about that other one. I think this ticket gives a better explanation though.

[sbrandhorst]

The real bug might be that quadratic form evaluation is broken. See #23977

[sbrandhorst]

I am sorry the evaluation is not broken. I just got confused with

sage: L=DiagonalQuadraticForm(ZZ,[4,-4])
sage: L.polynomial()
8*x0^2 - 8*x1^2

[sbrandhorst]

We are not restricted to p=2.

sage: L=DiagonalQuadraticForm(ZZ,[4,-4])
sage: bugs = []
....: for p in primes_first_n(200):
....:     if not L.is_locally_represented_number_at_place(44,p):
....:         bugs.append(p)
....:         
....:     
sage: bugs
[2, 11]

[sbrandhorst]

sage: ?L.local_representation_conditions

Signature:      L.local_representation_conditions(recompute_flag=False, silent_flag=False)
Docstring:     
   WARNING: THIS ONLY WORKS CORRECTLY FOR FORMS IN >=3 VARIABLES,
      WHICH ARE LOCALLY UNIVERSAL AT ALMOST ALL PRIMES!

[sbrandhorst]

sage: L.is_locally_represented_number_at_place(12,3)
False

Note that (2^2-12)*4 = 12. So the expected answer is True

[sbrandhorst]

L.local_representation_conditions(11,silent_flag=False)

This 2-dimensional form represents the p-adic integers of even
valuation for all primes p except [2].
For these and the reals, we have:
     Reals:   [0, 0]
     p = 2:   [1, 1, 1, 1, 2, 2, 2, 2]

It does not comment on the primes of odd valuation.

So my current hypothesis is that everything works as long as the valuation is even or the form has rank at least 3. We should also check if degenerate forms are handled.

[sbrandhorst]

def local_conditions_vector_for_prime(self, p):

    ...
    ...
    elif self.dim == 2:
        if p == 2:
             return [2, 0, 0, 0, 0, infinity, infinity, infinity, infinity]
        else:
            return [p, 0, 0, infinity, infinity, None, None, None, None]

This is certainly a bug.

[sbrandhorst]

It seems the theory for binary quadratic forms is different as usually. Do you know a good reference for this?

[sbrandhorst]

sage: D.is_locally_represented_number(44)
False
sage: D

Quadratic form in 3 variables over Integer Ring with coefficients: 
[ 4 0 0 ]
[ * -4 0 ]
[ * * 4 ]

Okay. Let's just say nothing is working.

[sbrandhorst]

I would recommend to deprecate and remove the function.