def TorsionQuadraticForm(q):
r"""
Create a torsion quadratic form module from a rational matrix.
The resulting quadratic form takes values in `\QQ / \ZZ`
or `\QQ / 2 \ZZ` (depending on ``q``).
If it takes values modulo `2`, then it is non-degenerate.
In any case the bilinear form is non-degenerate.
INPUT:
- ``q`` -- a symmetric rational matrix
EXAMPLES::
sage: q1 = Matrix(QQ,2,[1,1/2,1/2,1])
sage: TorsionQuadraticForm(q1)
Finite quadratic module over Integer Ring with invariants (2, 2)
Gram matrix of the quadratic form with values in Q/2Z:
[ 1 1/2]
[1/2 1]
In the following example the quadratic form is degenerate.
But the bilinear form is still non-degenerate::
sage: q2 = diagonal_matrix(QQ,[1/4,1/3])
sage: TorsionQuadraticForm(q2)
Finite quadratic module over Integer Ring with invariants (12,)
Gram matrix of the quadratic form with values in Q/Z:
[7/12]
TESTS::
sage: TorsionQuadraticForm(matrix.diagonal([3/4,3/8,3/8]))
Finite quadratic module over Integer Ring with invariants (4, 8, 8)
Gram matrix of the quadratic form with values in Q/2Z:
[3/4 0 0]
[ 0 3/8 0]
[ 0 0 3/8]
"""
q = matrix(QQ, q)
if q.nrows() != q.ncols():
raise ValueError("the input must be a square matrix")
if q != q.transpose():
raise ValueError("the input must be a symmetric matrix")
Q, d = q._clear_denom()
S, U, V = Q.smith_form()
D = U * q * V
Q = FreeQuadraticModule(ZZ, q.ncols(), inner_product_matrix=d**2 * q)
denoms = [D[i, i].denominator() for i in range(D.ncols())]
rels = Q.span(diagonal_matrix(ZZ, denoms) * U)
return TorsionQuadraticModule((1/d)*Q, (1/d)*rels, modulus=1)
def twist(self, s):
r"""
Return the torsion quadratic module with quadratic form scaled by ``s``.
If the old form was defined modulo `n`, then the new form is defined
modulo `n s`.
INPUT:
- ``s`` - a rational number
EXAMPLES::
sage: q = TorsionQuadraticForm(matrix.diagonal([3/9, 1/9]))
sage: q.twist(-1)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/Z:
[2/3 0]
[ 0 8/9]
This form is defined modulo `3`::
sage: q.twist(3)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/3Z:
[ 1 0]
[ 0 1/3]
The next form is defined modulo `4`::
sage: q.twist(4)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/4Z:
[4/3 0]
[ 0 4/9]
"""
s = self.base_ring().fraction_field()(s)
n = self.V().degree()
inner_product_matrix = s * self.V().inner_product_matrix()
ambient = FreeQuadraticModule(self.base_ring(), n,
inner_product_matrix)
V = ambient.span(self.V().basis())
W = ambient.span(self.W().basis())
return TorsionQuadraticModule(V, W)
def TorsionQuadraticForm(q):
r"""
Create a torsion quadratic form module from a rational matrix.
The resulting quadratic form takes values in `\QQ / \ZZ`
or `\QQ / 2 \ZZ` (depending on ``q``).
If it takes values modulo `2`, then it is non-degenerate.
In any case the bilinear form is non-degenerate.
INPUT:
- ``q`` -- a symmetric rational matrix
EXAMPLES::
sage: q1 = Matrix(QQ,2,[1,1/2,1/2,1])
sage: TorsionQuadraticForm(q1)
Finite quadratic module over Integer Ring with invariants (2, 2)
Gram matrix of the quadratic form with values in Q/2Z:
[ 1 1/2]
[1/2 1]
In the following example the quadratic form is degenerate.
But the bilinear form is still non-degenerate::
sage: q2 = diagonal_matrix(QQ,[1/4,1/3])
sage: TorsionQuadraticForm(q2)
Finite quadratic module over Integer Ring with invariants (12,)
Gram matrix of the quadratic form with values in Q/Z:
[7/12]
"""
q = matrix(QQ, q)
if q.nrows() != q.ncols():
raise ValueError("the input must be a square matrix")
if q != q.transpose():
raise ValueError("the input must be a symmetric matrix")
Q, d = q._clear_denom()
S, U, V = Q.smith_form()
D = U * q * V
Q = FreeQuadraticModule(ZZ, q.ncols(), inner_product_matrix=d**2 * q)
denoms = [D[i,i].denominator() for i in range(D.ncols())]
rels = Q.span(diagonal_matrix(ZZ, denoms) * U)
return TorsionQuadraticModule((1/d)*Q, (1/d)*rels)
def twist(self, s):
r"""
Return the torsion quadratic module with quadratic form scaled by ``s``.
If the old form was defined modulo `n`, then the new form is defined
modulo `n s`.
INPUT:
- ``s`` - a rational number
EXAMPLES::
sage: q = TorsionQuadraticForm(matrix.diagonal([3/9, 1/9]))
sage: q.twist(-1)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/Z:
[2/3 0]
[ 0 8/9]
This form is defined modulo `3`::
sage: q.twist(3)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/3Z:
[ 1 0]
[ 0 1/3]
The next form is defined modulo `4`::
sage: q.twist(4)
Finite quadratic module over Integer Ring with invariants (3, 9)
Gram matrix of the quadratic form with values in Q/4Z:
[4/3 0]
[ 0 4/9]
"""
s = self.base_ring().fraction_field()(s)
n = self.V().degree()
inner_product_matrix = s * self.V().inner_product_matrix()
ambient = FreeQuadraticModule(self.base_ring(), n, inner_product_matrix)
V = ambient.span(self.V().basis())
W = ambient.span(self.W().basis())
return TorsionQuadraticModule(V, W)
