Finite $\mathbf{Z}$-modules with bilinear and quadratic forms

AUTHORS:

• Simon Brandhorst (2017-09): First created

sage.modules.torsion_quadratic_module.TorsionQuadraticForm(q)

Create a torsion quadratic form module from a rational matrix.

The resulting quadratic form takes values in $\mathbf{Q}/\mathbf{Z}$ or $\mathbf{Q}/2\mathbf{Z}$ (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

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]

Python

>>> from sage.all import *
>>> q1 = Matrix(QQ, Integer(2), [Integer(1),Integer(1)/Integer(2),Integer(1)/Integer(2),Integer(1)])
>>> 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

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]

Python

>>> from sage.all import *
>>> q2 = diagonal_matrix(QQ, [Integer(1)/Integer(4),Integer(1)/Integer(3)])
>>> TorsionQuadraticForm(q2)
Finite quadratic module over Integer Ring with invariants (12,)
Gram matrix of the quadratic form with values in Q/Z:
[7/12]

class sage.modules.torsion_quadratic_module.TorsionQuadraticModule(V, W, gens, modulus, modulus_qf)

Bases: FGP_Module_class, CachedRepresentation

Finite quotients with a bilinear and a quadratic form.

Let $V$ be a symmetric FreeQuadraticModule and $W\subseteq V$ a submodule of the same rank as $V$. The quotient $V/W$ is a torsion quadratic module. It inherits a bilinear form $b$ and a quadratic form $q$.

$b:V\times V\to \mathbf{Q}/m\mathbf{Z}$, where $m\mathbf{Z}=(V,W)$ and $b(x,y)=(x,y)+m\mathbf{Z}$

$q:V\to \mathbf{Q}/n\mathbf{Z}$, where $n\mathbf{Z}=2(V,W)+\mathbf{Z}\{(w,w)|w\in W\}$

INPUT:

• V – a FreeModule with a symmetric inner product matrix

• W – a submodule of V of the same rank as V

• check – boolean (default: True)

• modulus – a rational number dividing $m$ (default: $m$); the inner product $b$ is defined in $\mathbf{Q}/$ modulus $\mathbf{Z}$

• modulus_qf – a rational number dividing $n$ (default: $n$); the quadratic form $q$ is defined in $\mathbf{Q}/$ modulus_qf $\mathbf{Z}$

EXAMPLES:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeModule(ZZ, 3)
sage: T = TorsionQuadraticModule(V, 5*V); T
Finite quadratic module over Integer Ring with invariants (5, 5, 5)
Gram matrix of the quadratic form with values in Q/5Z:
[1 0 0]
[0 1 0]
[0 0 1]

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> V = FreeModule(ZZ, Integer(3))
>>> T = TorsionQuadraticModule(V, Integer(5)*V); T
Finite quadratic module over Integer Ring with invariants (5, 5, 5)
Gram matrix of the quadratic form with values in Q/5Z:
[1 0 0]
[0 1 0]
[0 0 1]

Element

alias of TorsionQuadraticModuleElement

all_submodules()

Return a list of all submodules of self.

Warning

This method creates all submodules in memory. The number of submodules grows rapidly with the number of generators. For example consider a vector space of dimension $n$ over a finite field of prime order $p$. The number of subspaces is (very) roughly $p^{(n^2-n)/2}$.

EXAMPLES:

Sage

sage: D = IntegralLattice("D4").discriminant_group()                        # needs sage.combinat
sage: D.all_submodules()                                                    # needs sage.combinat
[Finite quadratic module over Integer Ring with invariants ()
  Gram matrix of the quadratic form with values in Q/2Z:
  [],
 Finite quadratic module over Integer Ring with invariants (2,)
  Gram matrix of the quadratic form with values in Q/2Z:
  [1],
 Finite quadratic module over Integer Ring with invariants (2,)
  Gram matrix of the quadratic form with values in Q/2Z:
  [1],
 Finite quadratic module over Integer Ring with invariants (2,)
  Gram matrix of the quadratic form with values in Q/2Z:
  [1],
 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]]

Python

>>> from sage.all import *
>>> D = IntegralLattice("D4").discriminant_group()                        # needs sage.combinat
>>> D.all_submodules()                                                    # needs sage.combinat
[Finite quadratic module over Integer Ring with invariants ()
  Gram matrix of the quadratic form with values in Q/2Z:
  [],
 Finite quadratic module over Integer Ring with invariants (2,)
  Gram matrix of the quadratic form with values in Q/2Z:
  [1],
 Finite quadratic module over Integer Ring with invariants (2,)
  Gram matrix of the quadratic form with values in Q/2Z:
  [1],
 Finite quadratic module over Integer Ring with invariants (2,)
  Gram matrix of the quadratic form with values in Q/2Z:
  [1],
 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]]

brown_invariant()

Return the Brown invariant of this torsion quadratic form.

Let $(D,q)$ be a torsion quadratic module with values in $\mathbf{Q}/2\mathbf{Z}$. The Brown invariant $Br(D,q)\in \mathbf{Z}/8\mathbf{Z}$ is defined by the equation

$$\exp (\frac{2\pi i}{8}Br(q))=\frac{1}{\sqrt{D}}\sum _{x\in D}\exp (i\pi q(x)).$$

The Brown invariant is additive with respect to direct sums of torsion quadratic modules.

OUTPUT: an element of $\mathbf{Z}/8\mathbf{Z}$

EXAMPLES:

Sage

sage: L = IntegralLattice("D4")                                             # needs sage.combinat
sage: D = L.discriminant_group()                                            # needs sage.combinat
sage: D.brown_invariant()                                                   # needs sage.combinat
4

Python

>>> from sage.all import *
>>> L = IntegralLattice("D4")                                             # needs sage.combinat
>>> D = L.discriminant_group()                                            # needs sage.combinat
>>> D.brown_invariant()                                                   # needs sage.combinat
4

We require the quadratic form to be defined modulo $2\mathbf{Z}$:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeQuadraticModule(ZZ, 3, matrix.identity(3))
sage: T = TorsionQuadraticModule((1/10)*V, V)
sage: T.brown_invariant()
Traceback (most recent call last):
...
ValueError: the torsion quadratic form must have values in QQ / 2 ZZ

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> V = FreeQuadraticModule(ZZ, Integer(3), matrix.identity(Integer(3)))
>>> T = TorsionQuadraticModule((Integer(1)/Integer(10))*V, V)
>>> T.brown_invariant()
Traceback (most recent call last):
...
ValueError: the torsion quadratic form must have values in QQ / 2 ZZ

gens()

Return generators of self.

There is no assumption on the generators except that they generate the module.

EXAMPLES:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeModule(ZZ, 3)
sage: T = TorsionQuadraticModule(V, 5*V)
sage: T.gens()
((1, 0, 0), (0, 1, 0), (0, 0, 1))

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> V = FreeModule(ZZ, Integer(3))
>>> T = TorsionQuadraticModule(V, Integer(5)*V)
>>> T.gens()
((1, 0, 0), (0, 1, 0), (0, 0, 1))

genus(signature_pair)

Return the genus defined by self and the signature_pair.

If no such genus exists, raise a ValueError.

REFERENCES:

[Nik1977] Corollary 1.9.4 and 1.16.3.

EXAMPLES:

Sage

sage: # needs sage.combinat
sage: L = IntegralLattice("D4").direct_sum(IntegralLattice("A2"))
sage: D = L.discriminant_group()
sage: genus = D.genus(L.signature_pair())                                   # needs sage.libs.pari
sage: genus                                                                 # needs sage.libs.pari
Genus of
None
Signature:  (6, 0)
Genus symbol at 2:    1^4:2^-2
Genus symbol at 3:     1^-5 3^-1
sage: genus == L.genus()                                                    # needs sage.libs.pari
True

Python

>>> from sage.all import *
>>> # needs sage.combinat
>>> L = IntegralLattice("D4").direct_sum(IntegralLattice("A2"))
>>> D = L.discriminant_group()
>>> genus = D.genus(L.signature_pair())                                   # needs sage.libs.pari
>>> genus                                                                 # needs sage.libs.pari
Genus of
None
Signature:  (6, 0)
Genus symbol at 2:    1^4:2^-2
Genus symbol at 3:     1^-5 3^-1
>>> genus == L.genus()                                                    # needs sage.libs.pari
True

Let $H$ be an even unimodular lattice of signature $(9,1)$. Then $L=D_4+A_2$ is primitively embedded in $H$. We compute the discriminant form of the orthogonal complement of $L$ in $H$:

Sage

sage: DK = D.twist(-1)                                                      # needs sage.combinat sage.libs.pari
sage: DK                                                                    # needs sage.combinat sage.libs.pari
Finite quadratic module over Integer Ring with invariants (2, 6)
Gram matrix of the quadratic form with values in Q/2Z:
[  1 1/2]
[1/2 1/3]

Python

>>> from sage.all import *
>>> DK = D.twist(-Integer(1))                                                      # needs sage.combinat sage.libs.pari
>>> DK                                                                    # needs sage.combinat sage.libs.pari
Finite quadratic module over Integer Ring with invariants (2, 6)
Gram matrix of the quadratic form with values in Q/2Z:
[  1 1/2]
[1/2 1/3]

We know that $K$ has signature $(5,1)$ and thus we can compute the genus of $K$ as:

Sage

sage: DK.genus((3,1))                                                       # needs sage.combinat sage.libs.pari
Genus of
None
Signature:  (3, 1)
Genus symbol at 2:    1^2:2^-2
Genus symbol at 3:     1^-3 3^1

Python

>>> from sage.all import *
>>> DK.genus((Integer(3),Integer(1)))                                                       # needs sage.combinat sage.libs.pari
Genus of
None
Signature:  (3, 1)
Genus symbol at 2:    1^2:2^-2
Genus symbol at 3:     1^-3 3^1

We can also compute the genus of an odd lattice from its discriminant form:

Sage

sage: L = IntegralLattice(matrix.diagonal(range(1, 5)))
sage: D = L.discriminant_group()
sage: D.genus((4,0))                                                        # needs sage.libs.pari
Genus of
None
Signature:  (4, 0)
Genus symbol at 2:    [1^-2 2^1 4^1]_6
Genus symbol at 3:     1^-3 3^1

Python

>>> from sage.all import *
>>> L = IntegralLattice(matrix.diagonal(range(Integer(1), Integer(5))))
>>> D = L.discriminant_group()
>>> D.genus((Integer(4),Integer(0)))                                                        # needs sage.libs.pari
Genus of
None
Signature:  (4, 0)
Genus symbol at 2:    [1^-2 2^1 4^1]_6
Genus symbol at 3:     1^-3 3^1

gram_matrix_bilinear()

Return the Gram matrix with respect to the generators.

OUTPUT:

A rational matrix G with G[i,j] given by the inner product of the $i$-th and $j$-th generator. Its entries are only well defined $\mathrm{mod}(V,W)$.

EXAMPLES:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeQuadraticModule(ZZ, 3, matrix.identity(3)*5)
sage: T = TorsionQuadraticModule((1/5)*V, V)
sage: T.gram_matrix_bilinear()
[1/5   0   0]
[  0 1/5   0]
[  0   0 1/5]

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> V = FreeQuadraticModule(ZZ, Integer(3), matrix.identity(Integer(3))*Integer(5))
>>> T = TorsionQuadraticModule((Integer(1)/Integer(5))*V, V)
>>> T.gram_matrix_bilinear()
[1/5   0   0]
[  0 1/5   0]
[  0   0 1/5]

gram_matrix_quadratic()

The Gram matrix of the quadratic form with respect to the generators.

OUTPUT:

• a rational matrix Gq with Gq[i,j] = gens[i]*gens[j] and G[i,i] = gens[i].q()

EXAMPLES:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: D4_gram = Matrix(ZZ, [[2,0,0,-1],[0,2,0,-1],[0,0,2,-1],[-1,-1,-1,2]])
sage: D4 = FreeQuadraticModule(ZZ, 4, D4_gram)
sage: D4dual = D4.span(D4_gram.inverse())
sage: discrForm = TorsionQuadraticModule(D4dual, D4)
sage: discrForm.gram_matrix_quadratic()
[  1 1/2]
[1/2   1]
sage: discrForm.gram_matrix_bilinear()
[  0 1/2]
[1/2   0]

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> D4_gram = Matrix(ZZ, [[Integer(2),Integer(0),Integer(0),-Integer(1)],[Integer(0),Integer(2),Integer(0),-Integer(1)],[Integer(0),Integer(0),Integer(2),-Integer(1)],[-Integer(1),-Integer(1),-Integer(1),Integer(2)]])
>>> D4 = FreeQuadraticModule(ZZ, Integer(4), D4_gram)
>>> D4dual = D4.span(D4_gram.inverse())
>>> discrForm = TorsionQuadraticModule(D4dual, D4)
>>> discrForm.gram_matrix_quadratic()
[  1 1/2]
[1/2   1]
>>> discrForm.gram_matrix_bilinear()
[  0 1/2]
[1/2   0]

is_genus(signature_pair, even=True)

Return True if there is a lattice with this signature and discriminant form.

Todo

implement the same for odd lattices

INPUT:

• signature_pair – tuple of nonnegative integers (s_plus, s_minus)

• even – boolean (default: True)

EXAMPLES:

Sage

sage: L3 = IntegralLattice(3 * Matrix(ZZ, 2, [2,1,1,2]))
sage: L = IntegralLattice("D4").direct_sum(L3)                              # needs sage.combinat
sage: D = L.discriminant_group()                                            # needs sage.combinat
sage: D.is_genus((6,0))                                                     # needs sage.combinat
True

Python

>>> from sage.all import *
>>> L3 = IntegralLattice(Integer(3) * Matrix(ZZ, Integer(2), [Integer(2),Integer(1),Integer(1),Integer(2)]))
>>> L = IntegralLattice("D4").direct_sum(L3)                              # needs sage.combinat
>>> D = L.discriminant_group()                                            # needs sage.combinat
>>> D.is_genus((Integer(6),Integer(0)))                                                     # needs sage.combinat
True

Let us see if there is a lattice in the genus defined by the same discriminant form but with a different signature:

Sage

sage: D.is_genus((4,2))                                                     # needs sage.combinat
False
sage: D.is_genus((16,2))                                                    # needs sage.combinat
True

Python

>>> from sage.all import *
>>> D.is_genus((Integer(4),Integer(2)))                                                     # needs sage.combinat
False
>>> D.is_genus((Integer(16),Integer(2)))                                                    # needs sage.combinat
True

normal_form(partial=False)

Return the normal form of this torsion quadratic module.

Two torsion quadratic modules are isomorphic if and only if they have the same value modules and the same normal form.

A torsion quadratic module $(T,q)$ with values in $\mathbf{Q}/n\mathbf{Z}$ is in normal form if the rescaled quadratic module $(T,q/n)$ with values in $\mathbf{Q}/\mathbf{Z}$ is in normal form.

For the definition of normal form see [MirMor2009] IV Definition 4.6. Below are some of its properties. Let $p$ be odd and $u$ be the smallest non-square modulo $p$. The normal form is a diagonal matrix with diagonal entries either $p^n$ or $u{p}^n$.

If $p=2$ is even, then the normal form consists of $1\times 1$ blocks of the form

$$(0),2^n(1),2^n(3),2^n(5),2^n(7)$$

or of $2\times 2$ blocks of the form

$$\begin{array}{r}{2}^n\left(\begin{array}{cc}2& 1\\ 1& 2\end{array}\right),2^n\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right).\end{array}$$

The blocks are ordered by their valuation.

INPUT:

• partial – boolean (default: False); return only a partial normal form. It is not unique but still useful to extract invariants.

OUTPUT: a torsion quadratic module

EXAMPLES:

Sage

sage: L1 = IntegralLattice(matrix([[-2,0,0], [0,1,0], [0,0,4]]))
sage: L1.discriminant_group().normal_form()                                 # needs sage.rings.padics
Finite quadratic module over Integer Ring with invariants (2, 4)
Gram matrix of the quadratic form with values in Q/Z:
[1/2   0]
[  0 1/4]
sage: L2 = IntegralLattice(matrix([[-2,0,0], [0,1,0], [0,0,-4]]))
sage: L2.discriminant_group().normal_form()                                 # needs sage.rings.padics
Finite quadratic module over Integer Ring with invariants (2, 4)
Gram matrix of the quadratic form with values in Q/Z:
[1/2   0]
[  0 1/4]

Python

>>> from sage.all import *
>>> L1 = IntegralLattice(matrix([[-Integer(2),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),Integer(4)]]))
>>> L1.discriminant_group().normal_form()                                 # needs sage.rings.padics
Finite quadratic module over Integer Ring with invariants (2, 4)
Gram matrix of the quadratic form with values in Q/Z:
[1/2   0]
[  0 1/4]
>>> L2 = IntegralLattice(matrix([[-Integer(2),Integer(0),Integer(0)], [Integer(0),Integer(1),Integer(0)], [Integer(0),Integer(0),-Integer(4)]]))
>>> L2.discriminant_group().normal_form()                                 # needs sage.rings.padics
Finite quadratic module over Integer Ring with invariants (2, 4)
Gram matrix of the quadratic form with values in Q/Z:
[1/2   0]
[  0 1/4]

We check that Issue #24864 is fixed:

Sage

sage: L1 = IntegralLattice(matrix([[-4,0,0], [0,4,0], [0,0,-2]]))
sage: AL1 = L1.discriminant_group()
sage: L2 = IntegralLattice(matrix([[-4,0,0], [0,-4,0], [0,0,2]]))
sage: AL2 = L2.discriminant_group()
sage: AL1.normal_form()                                                     # needs sage.rings.padics
Finite quadratic module over Integer Ring with invariants (2, 4, 4)
Gram matrix of the quadratic form with values in Q/2Z:
[1/2   0   0]
[  0 1/4   0]
[  0   0 5/4]
sage: AL2.normal_form()                                                     # needs sage.libs.pari sage.rings.padics
Finite quadratic module over Integer Ring with invariants (2, 4, 4)
Gram matrix of the quadratic form with values in Q/2Z:
[1/2   0   0]
[  0 1/4   0]
[  0   0 5/4]

Python

>>> from sage.all import *
>>> L1 = IntegralLattice(matrix([[-Integer(4),Integer(0),Integer(0)], [Integer(0),Integer(4),Integer(0)], [Integer(0),Integer(0),-Integer(2)]]))
>>> AL1 = L1.discriminant_group()
>>> L2 = IntegralLattice(matrix([[-Integer(4),Integer(0),Integer(0)], [Integer(0),-Integer(4),Integer(0)], [Integer(0),Integer(0),Integer(2)]]))
>>> AL2 = L2.discriminant_group()
>>> AL1.normal_form()                                                     # needs sage.rings.padics
Finite quadratic module over Integer Ring with invariants (2, 4, 4)
Gram matrix of the quadratic form with values in Q/2Z:
[1/2   0   0]
[  0 1/4   0]
[  0   0 5/4]
>>> AL2.normal_form()                                                     # needs sage.libs.pari sage.rings.padics
Finite quadratic module over Integer Ring with invariants (2, 4, 4)
Gram matrix of the quadratic form with values in Q/2Z:
[1/2   0   0]
[  0 1/4   0]
[  0   0 5/4]

Some exotic cases:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: D4_gram = Matrix(ZZ, 4, 4,[2,0,0,-1, 0,2,0,-1, 0,0,2,-1, -1,-1,-1,2])
sage: D4 = FreeQuadraticModule(ZZ, 4, D4_gram)
sage: D4dual = D4.span(D4_gram.inverse())
sage: T = TorsionQuadraticModule((1/6)*D4dual, D4); T
Finite quadratic module over Integer Ring with invariants (6, 6, 12, 12)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[ 1/18  1/12  5/36  1/36]
[ 1/12   1/6  1/36   1/9]
[ 5/36  1/36  1/36 11/72]
[ 1/36   1/9 11/72  1/36]
sage: T.normal_form()                                                       # needs sage.libs.pari sage.rings.padics
Finite quadratic module over Integer Ring with invariants (6, 6, 12, 12)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[ 1/6 1/12    0    0    0    0    0    0]
[1/12  1/6    0    0    0    0    0    0]
[   0    0 1/12 1/24    0    0    0    0]
[   0    0 1/24 1/12    0    0    0    0]
[   0    0    0    0  1/9    0    0    0]
[   0    0    0    0    0  1/9    0    0]
[   0    0    0    0    0    0  1/9    0]
[   0    0    0    0    0    0    0  1/9]

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> D4_gram = Matrix(ZZ, Integer(4), Integer(4),[Integer(2),Integer(0),Integer(0),-Integer(1), Integer(0),Integer(2),Integer(0),-Integer(1), Integer(0),Integer(0),Integer(2),-Integer(1), -Integer(1),-Integer(1),-Integer(1),Integer(2)])
>>> D4 = FreeQuadraticModule(ZZ, Integer(4), D4_gram)
>>> D4dual = D4.span(D4_gram.inverse())
>>> T = TorsionQuadraticModule((Integer(1)/Integer(6))*D4dual, D4); T
Finite quadratic module over Integer Ring with invariants (6, 6, 12, 12)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[ 1/18  1/12  5/36  1/36]
[ 1/12   1/6  1/36   1/9]
[ 5/36  1/36  1/36 11/72]
[ 1/36   1/9 11/72  1/36]
>>> T.normal_form()                                                       # needs sage.libs.pari sage.rings.padics
Finite quadratic module over Integer Ring with invariants (6, 6, 12, 12)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[ 1/6 1/12    0    0    0    0    0    0]
[1/12  1/6    0    0    0    0    0    0]
[   0    0 1/12 1/24    0    0    0    0]
[   0    0 1/24 1/12    0    0    0    0]
[   0    0    0    0  1/9    0    0    0]
[   0    0    0    0    0  1/9    0    0]
[   0    0    0    0    0    0  1/9    0]
[   0    0    0    0    0    0    0  1/9]

orthogonal_group(gens=None, check=False)

Orthogonal group of the associated torsion quadratic form.

Warning

This is can be smaller than the orthogonal group of the bilinear form.

INPUT:

• gens – a list of generators, for instance square matrices, something that acts on self, or an automorphism of the underlying abelian group

• check – perform additional checks on the generators

EXAMPLES:

You can provide generators to obtain a subgroup of the full orthogonal group:

Sage

sage: D = TorsionQuadraticForm(matrix.identity(2)/2)
sage: f = matrix(2, [0,1,1,0])
sage: D.orthogonal_group(gens=[f]).order()                                  # needs sage.groups
2

Python

>>> from sage.all import *
>>> D = TorsionQuadraticForm(matrix.identity(Integer(2))/Integer(2))
>>> f = matrix(Integer(2), [Integer(0),Integer(1),Integer(1),Integer(0)])
>>> D.orthogonal_group(gens=[f]).order()                                  # needs sage.groups
2

If no generators are given a slow brute force approach is used to calculate the full orthogonal group:

Sage

sage: D = TorsionQuadraticForm(matrix.identity(3)/2)
sage: OD = D.orthogonal_group()                                             # needs sage.groups
sage: OD.order()                                                            # needs sage.groups
6
sage: fd = D.hom([D.1, D.0, D.2])                                           # needs sage.symbolic
sage: OD(fd)                                                                # needs sage.groups sage.symbolic
[0 1 0]
[1 0 0]
[0 0 1]

Python

>>> from sage.all import *
>>> D = TorsionQuadraticForm(matrix.identity(Integer(3))/Integer(2))
>>> OD = D.orthogonal_group()                                             # needs sage.groups
>>> OD.order()                                                            # needs sage.groups
6
>>> fd = D.hom([D.gen(1), D.gen(0), D.gen(2)])                                           # needs sage.symbolic
>>> OD(fd)                                                                # needs sage.groups sage.symbolic
[0 1 0]
[1 0 0]
[0 0 1]

We compute the kernel of the action of the orthogonal group of $L$ on the discriminant group:

Sage

sage: # needs sage.combinat sage.groups
sage: L = IntegralLattice('A4')
sage: O = L.orthogonal_group()
sage: D = L.discriminant_group()
sage: Obar = D.orthogonal_group(O.gens())
sage: O.order()
240
sage: Obar.order()
2
sage: phi = O.hom(Obar.gens())
sage: phi.kernel().order()
120

Python

>>> from sage.all import *
>>> # needs sage.combinat sage.groups
>>> L = IntegralLattice('A4')
>>> O = L.orthogonal_group()
>>> D = L.discriminant_group()
>>> Obar = D.orthogonal_group(O.gens())
>>> O.order()
240
>>> Obar.order()
2
>>> phi = O.hom(Obar.gens())
>>> phi.kernel().order()
120

orthogonal_submodule_to(S)

Return the submodule orthogonal to S.

INPUT:

• S – a submodule, list, or tuple of generators

EXAMPLES:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeModule(ZZ, 10)
sage: T = TorsionQuadraticModule(V, 3*V)
sage: S = T.submodule(T.gens()[:5])
sage: O = T.orthogonal_submodule_to(S)
sage: O
Finite quadratic module over Integer Ring with invariants (3, 3, 3, 3, 3)
Gram matrix of the quadratic form with values in Q/3Z:
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
sage: O.V() + S.V() == T.V()
True

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> V = FreeModule(ZZ, Integer(10))
>>> T = TorsionQuadraticModule(V, Integer(3)*V)
>>> S = T.submodule(T.gens()[:Integer(5)])
>>> O = T.orthogonal_submodule_to(S)
>>> O
Finite quadratic module over Integer Ring with invariants (3, 3, 3, 3, 3)
Gram matrix of the quadratic form with values in Q/3Z:
[1 0 0 0 0]
[0 1 0 0 0]
[0 0 1 0 0]
[0 0 0 1 0]
[0 0 0 0 1]
>>> O.V() + S.V() == T.V()
True

primary_part(m)

Return the m-primary part of this torsion quadratic module as a submodule.

INPUT:

• m – integer

OUTPUT: a submodule

EXAMPLES:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: T = TorsionQuadraticModule((1/6)*ZZ^3, ZZ^3); T
Finite quadratic module over Integer Ring with invariants (6, 6, 6)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[1/36    0    0]
[   0 1/36    0]
[   0    0 1/36]
sage: T.primary_part(2)
Finite quadratic module over Integer Ring with invariants (2, 2, 2)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[1/4   0   0]
[  0 1/4   0]
[  0   0 1/4]

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> T = TorsionQuadraticModule((Integer(1)/Integer(6))*ZZ**Integer(3), ZZ**Integer(3)); T
Finite quadratic module over Integer Ring with invariants (6, 6, 6)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[1/36    0    0]
[   0 1/36    0]
[   0    0 1/36]
>>> T.primary_part(Integer(2))
Finite quadratic module over Integer Ring with invariants (2, 2, 2)
Gram matrix of the quadratic form with values in Q/(1/3)Z:
[1/4   0   0]
[  0 1/4   0]
[  0   0 1/4]

submodule_with_gens(gens)

Return a submodule with generators given by gens.

INPUT:

• gens – list of generators that convert into self

OUTPUT: a submodule with the specified generators

EXAMPLES:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = FreeQuadraticModule(ZZ, 3, matrix.identity(3)*10)
sage: T = TorsionQuadraticModule((1/10)*V, V)
sage: g = T.gens()
sage: new_gens = [2*g[0], 5*g[0]]
sage: T.submodule_with_gens(new_gens)
Finite quadratic module over Integer Ring with invariants (10,)
Gram matrix of the quadratic form with values in Q/2Z:
[2/5   0]
[  0 1/2]

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> V = FreeQuadraticModule(ZZ, Integer(3), matrix.identity(Integer(3))*Integer(10))
>>> T = TorsionQuadraticModule((Integer(1)/Integer(10))*V, V)
>>> g = T.gens()
>>> new_gens = [Integer(2)*g[Integer(0)], Integer(5)*g[Integer(0)]]
>>> T.submodule_with_gens(new_gens)
Finite quadratic module over Integer Ring with invariants (10,)
Gram matrix of the quadratic form with values in Q/2Z:
[2/5   0]
[  0 1/2]

The generators do not need to be independent:

Sage

sage: new_gens = [g[0], 2*g[1], g[0], g[1]]
sage: T.submodule_with_gens(new_gens)
Finite quadratic module over Integer Ring with invariants (10, 10)
Gram matrix of the quadratic form with values in Q/2Z:
[1/10    0 1/10    0]
[   0  2/5    0  1/5]
[1/10    0 1/10    0]
[   0  1/5    0 1/10]

Python

>>> from sage.all import *
>>> new_gens = [g[Integer(0)], Integer(2)*g[Integer(1)], g[Integer(0)], g[Integer(1)]]
>>> T.submodule_with_gens(new_gens)
Finite quadratic module over Integer Ring with invariants (10, 10)
Gram matrix of the quadratic form with values in Q/2Z:
[1/10    0 1/10    0]
[   0  2/5    0  1/5]
[1/10    0 1/10    0]
[   0  1/5    0 1/10]

twist(s)

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 $ns$.

INPUT:

• s – a rational number

EXAMPLES:

Sage

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]

Python

>>> from sage.all import *
>>> q = TorsionQuadraticForm(matrix.diagonal([Integer(3)/Integer(9), Integer(1)/Integer(9)]))
>>> q.twist(-Integer(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

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]

Python

>>> from sage.all import *
>>> q.twist(Integer(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

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]

Python

>>> from sage.all import *
>>> q.twist(Integer(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]

value_module()

Return $\mathbf{Q}/m\mathbf{Z}$ with $m=(V,W)$.

This is where the inner product takes values.

EXAMPLES:

Sage

sage: A2 = Matrix(ZZ, 2, 2, [2,-1,-1,2])
sage: L = IntegralLattice(2*A2)
sage: D = L.discriminant_group(); D
Finite quadratic module over Integer Ring with invariants (2, 6)
Gram matrix of the quadratic form with values in Q/2Z:
[  1 1/2]
[1/2 1/3]
sage: D.value_module()
Q/Z

Python

>>> from sage.all import *
>>> A2 = Matrix(ZZ, Integer(2), Integer(2), [Integer(2),-Integer(1),-Integer(1),Integer(2)])
>>> L = IntegralLattice(Integer(2)*A2)
>>> D = L.discriminant_group(); D
Finite quadratic module over Integer Ring with invariants (2, 6)
Gram matrix of the quadratic form with values in Q/2Z:
[  1 1/2]
[1/2 1/3]
>>> D.value_module()
Q/Z

value_module_qf()

Return $\mathbf{Q}/n\mathbf{Z}$ with $n\mathbf{Z}=(V,W)+\mathbf{Z}\{(w,w)|w\in W\}$.

This is where the torsion quadratic form takes values.

EXAMPLES:

Sage

sage: A2 = Matrix(ZZ, 2, 2, [2,-1,-1,2])
sage: L = IntegralLattice(2 * A2)
sage: D = L.discriminant_group(); D
Finite quadratic module over Integer Ring with invariants (2, 6)
Gram matrix of the quadratic form with values in Q/2Z:
[  1 1/2]
[1/2 1/3]
sage: D.value_module_qf()
Q/2Z

Python

>>> from sage.all import *
>>> A2 = Matrix(ZZ, Integer(2), Integer(2), [Integer(2),-Integer(1),-Integer(1),Integer(2)])
>>> L = IntegralLattice(Integer(2) * A2)
>>> D = L.discriminant_group(); D
Finite quadratic module over Integer Ring with invariants (2, 6)
Gram matrix of the quadratic form with values in Q/2Z:
[  1 1/2]
[1/2 1/3]
>>> D.value_module_qf()
Q/2Z

class sage.modules.torsion_quadratic_module.TorsionQuadraticModuleElement(parent, x, check=True)

Bases: FGP_Element

An element of a torsion quadratic module.

INPUT:

• parent – parent

• x – element of parent.V()

• check – boolean (default: True)

b(other)

Compute the inner product of two elements.

OUTPUT:

• an element of $\mathbf{Q}/m\mathbf{Z}$ with $m\mathbf{Z}=(V,W)$

EXAMPLES:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = (1/2)*ZZ^2; W = ZZ^2
sage: T = TorsionQuadraticModule(V, W)
sage: g = T.gens()
sage: x = g[0]; x
(1, 0)
sage: y = g[0] + g[1]
sage: x*y
1/4

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> V = (Integer(1)/Integer(2))*ZZ**Integer(2); W = ZZ**Integer(2)
>>> T = TorsionQuadraticModule(V, W)
>>> g = T.gens()
>>> x = g[Integer(0)]; x
(1, 0)
>>> y = g[Integer(0)] + g[Integer(1)]
>>> x*y
1/4

The inner product has further aliases:

Sage

sage: x.inner_product(y)
1/4
sage: x.b(y)
1/4

Python

>>> from sage.all import *
>>> x.inner_product(y)
1/4
>>> x.b(y)
1/4

inner_product(other)

Compute the inner product of two elements.

OUTPUT:

• an element of $\mathbf{Q}/m\mathbf{Z}$ with $m\mathbf{Z}=(V,W)$

EXAMPLES:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: V = (1/2)*ZZ^2; W = ZZ^2
sage: T = TorsionQuadraticModule(V, W)
sage: g = T.gens()
sage: x = g[0]; x
(1, 0)
sage: y = g[0] + g[1]
sage: x*y
1/4

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> V = (Integer(1)/Integer(2))*ZZ**Integer(2); W = ZZ**Integer(2)
>>> T = TorsionQuadraticModule(V, W)
>>> g = T.gens()
>>> x = g[Integer(0)]; x
(1, 0)
>>> y = g[Integer(0)] + g[Integer(1)]
>>> x*y
1/4

The inner product has further aliases:

Sage

sage: x.inner_product(y)
1/4
sage: x.b(y)
1/4

Python

>>> from sage.all import *
>>> x.inner_product(y)
1/4
>>> x.b(y)
1/4

q()

Compute the quadratic product of self.

OUTPUT:

• an element of $\mathbf{Q}/n\mathbf{Z}$ where $n\mathbf{Z}=2(V,W)+\mathbf{Z}\{(w,w)|w\in W\}$

EXAMPLES:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: W = FreeQuadraticModule(ZZ, 2, 2*matrix.identity(2))
sage: V = (1/2) * W
sage: T = TorsionQuadraticModule(V, W)
sage: x = T.gen(0)
sage: x
(1, 0)
sage: x.quadratic_product()
1/2
sage: x.quadratic_product().parent()
Q/2Z
sage: x*x
1/2
sage: (x*x).parent()
Q/Z

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> W = FreeQuadraticModule(ZZ, Integer(2), Integer(2)*matrix.identity(Integer(2)))
>>> V = (Integer(1)/Integer(2)) * W
>>> T = TorsionQuadraticModule(V, W)
>>> x = T.gen(Integer(0))
>>> x
(1, 0)
>>> x.quadratic_product()
1/2
>>> x.quadratic_product().parent()
Q/2Z
>>> x*x
1/2
>>> (x*x).parent()
Q/Z

quadratic_product()

Compute the quadratic product of self.

OUTPUT:

• an element of $\mathbf{Q}/n\mathbf{Z}$ where $n\mathbf{Z}=2(V,W)+\mathbf{Z}\{(w,w)|w\in W\}$

EXAMPLES:

Sage

sage: from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
sage: W = FreeQuadraticModule(ZZ, 2, 2*matrix.identity(2))
sage: V = (1/2) * W
sage: T = TorsionQuadraticModule(V, W)
sage: x = T.gen(0)
sage: x
(1, 0)
sage: x.quadratic_product()
1/2
sage: x.quadratic_product().parent()
Q/2Z
sage: x*x
1/2
sage: (x*x).parent()
Q/Z

Python

>>> from sage.all import *
>>> from sage.modules.torsion_quadratic_module import TorsionQuadraticModule
>>> W = FreeQuadraticModule(ZZ, Integer(2), Integer(2)*matrix.identity(Integer(2)))
>>> V = (Integer(1)/Integer(2)) * W
>>> T = TorsionQuadraticModule(V, W)
>>> x = T.gen(Integer(0))
>>> x
(1, 0)
>>> x.quadratic_product()
1/2
>>> x.quadratic_product().parent()
Q/2Z
>>> x*x
1/2
>>> (x*x).parent()
Q/Z