Discriminant Groups

Torsion Quadratic Modules

A torsion quadratic module is the quotient $M/N$ of two quadratic integer lattices $N \subseteq M$ in the quadratic space $(V,\Phi)$. It inherits a bilinear form

\[b: M/N \times M/N \to \mathbb{Q} / n \mathbb{Z}\]

as well as a quadratic form

\[q: M/N \to \mathbb{Q} / m \mathbb{Z}.\]

where $n \mathbb{Z} = \Phi(M,N)$ and $m \mathbb{Z} = 2n\mathbb{Z} + \sum_{x \in N} \mathbb{Z} \Phi (x,x)$.

torsion_quadratic_module — Method

torsion_quadratic_module(M::ZZLat, N::ZZLat; gens::Union{Nothing, Vector{<:Vector}} = nothing,
                                                snf::Bool = true,
                                                modulus::QQFieldElem = QQFieldElem(0),
                                                check::Bool = true) -> TorQuadModule

Given a Z-lattice $M$ and a sublattice $N$ of $M$, return the torsion quadratic module $M/N$.

If gens is set, the images of gens will be used as the generators of the abelian group $M/N$.

If snf is true, the underlying abelian group will be in Smith normal form. Otherwise, the images of the basis of $M$ will be used as the generators.

The underlying Type

TorQuadModule — Type

TorQuadModule

Examples

julia> A = matrix(ZZ, [[2,0,0,-1],[0,2,0,-1],[0,0,2,-1],[-1,-1,-1,2]]);

julia> L = integer_lattice(gram = A);

julia> T = Hecke.discriminant_group(L)
Finite quadratic module over Integer Ring with underlying abelian group
GrpAb: (Z/2)^2
Gram matrix of the quadratic form with values in Q/2Z
[   1   1//2]
[1//2      1]

We represent torsion quadratic modules as quotients of $\Z$-lattices by a full rank sublattice.

We store them as a $\Z$-lattice M together with a projection p : M -> A onto an abelian group A. The bilinear structure of A is induced via p, that is <a, b> = <p^-1(a), p^-1(a)> with values in $\Q/n\Z$, where $n$ is the modulus and depends on the kernel of p.

Elements of A are basically just elements of the underlying abelian group. To move between M and A, we use the lift function lift : M -> A and coercion A(m).

Examples

julia> R = rescale(root_lattice(:D,4),2);

julia> D = discriminant_group(R);

julia> A = abelian_group(D)
GrpAb: (Z/2)^2 x (Z/4)^2

julia> d = D[1]
[1, 0, 0, 0]

julia> d == D(A(d))
true

julia> lift(d)
4-element Vector{QQFieldElem}:
 1
 1
 3//2
 1

N.B. Since there are no elements of $\Z$-latties, we think of elements of M as elements of the ambient vector space. Thus if v::Vector is such an element then the coordinates with respec to the basis of M are given by solve_left(basis_matrix(M), v).

Most of the functionality mirrors that of AbGrp its elements and homomorphisms. Here we display the part that is specific to elements of torsion quadratic modules.

Attributes

abelian_group — Method

abelian_group(T::TorQuadModule) -> GrpAbFinGen

Return the underlying abelian group of T.

cover — Method

cover(T::TorQuadModule) -> ZZLat

For $T=M/N$ this returns $M$.

relations — Method

relations(T::TorQuadModule) -> ZZLat

For $T=M/N$ this returns $N$.

value_module — Method

value_module(T::TorQuadModule)

Return the value module Q/nZ of the bilinear form of T.

value_module_quadratic_form — Method

value_module_quadratic_form(T::TorQuadModule)

Return the value module Q/mZ of the quadratic form of T.

gram_matrix_bilinear — Method

gram_matrix_bilinear(T::TorQuadModule) -> QQMatrix

Return the gram matrix of the bilinear form of T.

gram_matrix_quadratic — Method

gram_matrix_quadratic(T::TorQuadModule) -> QQMatrix

Return the 'gram matrix' of the quadratic form of T.

The off diagonal entries are given by the bilinear form whereas the diagonal entries are given by the quadratic form.

modulus_bilinear_form — Method

modulus_bilinear_form(T::TorQuadModule) -> QQFieldElem

Return the modulus of the value module of the bilinear form ofT.

modulus_quadratic_form — Method

modulus_quadratic_form(T::TorQuadModule) -> QQFieldElem

Return the modulus of the value module of the quadratic form of T.

Elements

quadratic_product — Method

quadratic_product(a::TorQuadModuleElem)

Return the quadratic product of a.

It is defined in terms of a representative: for $b + M \in M/N=T$ this returns $\Phi(b,b) + n \Z$..

inner_product — Method

inner_product(a::TorQuadModuleElem, b::TorQuadModuleElem)

Return the inner product of a and b.

Lift to the cover

lift — Method

lift(a::TorQuadModuleElem) -> Vector{QQFieldElem}

Lift a to the ambient space of cover(parent(a)).

For $a + N \in M/N$ this returns the representative $a$.

representative — Method

representative(a::TorQuadModuleElem) -> Vector{QQFieldElem}

For $a + N \in M/N$ this returns the representative $a$. An alias for lift(a).

Orthogonal submodules

orthogonal_submodule — Method

orthogonal_submodule(T::TorQuadModule, S::TorQuadModule)-> TorQuadModule

Return the orthogonal submodule to the submodule S of T.

Isometry

is_isometric_with_isometry — Method

is_isometric_with_isometry(T::TorQuadModule, U::TorQuadModule)
                                               -> Bool, TorQuadModuleMor

Return whether the torsion quadratic modules T and U are isometric. If yes, it also returns an isometry $T \to U$.

If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).

It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).

Examples

julia> T = torsion_quadratic_module(QQ[2//3 2//3    0    0    0;
                                       2//3 2//3 2//3    0 2//3;
                                          0 2//3 2//3 2//3    0;
                                          0    0 2//3 2//3    0;
                                          0 2//3    0    0 2//3])
Finite quadratic module over Integer Ring with underlying abelian group
GrpAb: (Z/3)^5
Gram matrix of the quadratic form with values in Q/2Z
[2//3   2//3      0      0      0]
[2//3   2//3   2//3      0   2//3]
[   0   2//3   2//3   2//3      0]
[   0      0   2//3   2//3      0]
[   0   2//3      0      0   2//3]

julia> U = torsion_quadratic_module(QQ[4//3    0    0    0    0;
                                          0 4//3    0    0    0;
                                          0    0 4//3    0    0;
                                          0    0    0 4//3    0;
                                          0    0    0    0 4//3])
Finite quadratic module over Integer Ring with underlying abelian group
GrpAb: (Z/3)^5
Gram matrix of the quadratic form with values in Q/2Z
[4//3      0      0      0      0]
[   0   4//3      0      0      0]
[   0      0   4//3      0      0]
[   0      0      0   4//3      0]
[   0      0      0      0   4//3]

julia> bool, phi = is_isometric_with_isometry(T,U)
(true, Map with following data
Domain:
=======
TorQuadModule [2//3 2//3 0 0 0; 2//3 2//3 2//3 0 2//3; 0 2//3 2//3 2//3 0; 0 0 2//3 2//3 0; 0 2//3 0 0 2//3]
Codomain:
=========
TorQuadModule [4//3 0 0 0 0; 0 4//3 0 0 0; 0 0 4//3 0 0; 0 0 0 4//3 0; 0 0 0 0 4//3])

julia> is_bijective(phi)
true

julia> T2, _ = sub(T, [-T[4], T[2]+T[3]+T[5]])
(TorQuadModule: (Z/3)^2 [2//3 1//3; 1//3 2//3], Map with following data
Domain:
=======
TorQuadModule [2//3 1//3; 1//3 2//3]
Codomain:
=========
TorQuadModule [2//3 2//3 0 0 0; 2//3 2//3 2//3 0 2//3; 0 2//3 2//3 2//3 0; 0 0 2//3 2//3 0; 0 2//3 0 0 2//3])

julia> U2, _ = sub(T, [T[4], T[2]+T[3]+T[5]])
(TorQuadModule: (Z/3)^2 [2//3 2//3; 2//3 2//3], Map with following data
Domain:
=======
TorQuadModule [2//3 2//3; 2//3 2//3]
Codomain:
=========
TorQuadModule [2//3 2//3 0 0 0; 2//3 2//3 2//3 0 2//3; 0 2//3 2//3 2//3 0; 0 0 2//3 2//3 0; 0 2//3 0 0 2//3])

julia> bool, phi = is_isometric_with_isometry(U2, T2)
(true, Map with following data
Domain:
=======
TorQuadModule [2//3 2//3; 2//3 2//3]
Codomain:
=========
TorQuadModule [2//3 1//3; 1//3 2//3])

julia> is_bijective(phi)
true

is_anti_isometric_with_anti_isometry — Method

is_anti_isometric_with_anti_isometry(T::TorQuadModule, U::TorQuadModule)
                                                 -> Bool, TorQuadModuleMor

Return whether there exists an anti-isometry between the torsion quadratic modules T and U. If yes, it returns such an anti-isometry $T \to U$.

If T and U are not semi-regular it requires that they both split into a direct sum of their respective quadratic radical (see radical_quadratic).

It requires that both T and U have modulus 1: in case one of them do not, they should be rescaled (see rescale).

Examples

julia> T = torsion_quadratic_module(QQ[4//5;])
Finite quadratic module over Integer Ring with underlying abelian group
GrpAb: Z/5
Gram matrix of the quadratic form with values in Q/2Z
[4//5]

julia> bool, phi = is_anti_isometric_with_anti_isometry(T, T)
(true, Map with following data
Domain:
=======
TorQuadModule [4//5]
Codomain:
=========
TorQuadModule [4//5])

julia> a = gens(T)[1];

julia> a*a == -phi(a)*phi(a)
true

julia> G = matrix(FlintQQ, 6, 6 , [3 3 0 0 0  0;
                                   3 3 3 0 3  0;
                                   0 3 3 3 0  0;
                                   0 0 3 3 0  0;
                                   0 3 0 0 3  0;
                                   0 0 0 0 0 10]);

julia> V = quadratic_space(QQ, G);

julia> B = matrix(QQ, 6, 6 , [1    0    0    0    0    0;
                              0 1//3 1//3 2//3 1//3    0;
                              0    0    1    0    0    0;
                              0    0    0    1    0    0;
                              0    0    0    0    1    0;
                              0    0    0    0    0 1//5]);


julia> M = lattice(V, B);

julia> B2 = matrix(FlintQQ, 6, 6 , [ 1  0 -1  1  0 0;
                                     0  0  1 -1  0 0;
                                    -1  1  1 -1 -1 0;
                                     1 -1 -1  2  1 0;
                                     0  0 -1  1  1 0;
                                     0  0  0  0  0 1]);

julia> N = lattice(V, B2);

julia> T = torsion_quadratic_module(M, N)
Finite quadratic module over Integer Ring with underlying abelian group
GrpAb: Z/15
Gram matrix of the quadratic form with values in Q/Z
[3//5]

julia> bool, phi = is_anti_isometric_with_anti_isometry(T,T)
(true, Map with following data
Domain:
=======
TorQuadModule [3//5]
Codomain:
=========
TorQuadModule [3//5])

julia> a = gens(T)[1];

julia> a*a == -phi(a)*phi(a)
true

Primary and elementary modules

is_primary_with_prime(T::TorQuadModule)
is_primary(T::TorQuadModule, p::Union{Integer, ZZRingElem})
is_elementary_with_prime(T::TorQuadModule)
is_elementary(T::TorQuadModule, p::Union{Integer, ZZRingElem})

Smith normal form

snf(T::TorQuadModule)
is_snf(T::TorQuadModule)

Discriminant Groups

See V. V. Nikulin (1979) for the general theory of discriminant groups. They are particularly useful to work with primitive embeddings of integral integer quadratic lattices.

From a lattice

discriminant_group — Method

discriminant_group(L::ZZLat) -> TorQuadModule

Return the discriminant group of L.

The discriminant group of an integral lattice L is the finite abelian group D = dual(L)/L.

It comes equipped with the discriminant bilinear form

\[D \times D \to \Q / \Z \qquad (x,y) \mapsto \Phi(x,y) + \Z.\]

If L is even, then the discriminant group is equipped with the discriminant quadratic form $D \to \Q / 2 \Z, x \mapsto \Phi(x,x) + 2\Z$.

From a Matrix

torsion_quadratic_module — Method

torsion_quadratic_module(q::QQMatrix) -> TorQuadModule

Return a torsion quadratic module with gram matrix given by q and value module Q/Z. If all the diagonal entries of q have: either even numerator or even denominator, then the value module of the quadratic form is Q/2Z

Example

julia> torsion_quadratic_module(QQ[1//6;])
Finite quadratic module over Integer Ring with underlying abelian group
GrpAb: Z/6
Gram matrix of the quadratic form with values in Q/2Z
[1//6]

julia> torsion_quadratic_module(QQ[1//2;])
Finite quadratic module over Integer Ring with underlying abelian group
GrpAb: Z/2
Gram matrix of the quadratic form with values in Q/2Z
[1//2]

julia> torsion_quadratic_module(QQ[3//2;])
Finite quadratic module over Integer Ring with underlying abelian group
GrpAb: Z/2
Gram matrix of the quadratic form with values in Q/2Z
[3//2]

julia> torsion_quadratic_module(QQ[1//3;])
Finite quadratic module over Integer Ring with underlying abelian group
GrpAb: Z/3
Gram matrix of the quadratic form with values in Q/Z
[1//3]

Rescaling the form

rescale — Method

rescale(T::TorQuadModule, k::RingElement) -> TorQuadModule

Return the torsion quadratic module with quadratic form scaled by $k$, where k is a non-zero rational number. If the old form was defined modulo n, then the new form is defined modulo n k.

Invariants

is_degenerate — Method

is_degenerate(T::TorQuadModule) -> Bool

Return true if the underlying bilinear form is degenerate.

is_semi_regular — Method

is_semi_regular(T::TorQuadModule) -> Bool

Return whether T is semi-regular, that is its quadratic radical is trivial (see radical_quadratic).

radical_bilinear — Method

radical_bilinear(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMor

Return the radical \{x \in T | b(x,T) = 0\} of the bilinear form b on T.

radical_quadratic — Method

radical_quadratic(T::TorQuadModule) -> TorQuadModule, TorQuadModuleMor

Return the radical \{x \in T | b(x,T) = 0 and q(x)=0\} of the quadratic form q on T.

normal_form — Method

normal_form(T::TorQuadModule; partial=false) -> TorQuadModule, TorQuadModuleMor

Return the normal form N of the given torsion quadratic module T along with the projection T -> N.

Let K be the radical of the quadratic form of T. Then N = T/K is half-regular. Two half-regular torsion quadratic modules are isometric if and only if they have equal normal forms.

Genus

genus — Method

genus(T::TorQuadModule, signature_pair::Tuple{Int, Int}) -> ZZGenus

Return the genus of an integer lattice with discriminant group T and the given signature_pair. If no such genus exists, raise an error.

Reference

V. V. Nikulin (1979) Corollary 1.9.4 and 1.16.3.

brown_invariant — Method

brown_invariant(self::TorQuadModule) -> Nemo.zzModRingElem

Return the Brown invariant of this torsion quadratic form.

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

\[\exp \left( \frac{2 \pi i }{8} Br(q)\right) = \frac{1}{\sqrt{D}} \sum_{x \in D} \exp(i \pi q(x)).\]

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

Examples

julia> L = integer_lattice(gram=matrix(ZZ, [[2,-1,0,0],[-1,2,-1,-1],[0,-1,2,0],[0,-1,0,2]]));

julia> T = Hecke.discriminant_group(L);

julia> brown_invariant(T)
4

is_genus — Method

is_genus(T::TorQuadModule, signature_pair::Tuple{Int, Int}) -> Bool

Return if there is an integral lattice with this signature and discriminant form.

Categorical constructions

direct_sum — Method

direct_sum(x::Vararg{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMor}
direct_sum(x::Vector{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMor}

Given a collection of torsion quadratic modules $T_1, \ldots, T_n$, return their direct sum $T := T_1\oplus \ldots \oplus T_n$, together with the injections $T_i \to T$.

For objects of type TorQuadModule, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain T as a direct product with the projections $T \to T_i$, one should call direct_product(x). If one wants to obtain T as a biproduct with the injections $T_i \to T$ and the projections $T \to T_i$, one should call biproduct(x).

direct_product — Method

direct_product(x::Vararg{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMor}
direct_product(x::Vector{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMor}

Given a collection of torsion quadratic modules $T_1, \ldots, T_n$, return their direct product $T := T_1\times \ldots \times T_n$, together with the projections $T \to T_i$.

For objects of type TorQuadModule, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain T as a direct sum with the inctions $T_i \to T$, one should call direct_sum(x). If one wants to obtain T as a biproduct with the injections $T_i \to T$ and the projections $T \to T_i$, one should call biproduct(x).

biproduct — Method

biproduct(x::Vararg{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMor}, Vector{TorQuadModuleMor}
biproduct(x::Vector{TorQuadModule}) -> TorQuadModule, Vector{TorQuadModuleMor}, Vector{TorQuadModuleMor}

Given a collection of torsion quadratic modules $T_1, \ldots, T_n$, return their biproduct $T := T_1\oplus \ldots \oplus T_n$, together with the injections $T_i \to T$ and the projections $T \to T_i$.

For objects of type TorQuadModule, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain T as a direct sum with the inctions $T_i \to T$, one should call direct_sum(x). If one wants to obtain T as a direct product with the projections $T \to T_i$, one should call direct_product(x).