# Discriminant Groups

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(M::ZZLat, N::ZZLat; gens::Union{Nothing, Vector{<:Vector}} = nothing,
snf::Bool = true,
modulus::RationalUnion = QQFieldElem(0),
modulus_qf::RationalUnion = QQFieldElem(0),

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.

One can decide on the modulus for the associated finite bilinear and quadratic forms by setting modulus and modulus_qf respectively to the desired values.

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### The underlying Type

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)
over integer ring
Abelian group: (Z/2)^2
Bilinear value module: Q/Z
[   1   1//2]
[1//2      1]

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

We store them as a $\mathbb{Z}$-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 $\mathbb{Q}/n\mathbb{Z}$, 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)
(Z/2)^2 x (Z/4)^2

julia> d = D[1]
Element
of finite quadratic module: (Z/2)^2 x (Z/4)^2 -> Q/2Z
with components [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 $\mathbb{Z}$-lattices, 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(basis_matrix(M), v; side = :left).

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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_groupMethod

Return the underlying abelian group of T.

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coverMethod

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

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relationsMethod

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

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value_moduleMethod

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

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gram_matrix_bilinearMethod

Return the gram matrix of the bilinear form of T.

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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.

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modulus_bilinear_formMethod

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

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

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### Elements

Return the quadratic product of a.

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

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inner_productMethod

Return the inner product of a and b.

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### Lift to the cover

liftMethod

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

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

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representativeMethod

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

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### Orthogonal submodules

orthogonal_submoduleMethod

Return the orthogonal submodule to the submodule S of T.

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### Isometry

is_isometric_with_isometryMethod

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])
over integer ring
Abelian group: (Z/3)^5
Bilinear value module: Q/Z
[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])
over integer ring
Abelian group: (Z/3)^5
Bilinear value module: Q/Z
[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)

julia> is_bijective(phi)
true

julia> T2, _ = sub(T, [-T[4], T[2]+T[3]+T[5]])

julia> U2, _ = sub(T, [T[4], T[2]+T[3]+T[5]])

julia> bool, phi = is_isometric_with_isometry(U2, T2)

julia> is_bijective(phi)
true
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is_anti_isometric_with_anti_isometryMethod

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

over integer ring
Abelian group: Z/5
Bilinear value module: Q/Z
[4//5]

julia> bool, phi = is_anti_isometric_with_anti_isometry(T, T)

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> 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);

over integer ring
Abelian group: Z/15
Bilinear value module: Q/Z
[3//5]

julia> bool, phi = is_anti_isometric_with_anti_isometry(T,T)

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

julia> a*a == -phi(a)*phi(a)
true
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### Primary and elementary modules

is_primary_with_primeMethod

Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group for some prime number p. In case it is, p is also returned as second output.

Note that in the case of trivial groups, this function returns (true, 1). If T is not primary, the second return value is -1 by default.

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is_primaryMethod

Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is a p-group.

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is_elementary_with_primeMethod

Given a torsion quadratic module T, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group, for some prime number p. In case it is, p is also returned as second output.

Note that in the case of trivial groups, this function returns (true, 1). If T is not elementary, the second return value is -1 by default.

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is_elementaryMethod

Given a torsion quadratic module T and a prime number p, return whether the underlying (finite) abelian group of T (see abelian_group) is an elementary p-group.

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### Smith normal form

snfMethod

Given a torsion quadratic module T, return a torsion quadratic module S, isometric to T, such that the underlying abelian group of S is in canonical Smith normal form. It comes with an isometry $f : S \to T$.

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is_snfMethod

Given a torsion quadratic module T, return whether its underlying abelian group is in Smith normal form.

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## Discriminant Groups

See [Nik79] 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_groupMethod

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

$$$D \times D \to \mathbb{Q} / \mathbb{Z} \qquad (x,y) \mapsto \Phi(x,y) + \mathbb{Z}.$$$

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

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### From a matrix

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

over integer ring
Abelian group: Z/6
Bilinear value module: Q/Z
[1//6]

over integer ring
Abelian group: Z/2
Bilinear value module: Q/Z
[1//2]

over integer ring
Abelian group: Z/2
Bilinear value module: Q/Z
[3//2]

over integer ring
Abelian group: Z/3
Bilinear value module: Q/Z
[1//3]
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### Rescaling the form

rescaleMethod

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.

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### Invariants

is_degenerateMethod

Return true if the underlying bilinear form is degenerate.

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

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

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normal_formMethod

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.

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### Genus

genusMethod
-> ZZGenus

Return the genus of an integer lattice whose discriminant group has the bilinear form of T, the given signature_pair and the given parity.

The argument parity is one of the following: either parity == 1 for genera of odd lattices, or parity == 2 for even lattices. By default, parity is set to be as the parity of the quadratic form on T

If no such genus exists, raise an error.

Reference

[Nik79] Corollary 1.9.4 and 1.16.3.

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brown_invariantMethod

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
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is_genusMethod

Return if there is an integral lattice whose discriminant form has the bilinear form of T, whose signatures match signature_pair and which is of parity parity.

The argument parity is one of the following: either parity == 1 for genera of odd lattices, or parity == 2 for even lattices. By default, parity is set to be as the parity of the quadratic form on T

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### Categorical constructions

direct_sumMethod

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).

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direct_productMethod

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).

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biproductMethod

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).

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### Submodules

submodulesMethod

Return the submodules of T as an iterator. Possible keyword arguments to restrict the submodules:

• order::Int: only submodules of order order,
• index::Int: only submodules of index index,
• subtype::Vector{Int}: only submodules which are isomorphic as an abelian group to abelian_group(subtype),
• quotype::Vector{Int}: only submodules whose quotient are isomorphic as an abelian to abelian_group(quotype).
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stable_submodulesMethod