Integer Lattices
An integer lattice
A ZZLat
. It is given in terms of its ambient quadratic space
To access ambient_space(L::ZZLat)
and basis_matrix(L::ZZLat)
.
Creation of integer lattices
From a gram matrix
integer_lattice([B::MatElem]; gram) -> ZZLat
Return the Z-lattice with basis matrix gram
.
If the keyword gram
is not specified, the Gram matrix is the identity matrix. If
Examples
julia> L = integer_lattice(matrix(QQ, 2, 2, [1//2, 0, 0, 2]));
julia> gram_matrix(L) == matrix(QQ, 2, 2, [1//4, 0, 0, 4])
true
julia> L = integer_lattice(gram = matrix(ZZ, [2 -1; -1 2]));
julia> gram_matrix(L) == matrix(ZZ, [2 -1; -1 2])
true
In a quadratic space
lattice(V::AbstractSpace, basis::MatElem ; check::Bool = true) -> AbstractLat
Given an ambient space V
and a matrix basis
, return the lattice spanned by the rows of basis
inside V
. If V
is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.
By default, basis
is checked to be of full rank. This test can be disabled by setting check
to false.
Special lattices
root_lattice(R::Symbol, n::Int) -> ZZLat
Return the root lattice of type R
given by :A
, :D
or :E
with parameter n
.
The type :I
with parameter n = 1
is also allowed and denotes the odd unimodular lattice of rank 1.
hyperbolic_plane_lattice(n::RationalUnion = 1) -> ZZLat
Return the hyperbolic plane with intersection form of scale n
, that is, the unique (up to isometry) even unimodular hyperbolic n
.
Examples
julia> L = hyperbolic_plane_lattice(6);
julia> gram_matrix(L)
[0 6]
[6 0]
julia> L = hyperbolic_plane_lattice(ZZ(-13));
julia> gram_matrix(L)
[ 0 -13]
[-13 0]
integer_lattice(S::Symbol, n::RationalUnion = 1) -> ZZlat
Given S = :H
or S = :U
, return a
leech_lattice() -> ZZLat
Return the Leech lattice.
leech_lattice(niemeier_lattice::ZZLat) -> ZZLat, QQMatrix, Int
Return a triple L, v, h
where L
is the Leech lattice.
L is an h
-neighbor of the Niemeier lattice N
with respect to v
. This means that L / L ∩ N ≅ ℤ / h ℤ
. Here h
is the Coxeter number of the Niemeier lattice.
This implements the 23 holy constructions of the Leech lattice in [5].
Examples
julia> R = integer_lattice(gram=2 * identity_matrix(ZZ, 24));
julia> N = maximal_even_lattice(R) # Some Niemeier lattice
Integer lattice of rank 24 and degree 24
with gram matrix
[2 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 1 0 0 0 0]
[1 2 1 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0]
[1 1 2 1 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0]
[1 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 2 1 1 1 0 0 0 0 1 0 1 1 0 0 0 0 0 0 0 0]
[0 0 0 0 1 2 1 1 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 2 1 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 2 1 1 1 0 0 0 0 0 0 0 0 1 1 1 0]
[0 0 0 0 0 0 0 0 1 2 1 1 0 0 0 0 0 0 0 0 1 0 1 1]
[0 0 0 0 0 0 0 0 1 1 2 1 0 0 0 0 0 0 0 0 1 1 0 1]
[0 0 0 0 0 0 0 0 1 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 1 0 0 0 0 0 2 1 1 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 1 1 0 0 0 0 0 1 2 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 0 1 0 0 0 0 0 1 0 2 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 1 0 0 0 0 0 0 1 0 0 2 0 0 0 0 0 0 0 0]
[1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 2 1 1 1 0 0 0 0]
[0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 0 0 0 0 0 0]
[1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 2 0 0 0 0 0]
[1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 2 0 0 0 0]
[0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 2 1 1 1]
[0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 1 2 0 0]
[0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 0 1 0 2 0]
[0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 0 1 0 0 2]
julia> minimum(N)
2
julia> det(N)
1
julia> L, v, h = leech_lattice(N);
julia> minimum(L)
4
julia> det(L)
1
julia> h == index(L, intersect(L, N))
true
We illustrate how the Leech lattice is constructed from N
, h
and v
.
julia> Zmodh, _ = residue_ring(ZZ, h);
julia> V = ambient_space(N);
julia> vG = map_entries(x->Zmodh(ZZ(x)), inner_product(V, v, basis_matrix(N)));
julia> LN = transpose(lift(Hecke.kernel(vG; side = :right)))*basis_matrix(N); # vectors whose inner product with `v` is divisible by `h`.
julia> lattice(V, LN) == intersect(L, N)
true
julia> gensL = vcat(LN, 1//h * v);
julia> lattice(V, gensL, isbasis=false) == L
true
k3_lattice()
Return the integer lattice corresponding to the Beauville-Bogomolov-Fujiki form associated to a K3 surface.
Examples
julia> L = k3_lattice();
julia> is_unimodular(L)
true
julia> signature_tuple(L)
(3, 0, 19)
mukai_lattice(S::Symbol = :K3; extended::Bool = false)
Return the (extended) Mukai lattice.
If S == :K3
, it returns the (extended) Mukai lattice associated to hyperkaehler manifolds which are deformation equivalent to a moduli space of stable sheaves on a K3 surface.
If S == :Ab
, it returns the (extended) Mukai lattice associated to hyperkaehler manifolds which are deformation equivalent to a moduli space of stable sheaves on an abelian surface.
Examples
julia> L = mukai_lattice();
julia> genus(L)
Genus symbol for integer lattices
Signatures: (4, 0, 20)
Local symbol:
Local genus symbol at 2: 1^24
julia> L = mukai_lattice(; extended = true);
julia> genus(L)
Genus symbol for integer lattices
Signatures: (5, 0, 21)
Local symbol:
Local genus symbol at 2: 1^26
julia> L = mukai_lattice(:Ab);
julia> genus(L)
Genus symbol for integer lattices
Signatures: (4, 0, 4)
Local symbol:
Local genus symbol at 2: 1^8
julia> L = mukai_lattice(:Ab; extended = true);
julia> genus(L)
Genus symbol for integer lattices
Signatures: (5, 0, 5)
Local symbol:
Local genus symbol at 2: 1^10
hyperkaehler_lattice(S::Symbol; n::Int = 2)
Return the integer lattice corresponding to the Beauville-Bogomolov-Fujiki form on a hyperkaehler manifold whose deformation type is determined by S
and n
.
If
S == :K3
orS == :Kum
, thenn
must be an integer bigger than 2;If
S == :OG6
orS == :OG10
, the value ofn
has no effect.
Examples
julia> L = hyperkaehler_lattice(:Kum; n = 3)
Integer lattice of rank 7 and degree 7
with gram matrix
[0 1 0 0 0 0 0]
[1 0 0 0 0 0 0]
[0 0 0 1 0 0 0]
[0 0 1 0 0 0 0]
[0 0 0 0 0 1 0]
[0 0 0 0 1 0 0]
[0 0 0 0 0 0 -8]
julia> L = hyperkaehler_lattice(:OG6)
Integer lattice of rank 8 and degree 8
with gram matrix
[0 1 0 0 0 0 0 0]
[1 0 0 0 0 0 0 0]
[0 0 0 1 0 0 0 0]
[0 0 1 0 0 0 0 0]
[0 0 0 0 0 1 0 0]
[0 0 0 0 1 0 0 0]
[0 0 0 0 0 0 -2 0]
[0 0 0 0 0 0 0 -2]
julia> L = hyperkaehler_lattice(:OG10);
julia> genus(L)
Genus symbol for integer lattices
Signatures: (3, 0, 21)
Local symbols:
Local genus symbol at 2: 1^-24
Local genus symbol at 3: 1^-23 3^1
julia> L = hyperkaehler_lattice(:K3; n = 3);
julia> genus(L)
Genus symbol for integer lattices
Signatures: (3, 0, 20)
Local symbol:
Local genus symbol at 2: 1^22 4^1_7
From a genus
Integer lattices can be created as representatives of a genus. See (representative(L::ZZGenus)
)
Rescaling the Quadratic Form
rescale(L::ZZLat, r::RationalUnion) -> ZZLat
Return the lattice L
in the quadratic space with form r \Phi
.
Examples
This can be useful to apply methods intended for positive definite lattices.
julia> L = integer_lattice(gram=ZZ[-1 0; 0 -1])
Integer lattice of rank 2 and degree 2
with gram matrix
[-1 0]
[ 0 -1]
julia> shortest_vectors(rescale(L, -1))
2-element Vector{Vector{ZZRingElem}}:
[0, 1]
[1, 0]
Attributes
ambient_space(L::AbstractLat) -> AbstractSpace
Return the ambient space of the lattice L
. If the ambient space is not known, an error is raised.
basis_matrix(L::ZZLat) -> QQMatrix
Return the basis matrix
The lattice is given by the row span of
gram_matrix(L::ZZLat) -> QQMatrix
Return the gram matrix of
Examples
julia> L = integer_lattice(matrix(ZZ, [2 0; -1 2]));
julia> gram_matrix(L)
[ 4 -2]
[-2 5]
rational_span(L::ZZLat) -> QuadSpace
Return the rational span of gram_matrix(L)
.
Examples
julia> L = integer_lattice(matrix(ZZ, [2 0; -1 2]));
julia> rational_span(L)
Quadratic space of dimension 2
over rational field
with gram matrix
[ 4 -2]
[-2 5]
Invariants
rank(L::AbstractLat) -> Int
Return the rank of the underlying module of the lattice L
.
scale(L::ZZLat) -> QQFieldElem
Return the scale of L
.
The scale of L
is defined as the positive generator of the
norm(L::ZZLat) -> QQFieldElem
Return the norm of L
.
The norm of L
is defined as the positive generator of the
iseven(L::ZZLat) -> Bool
Return whether L
is even.
An integer lattice L
in the rational quadratic space
is_integral(L::AbstractLat) -> Bool
Return whether the lattice L
is integral.
is_primary_with_prime(L::ZZLat) -> Bool, ZZRingElem
Given a L
, return whether L
is primary, that is whether L
is integral and its discriminant group (see discriminant_group
) is a p
-group for some prime number p
. In case it is, p
is also returned as second output.
Note that for unimodular lattices, this function returns (true, 1)
. If the lattice is not primary, the second return value is -1
by default.
is_primary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
Given an integral L
and a prime number p
, return whether L
is p
-primary, that is whether its discriminant group (see discriminant_group
) is a p
-group.
is_elementary_with_prime(L::ZZLat) -> Bool, ZZRingElem
Given a L
, return whether L
is elementary, that is whether L
is integral and its discriminant group (see discriminant_group
) is an elemenentary p
-group for some prime number p
. In case it is, p
is also returned as second output.
Note that for unimodular lattices, this function returns (true, 1)
. If the lattice is not elementary, the second return value is -1
by default.
is_elementary(L::ZZLat, p::Union{Integer, ZZRingElem}) -> Bool
Given an integral L
and a prime number p
, return whether L
is p
-elementary, that is whether its discriminant group (see discriminant_group
) is an elementary p
-group.
The Genus
For an integral lattice The genus of an integer lattice collects its local invariants. genus(::ZZLat)
genus_representatives(L::ZZLat) -> Vector{ZZLat}
Return representatives for the isometry classes in the genus of L
.
Real invariants
signature_tuple(L::ZZLat) -> Tuple{Int,Int,Int}
Return the number of (positive, zero, negative) inertia of L
.
is_positive_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L
is positive definite.
is_negative_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L
is negative definite.
is_definite(L::AbstractLat) -> Bool
Return whether the rational span of the lattice L
is definite.
Isometries
automorphism_group_generators(E::EllipticCurve) -> Vector{EllCrvIso}
Return generators of the automorphism group of
automorphism_group_generators(L::AbstractLat; ambient_representation::Bool = true,
depth::Int = -1, bacher_depth::Int = 0)
-> Vector{MatElem}
Given a definite lattice L
, return generators for the automorphism group of L
. If ambient_representation == true
(the default), the transformations are represented with respect to the ambient space of L
. Otherwise, the transformations are represented with respect to the (pseudo-)basis of L
.
Setting the parameters depth
and bacher_depth
to a positive value may improve performance. If set to -1
(default), the used value of depth
is chosen heuristically depending on the rank of L
. By default, bacher_depth
is set to 0
.
automorphism_group_order(L::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Int
Given a definite lattice L
, return the order of the automorphism group of L
.
Setting the parameters depth
and bacher_depth
to a positive value may improve performance. If set to -1
(default), the used value of depth
is chosen heuristically depending on the rank of L
. By default, bacher_depth
is set to 0
.
is_isometric(L::AbstractLat, M::AbstractLat; depth::Int = -1, bacher_depth::Int = 0) -> Bool
Return whether the lattices L
and M
are isometric.
Setting the parameters depth
and bacher_depth
to a positive value may improve performance. If set to -1
(default), the used value of depth
is chosen heuristically depending on the rank of L
. By default, bacher_depth
is set to 0
.
is_locally_isometric(L::ZZLat, M::ZZLat, p::Int) -> Bool
Return whether L
and M
are isometric over the p
-adic integers.
i.e. whether
Root lattices
root_lattice_recognition(L::ZZLat)
Return the ADE type of the root sublattice of L
.
The root sublattice is the lattice spanned by the vectors of squared length (:I, 1)
.
Input:
L
– a definite and integral
Output:
Two lists, the first one containing the ADE types and the second one the irreducible root sublattices.
For more recognizable gram matrices use root_lattice_recognition_fundamental
.
Examples
julia> L = integer_lattice(gram=ZZ[4 0 0 0 3 0 3 0;
0 16 8 12 2 12 6 10;
0 8 8 6 2 8 4 5;
0 12 6 10 2 9 5 8;
3 2 2 2 4 2 4 2;
0 12 8 9 2 12 6 9;
3 6 4 5 4 6 6 5;
0 10 5 8 2 9 5 8])
Integer lattice of rank 8 and degree 8
with gram matrix
[4 0 0 0 3 0 3 0]
[0 16 8 12 2 12 6 10]
[0 8 8 6 2 8 4 5]
[0 12 6 10 2 9 5 8]
[3 2 2 2 4 2 4 2]
[0 12 8 9 2 12 6 9]
[3 6 4 5 4 6 6 5]
[0 10 5 8 2 9 5 8]
julia> R = root_lattice_recognition(L)
([(:A, 1), (:D, 6)], ZZLat[Integer lattice of rank 1 and degree 8, Integer lattice of rank 6 and degree 8])
julia> L = integer_lattice(; gram = QQ[1 0 0 0;
0 9 3 3;
0 3 2 1;
0 3 1 11])
Integer lattice of rank 4 and degree 4
with gram matrix
[1 0 0 0]
[0 9 3 3]
[0 3 2 1]
[0 3 1 11]
julia> root_lattice_recognition(L)
([(:A, 1), (:I, 1)], ZZLat[Integer lattice of rank 1 and degree 4, Integer lattice of rank 1 and degree 4])
root_lattice_recognition_fundamental(L::ZZLat)
Return the ADE type of the root sublattice of L
as well as the corresponding irreducible root sublattices with basis given by a fundamental root system.
The type (:I, 1)
corresponds to the odd unimodular root lattice of rank 1.
Input:
L
– a definite and integral
Output:
the root sublattice, with basis given by a fundamental root system
the ADE types
a Vector consisting of the irreducible root sublattices.
Examples
julia> L = integer_lattice(gram=ZZ[4 0 0 0 3 0 3 0;
0 16 8 12 2 12 6 10;
0 8 8 6 2 8 4 5;
0 12 6 10 2 9 5 8;
3 2 2 2 4 2 4 2;
0 12 8 9 2 12 6 9;
3 6 4 5 4 6 6 5;
0 10 5 8 2 9 5 8])
Integer lattice of rank 8 and degree 8
with gram matrix
[4 0 0 0 3 0 3 0]
[0 16 8 12 2 12 6 10]
[0 8 8 6 2 8 4 5]
[0 12 6 10 2 9 5 8]
[3 2 2 2 4 2 4 2]
[0 12 8 9 2 12 6 9]
[3 6 4 5 4 6 6 5]
[0 10 5 8 2 9 5 8]
julia> R = root_lattice_recognition_fundamental(L);
julia> gram_matrix(R[1])
[2 0 0 0 0 0 0]
[0 2 0 -1 0 0 0]
[0 0 2 -1 0 0 0]
[0 -1 -1 2 -1 0 0]
[0 0 0 -1 2 -1 0]
[0 0 0 0 -1 2 -1]
[0 0 0 0 0 -1 2]
ADE_type(G::MatrixElem) -> Tuple{Symbol,Int64}
Return the type of the irreducible root lattice with gram matrix G
.
See also root_lattice_recognition
.
Examples
julia> Hecke.ADE_type(gram_matrix(root_lattice(:A,3)))
(:A, 3)
coxeter_number(ADE::Symbol, n) -> Int
Return the Coxeter number of the corresponding ADE root lattice.
If n
is the rank of
Examples
julia> coxeter_number(:D, 4)
6
highest_root(ADE::Symbol, n) -> ZZMatrix
Return coordinates of the highest root of root_lattice(ADE, n)
.
Examples
julia> highest_root(:E, 6)
[1 2 3 2 1 2]
Module operations
Most module operations assume that the lattices live in the same ambient space. For instance only lattices in the same ambient space compare.
Return true
if both lattices have the same ambient quadratic space and the same underlying module.
is_sublattice(L::AbstractLat, M::AbstractLat) -> Bool
Return whether M
is a sublattice of the lattice L
.
is_sublattice_with_relations(M::ZZLat, N::ZZLat) -> Bool, QQMatrix
Returns whether
+(L::AbstractLat, M::AbstractLat) -> AbstractLat
Return the sum of the lattices L
and M
.
The lattices L
and M
must have the same ambient space.
*(a::RationalUnion, L::ZZLat) -> ZZLat
Return the lattice
intersect(L::AbstractLat, M::AbstractLat) -> AbstractLat
Return the intersection of the lattices L
and M
.
The lattices L
and M
must have the same ambient space.
Base.in(v::Vector, L::ZZLat) -> Bool
Return whether the vector v
lies in the lattice L
.
Base.in(v::QQMatrix, L::ZZLat) -> Bool
Return whether the row span of v
lies in the lattice L
.
primitive_closure(M::ZZLat, N::ZZLat) -> ZZLat
Given two M
and N
with N
in M
.
Examples
julia> M = root_lattice(:D, 6);
julia> N = lattice_in_same_ambient_space(M, 3*basis_matrix(M)[1:1,:]);
julia> basis_matrix(N)
[3 0 0 0 0 0]
julia> N2 = primitive_closure(M, N)
Integer lattice of rank 1 and degree 6
with gram matrix
[2]
julia> basis_matrix(N2)
[1 0 0 0 0 0]
julia> M2 = primitive_closure(dual(M), M);
julia> is_integral(M2)
false
is_primitive(M::ZZLat, N::ZZLat) -> Bool
Given two N
is a primitive sublattice of M
.
Examples
julia> U = hyperbolic_plane_lattice(3);
julia> bU = basis_matrix(U);
julia> e1, e2 = bU[1:1,:], bU[2:2,:]
([1 0], [0 1])
julia> N = lattice_in_same_ambient_space(U, e1 + e2)
Integer lattice of rank 1 and degree 2
with gram matrix
[6]
julia> is_primitive(U, N)
true
julia> M = root_lattice(:A, 3);
julia> f = matrix(QQ, 3, 3, [0 1 1; -1 -1 -1; 1 1 0]);
julia> N = kernel_lattice(M, f+1)
Integer lattice of rank 1 and degree 3
with gram matrix
[4]
julia> is_primitive(M, N)
true
is_primitive(L::ZZLat, v::Union{Vector, QQMatrix}) -> Bool
Return whether the vector v
is primitive in L
.
A vector v
in a L
is called primitive if for all w
in L
such that d
, then
divisibility(L::ZZLat, v::Union{Vector, QQMatrix}) -> QQFieldElem
Return the divisibility of v
with respect to L
.
For a vector v
in the ambient quadratic space L
, we call the divisibility of v
with the respect to L
the non-negative generator of the fractional
Embeddings
Categorical constructions
direct_sum(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
direct_sum(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
Given a collection of
For objects of type ZZLat
, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain L
as a direct product with the projections direct_product(x)
. If one wants to obtain L
as a biproduct with the injections biproduct(x)
.
direct_product(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
direct_product(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}
Given a collection of
For objects of type ZZLat
, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain L
as a direct sum with the injections direct_sum(x)
. If one wants to obtain L
as a biproduct with the injections biproduct(x)
.
biproduct(x::Vararg{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
biproduct(x::Vector{ZZLat}) -> ZZLat, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
Given a collection of
For objects of type ZZLat
, finite direct sums and finite direct products agree and they are therefore called biproducts. If one wants to obtain L
as a direct sum with the injections direct_sum(x)
. If one wants to obtain L
as a direct product with the projections direct_product(x)
.
Orthogonal sublattices
orthogonal_submodule(L::ZZLat, S::ZZLat) -> ZZLat
Return the largest submodule of
irreducible_components(L::ZZLat) -> Vector{ZZLat}
Return the irreducible components
This yields a maximal orthogonal splitting of L
as
Dual lattice
Discriminant group
See discriminant_group(L::ZZLat)
.
Overlattices
glue_map(L::ZZLat, S::ZZLat, R::ZZLat; check=true)
-> Tuple{TorQuadModuleMap, TorQuadModuleMap, TorQuadModuleMap}
Given three integral L
, S
and R
, with S
and R
primitive sublattices of L
and such that the sum of the ranks of S
and R
is equal to the rank of L
, return the glue map S
and R
.
Example
julia> M = root_lattice(:E,8);
julia> f = matrix(QQ, 8, 8, [-1 -1 0 0 0 0 0 0;
1 0 0 0 0 0 0 0;
0 1 1 0 0 0 0 0;
0 0 0 1 0 0 0 0;
0 0 0 0 1 0 0 0;
0 0 0 0 0 1 1 0;
-2 -4 -6 -5 -4 -3 -2 -3;
0 0 0 0 0 0 0 1]);
julia> S = kernel_lattice(M ,f-1)
Integer lattice of rank 4 and degree 8
with gram matrix
[12 -3 0 -3]
[-3 2 -1 0]
[ 0 -1 2 0]
[-3 0 0 2]
julia> R = kernel_lattice(M , f^2+f+1)
Integer lattice of rank 4 and degree 8
with gram matrix
[ 2 -1 0 0]
[-1 2 -6 0]
[ 0 -6 30 -3]
[ 0 0 -3 2]
julia> glue, iS, iR = glue_map(M, S, R)
(Map: finite quadratic module -> finite quadratic module, Map: finite quadratic module -> finite quadratic module, Map: finite quadratic module -> finite quadratic module)
julia> is_bijective(glue)
true
overlattice(glue_map::TorQuadModuleMap) -> ZZLat
Given the glue map of a primitive extension of L
.
Example
julia> M = root_lattice(:E,8);
julia> f = matrix(QQ, 8, 8, [ 1 0 0 0 0 0 0 0;
0 1 0 0 0 0 0 0;
1 2 4 4 3 2 1 2;
-2 -4 -6 -5 -4 -3 -2 -3;
2 4 6 4 3 2 1 3;
-1 -2 -3 -2 -1 0 0 -2;
0 0 0 0 0 -1 0 0;
-1 -2 -3 -3 -2 -1 0 -1]);
julia> S = kernel_lattice(M ,f-1)
Integer lattice of rank 4 and degree 8
with gram matrix
[ 2 -1 0 0]
[-1 2 -1 0]
[ 0 -1 12 -15]
[ 0 0 -15 20]
julia> R = kernel_lattice(M , f^4+f^3+f^2+f+1)
Integer lattice of rank 4 and degree 8
with gram matrix
[10 -4 0 1]
[-4 2 -1 0]
[ 0 -1 4 -3]
[ 1 0 -3 4]
julia> glue, iS, iR = glue_map(M, S, R);
julia> overlattice(glue) == M
true
local_modification(M::ZZLat, L::ZZLat, p)
Return a local modification of M
that matches L
at p
.
INPUT:
– a \mathbb{Z}_p
-maximal lattice– the a lattice isomorphic to M
over\QQ_p
– a prime number
OUTPUT:
an integral lattice M'
in the ambient space of M
such that M
and M'
are locally equal at all completions except at p
where M'
is locally isometric to the lattice L
.
maximal_integral_lattice(L::AbstractLat) -> AbstractLat
Given a lattice L
with integral norm, return a maximal integral overlattice M
of L
.
Sublattices defined by endomorphisms
kernel_lattice(L::ZZLat, f::MatElem;
ambient_representation::Bool = true) -> ZZLat
Given a
If ambient_representation
is true
(the default), the endomorphism is represented with respect to the ambient space of
invariant_lattice(L::ZZLat, G::Vector{MatElem};
ambient_representation::Bool = true) -> ZZLat
invariant_lattice(L::ZZLat, G::MatElem;
ambient_representation::Bool = true) -> ZZLat
Given a
If ambient_representation
is true
(the default), the endomorphism is represented with respect to the ambient space of
coinvariant_lattice(L::ZZLat, G::Vector{MatElem};
ambient_representation::Bool = true) -> ZZLat
coinvariant_lattice(L::ZZLat, G::MatElem;
ambient_representation::Bool = true) -> ZZLat
Given a invariant_lattice
).
If ambient_representation
is true
(the default), the endomorphism is represented with respect to the ambient space of
Computing embeddings
embed(S::ZZLat, G::Genus, primitive::Bool=true) -> Bool, embedding
Return a (primitive) embedding of the integral lattice S
into some lattice in the genus of G
.
julia> G = integer_genera((8,0), 1, even=true)[1];
julia> L, S, i = embed(root_lattice(:A,5), G);
embed_in_unimodular(S::ZZLat, pos::Int, neg::Int, primitive=true, even=true) -> Bool, L, S', iS, iR
Return a (primitive) embedding of the integral lattice S
into some (even) unimodular lattice of signature (pos, neg)
.
For now this works only for even lattices.
julia> NS = direct_sum(integer_lattice(:U), rescale(root_lattice(:A, 16), -1))[1];
julia> LK3, iNS, i = embed_in_unimodular(NS, 3, 19);
julia> genus(LK3)
Genus symbol for integer lattices
Signatures: (3, 0, 19)
Local symbol:
Local genus symbol at 2: 1^22
julia> iNS
Integer lattice of rank 18 and degree 22
with gram matrix
[0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 -2 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 1 -2 1 0 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -2 1 0]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -2 1]
[0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 -2]
julia> is_primitive(LK3, iNS)
true
LLL, Short and Close Vectors
LLL and indefinite LLL
lll(L::ZZLat, same_ambient::Bool = true) -> ZZLat
Given an integral L
with basis matrix B
, compute a basis C
of L
such that the gram matrix L
with respect to C
is LLL-reduced.
By default, it creates the lattice in the same ambient space as L
. This can be disabled by setting same_ambient = false
. Works with both definite and indefinite lattices.
Short Vectors
short_vectors(L::ZZLat, [lb = 0], ub, [elem_type = ZZRingElem]; check::Bool = true)
-> Vector{Tuple{Vector{elem_type}, QQFieldElem}}
Return all tuples (v, n)
such that lb <= n <= ub
, where G
is the Gram matrix of L
and v
is non-zero.
Note that the vectors are computed up to sign (so only one of v
and -v
appears).
It is assumed and checked that L
is definite.
See also short_vectors_iterator
for an iterator version.
shortest_vectors(L::ZZLat, [elem_type = ZZRingElem]; check::Bool = true)
-> QQFieldElem, Vector{elem_type}, QQFieldElem}
Return the list of shortest non-zero vectors in absolute value. Note that the vectors are computed up to sign (so only one of v
and -v
appears).
It is assumed and checked that L
is definite.
See also minimum
.
short_vectors_iterator(L::ZZLat, [lb = 0], ub,
[elem_type = ZZRingElem]; check::Bool = true)
-> Tuple{Vector{elem_type}, QQFieldElem} (iterator)
Return an iterator for all tuples (v, n)
such that lb <= n <= ub
, where G
is the Gram matrix of L
and v
is non-zero.
Note that the vectors are computed up to sign (so only one of v
and -v
appears).
It is assumed and checked that L
is definite.
See also short_vectors
.
minimum(L::ZZLat) -> QQFieldElem
Return the minimum absolute squared length among the non-zero vectors in L
.
kissing_number(L::ZZLat) -> Int
Return the Kissing number of the sphere packing defined by L
.
This is the number of non-overlapping spheres touching any other given sphere.
Close Vectors
close_vectors(L:ZZLat, v:Vector, [lb,], ub; check::Bool = false)
-> Vector{Tuple{Vector{Int}}, QQFieldElem}
Return all tuples (x, d)
where x
is an element of L
such that d = b(v - x, v - x) <= ub
. If lb
is provided, then also lb <= d
.
If filter
is not nothing
, then only those x
with filter(x)
evaluating to true
are returned.
By default, it will be checked whether L
is positive definite. This can be disabled setting check = false
.
Both input and output are with respect to the basis matrix of L
.
Examples
julia> L = integer_lattice(matrix(QQ, 2, 2, [1, 0, 0, 2]));
julia> close_vectors(L, [1, 1], 1)
3-element Vector{Tuple{Vector{ZZRingElem}, QQFieldElem}}:
([2, 1], 1)
([0, 1], 1)
([1, 1], 0)
julia> close_vectors(L, [1, 1], 1, 1)
2-element Vector{Tuple{Vector{ZZRingElem}, QQFieldElem}}:
([2, 1], 1)
([0, 1], 1)