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Lattices

Creation of lattices

Inside a given ambient space

lattice — Method
lattice(V::AbstractSpace) -> AbstractLat

Given an ambient space V, return the lattice with the standard basis matrix. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.

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lattice — Method
lattice(V::AbstractSpace, B::PMat ; check::Bool = true) -> AbstractLat

Given an ambient space V and a pseudo-matrix B, return the lattice spanned by the pseudo-matrix B inside V. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.

By default, B is checked to be of full rank. This test can be disabled by setting check to false.

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lattice — Method
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.

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lattice — Method
lattice(V::AbstractSpace, gens::Vector) -> AbstractLat

Given an ambient space V and a list of generators gens, return the lattice spanned by gens in V. If V is hermitian (resp. quadratic) then the output is a hermitian (resp. quadratic) lattice.

If gens is empty, the function returns the zero lattice in V.

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Quadratic lattice over a number field

quadratic_lattice — Method
quadratic_lattice(K::Field ; gram::MatElem) -> Union{ZZLat, QuadLat}

Given a matrix gram and a field K, return the free quadratic lattice inside the quadratic space over K with Gram matrix gram.

If $K = \mathbb{Q}$, then the output lattice is of type ZZLat, seen as a lattice over the ring $\mathbb{Z}$.

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quadratic_lattice — Method
quadratic_lattice(K::Field, B::PMat ; gram = nothing,
                                      check:::Bool = true) -> QuadLat

Given a pseudo-matrix B with entries in a field K return the quadratic lattice spanned by the pseudo-matrix B inside the quadratic space over K with Gram matrix gram.

If gram is not supplied, the Gram matrix of the ambient space will be the identity matrix over K of size the number of columns of B.

By default, B is checked to be of full rank. This test can be disabled by setting check to false.

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quadratic_lattice — Method
quadratic_lattice(K::Field, basis::MatElem ; gram = nothing,
                                             check::Bool = true)
                                                      -> Union{ZZLat, QuadLat}

Given a matrix basis and a field K, return the quadratic lattice spanned by the rows of basis inside the quadratic space over K with Gram matrix gram.

If gram is not supplied, the Gram matrix of the ambient space will be the identity matrix over K of size the number of columns of basis.

By default, basis is checked to be of full rank. This test can be disabled by setting check to false.

If $K = \mathbb{Q}$, then the output lattice is of type ZZLat, seen as a lattice over the ring $\mathbb{Z}$.

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quadratic_lattice — Method
quadratic_lattice(K::Field, gens::Vector ; gram = nothing) -> Union{ZZLat, QuadLat}

Given a list of vectors gens and a field K, return the quadratic lattice spanned by the elements of gens inside the quadratic space over K with Gram matrix gram.

If gram is not supplied, the Gram matrix of the ambient space will be the identity matrix over K of size the length of the elements of gens.

If gens is empty, gram must be supplied and the function returns the zero lattice in the quadratic space over K with gram matrix gram.

If $K = \mathbb{Q}$, then the output lattice is of type ZZLat, seen as a lattice over the ring $\mathbb{Z}$.

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Hermitian lattice over a degree 2 extension

hermitian_lattice — Method
hermitian_lattice(E::NumField; gram::MatElem) -> HermLat

Given a matrix gram and a number field E of degree 2, return the free hermitian lattice inside the hermitian space over E with Gram matrix gram.

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hermitian_lattice — Method
hermitian_lattice(E::NumField, B::PMat; gram = nothing,
			             check::Bool = true) -> HermLat

Given a pseudo-matrix B with entries in a number field E of degree 2, return the hermitian lattice spanned by the pseudo-matrix B inside the hermitian space over E with Gram matrix gram.

If gram is not supplied, the Gram matrix of the ambient space will be the identity matrix over E of size the number of columns of B.

By default, B is checked to be of full rank. This test can be disabled by setting check to false.

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hermitian_lattice — Method
hermitian_lattice(E::NumField, basis::MatElem; gram = nothing,
			                    check::Bool = true) -> HermLat

Given a matrix basis and a number field E of degree 2, return the hermitian lattice spanned by the rows of basis inside the hermitian space over E with Gram matrix gram.

If gram is not supplied, the Gram matrix of the ambient space will be the identity matrix over E of size the number of columns of basis.

By default, basis is checked to be of full rank. This test can be disabled by setting check to false.

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hermitian_lattice — Method
hermitian_lattice(E::NumField, gens::Vector ; gram = nothing) -> HermLat

Given a list of vectors gens and a number field E of degree 2, return the hermitian lattice spanned by the elements of gens inside the hermitian space over E with Gram matrix gram.

If gram is not supplied, the Gram matrix of the ambient space will be the identity matrix over E of size the length of the elements of gens.

If gens is empty, gram must be supplied and the function returns the zero lattice in the hermitan space over E with Gram matrix gram.

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Examples

The two following examples will be used all along this section:


julia> K, a = rationals_as_number_field();
julia> Kt, t = K["t"];
julia> g = t^2 + 7;
julia> E, b = number_field(g, "b");
julia> D = matrix(K, 3, 3, [2, 0, 0, 0, 2, 0, 0, 0, 2]);
julia> gens = Vector{AbsSimpleNumFieldElem}[map(K, [1, 1, 0]), map(K, [1, 0, 1]), map(K, [2, 0, 0])];
julia> Lquad = quadratic_lattice(K, gens, gram = D)Quadratic lattice of rank 3 and degree 3
  over maximal order of Number field of degree 1 over QQ
  with basis AbsSimpleNumFieldElem[1]
julia> D = matrix(E, 4, 4, [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [2, -1, 0, 0]), map(E, [-3, 0, -1, 0]), map(E, [0, 0, 0, -1]), map(E, [b, 0, 0, 0])];
julia> Lherm = hermitian_lattice(E, gens, gram = D)Hermitian lattice of rank 4 and degree 4
  over relative maximal order of Relative number field of degree 2 over number field
  with pseudo-basis
  (1, 1//1 * <1, 1>)
  (b + 1, 1//2 * <1, 1>)

Note that the format used here is the one given by the internal function Hecke.to_hecke() which prints REPL commands to get back the input lattice.


julia> K, a = rationals_as_number_field();
julia> Kt, t = K["t"];
julia> g = t^2 + 7;
julia> E, b = number_field(g, "b");
julia> D = matrix(E, 4, 4, [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [2, -1, 0, 0]), map(E, [-3, 0, -1, 0]), map(E, [0, 0, 0, -1]), map(E, [b, 0, 0, 0])];
julia> Lherm = hermitian_lattice(E, gens, gram = D);
julia> Hecke.to_hecke(Lherm)Qx, x = polynomial_ring(QQ, :x)
f = x - 1
K, a = number_field(f, :a, cached = false)
Kt, t = polynomial_ring(K, :t)
g = t^2 + 7
E, b = number_field(g, :b, cached = false)
D = matrix(E, 4, 4, [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1])
gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [2, -1, 0, 0]), map(E, [-3, 0, -1, 0]), map(E, [0, 0, 0, -1]), map(E, [b, 0, 0, 0])]
L = hermitian_lattice(E, gens, gram = D)

Finally, one can access some databases in which are stored several quadratic and hermitian lattices. Up to now, these are not automatically available while running Hecke. It can nonethelss be used in the following way:


julia> qld = Hecke.quadratic_lattice_database()Quadratic lattices of rank >= 3 with class number 1 or 2
Author: Markus Kirschmer
Source: http://www.math.rwth-aachen.de/~Markus.Kirschmer/forms/
Version: 0.0.1
Number of lattices: 30250
julia> lattice(qld, 1)Quadratic lattice of rank 3 and degree 3
  over maximal order of Number field of degree 1 over QQ
  with basis AbsSimpleNumFieldElem[1]
julia> hlb = Hecke.hermitian_lattice_database()Hermitian lattices of rank >= 3 with class number 1 or 2
Author: Markus Kirschmer
Source: http://www.math.rwth-aachen.de/~Markus.Kirschmer/forms/
Version: 0.0.1
Number of lattices: 570
julia> lattice(hlb, 426)Hermitian lattice of rank 4 and degree 4
  over relative maximal order of Relative number field of degree 2 over number field
  with pseudo-basis
  (1, 1//1 * <1, 1>)
  (b + 1, 1//2 * <1, 1>)

Ambient space and rational span

ambient_space — Method
ambient_space(L::AbstractLat) -> AbstractSpace

Return the ambient space of the lattice L. If the ambient space is not known, an error is raised.

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rational_span — Method
rational_span(L::AbstractLat) -> AbstractSpace

Return the rational span of the lattice L.

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Examples


julia> K, a = rationals_as_number_field();
julia> Kt, t = K["t"];
julia> g = t^2 + 7;
julia> E, b = number_field(g, "b");
julia> D = matrix(K, 3, 3, [2, 0, 0, 0, 2, 0, 0, 0, 2]);
julia> gens = Vector{AbsSimpleNumFieldElem}[map(K, [1, 1, 0]), map(K, [1, 0, 1]), map(K, [2, 0, 0])];
julia> Lquad = quadratic_lattice(K, gens, gram = D);
julia> D = matrix(E, 4, 4, [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [2, -1, 0, 0]), map(E, [-3, 0, -1, 0]), map(E, [0, 0, 0, -1]), map(E, [b, 0, 0, 0])];
julia> Lherm = hermitian_lattice(E, gens, gram = D);
julia> ambient_space(Lherm)Hermitian space of dimension 4
  over relative number field with defining polynomial t^2 + 7
    over number field with defining polynomial x - 1
      over rational field
with gram matrix
[1   0   0   0]
[0   1   0   0]
[0   0   1   0]
[0   0   0   1]
julia> rational_span(Lquad)Quadratic space of dimension 3
  over number field of degree 1 over QQ
with gram matrix
[2   2   2]
[2   4   2]
[2   2   4]
julia> basis_matrix_of_rational_span(Lherm)[1   0   0   0]
[5   1   0   0]
[3   0   1   0]
[0   0   0   1]
julia> gram_matrix_of_rational_span(Lherm)[1    5    3   0]
[5   26   15   0]
[3   15   10   0]
[0    0    0   1]
julia> diagonal_of_rational_span(Lquad)3-element Vector{AbsSimpleNumFieldElem}:
 2
 2
 2

Rational equivalence

hasse_invariant — Method
hasse_invariant(L::AbstractLat, p::Union{InfPlc, AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Int

Return the Hasse invariant of the rational span of the lattice L at the place p. The lattice must be quadratic.

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witt_invariant — Method
witt_invariant(L::AbstractLat, p::Union{InfPlc, AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}) -> Int

Return the Witt invariant of the rational span of the lattice L at the place p. The lattice must be quadratic.

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is_rationally_isometric — Method
is_rationally_isometric(L::AbstractLat, M::AbstractLat, p::Union{InfPlc, AbsNumFieldOrderIdeal})
                                                                     -> Bool

Return whether the rational spans of the lattices L and M are isometric over the completion at the place p.

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is_rationally_isometric — Method
is_rationally_isometric(L::AbstractLat, M::AbstractLat) -> Bool

Return whether the rational spans of the lattices L and M are isometric.

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Examples

For now and for the rest of this section, the examples will include the new lattice Lquad2 which is quadratic. Moreover, all the completions are going to be done at the prime ideal $p = 7*\mathcal O_K$.


julia> K, a = rationals_as_number_field();
julia> D = matrix(K, 3, 3, [2, 0, 0, 0, 2, 0, 0, 0, 2]);
julia> gens = Vector{AbsSimpleNumFieldElem}[map(K, [1, 1, 0]), map(K, [1, 0, 1]), map(K, [2, 0, 0])];
julia> Lquad = quadratic_lattice(K, gens, gram = D);
julia> D = matrix(K, 3, 3, [2, 0, 0, 0, 2, 0, 0, 0, 2]);
julia> gens = Vector{AbsSimpleNumFieldElem}[map(K, [-35, 25, 0]), map(K, [30, 40, -20]), map(K, [5, 10, -5])];
julia> Lquad2 = quadratic_lattice(K, gens, gram = D)Quadratic lattice of rank 3 and degree 3
  over maximal order of Number field of degree 1 over QQ
  with basis AbsSimpleNumFieldElem[1]
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 7)[1][1]<7, 7>
Norm: 7
Minimum: 7
principal generator 7
two normal wrt: 7
julia> hasse_invariant(Lquad, p), witt_invariant(Lquad, p)(1, 1)
julia> is_rationally_isometric(Lquad, Lquad2, p)true
julia> is_rationally_isometric(Lquad, Lquad2)true

Attributes

Let $L$ be a lattice over $E/K$. We call a pseudo-basis of $L$ any sequence of pairs $(\mathfrak A_i, x_i)_{1 \leq i \leq n}$ where the $\mathfrak A_i$'s are fractional (left) ideals of $\mathcal O_E$ and $(x_i)_{1 \leq i \leq n}$ is a basis of the rational span of $L$, and such that

\[ L = \bigoplus_{i = 1}^n \mathfrak A_ix_i.\]

Note that a pseudo-basis is not unique. Given a pseudo-basis $(\mathfrak A_i, x_i)_{1 \leq i \leq n}$ of $L$, we define the corresponding pseudo-matrix of $L$ to be the datum consisting of a list of coefficient ideals corresponding to the ideals $\mathfrak A_i$'s and a matrix whose rows are the coordinates of the $x_i$'s in the canonical basis of the ambient space of $L$ (conversely, given any such pseudo-matrix, one can define the corresponding pseudo-basis).

rank — Method
rank(L::AbstractLat) -> Int

Return the rank of the underlying module of the lattice L.

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degree — Method
degree(L::AbstractLat) -> Int

Return the dimension of the ambient space of the lattice L.

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discriminant — Method
discriminant(L::AbstractLat) -> AbsSimpleNumFieldOrderFractionalIdeal

Return the discriminant of the lattice L, that is, the generalized index ideal $[L^\# : L]$.

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base_field — Method
base_field(L::AbstractLat) -> Field

Return the algebra over which the rational span of the lattice L is defined.

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base_ring — Method
base_ring(L::AbstractLat) -> Ring

Return the order over which the lattice L is defined.

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fixed_field — Method
fixed_field(L::AbstractLat) -> Field

Returns the fixed field of the involution of the lattice L.

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fixed_ring — Method
fixed_ring(L::AbstractLat) -> Ring

Return the maximal order in the fixed field of the lattice L.

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involution — Method
involution(L::AbstractLat) -> Map

Return the involution of the rational span of the lattice L.

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pseudo_matrix — Method
pseudo_matrix(L::AbstractLat) -> PMat

Return a basis pseudo-matrix of the lattice L.

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pseudo_basis — Method
pseudo_basis(L::AbstractLat) -> Vector{Tuple{Vector, Ideal}}

Return a pseudo-basis of the lattice L.

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coefficient_ideals — Method
coefficient_ideals(L::AbstractLat) -> Vector{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}}

Return the coefficient ideals of a pseudo-basis of the lattice L.

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absolute_basis_matrix — Method
absolute_basis_matrix(L::AbstractLat) -> MatElem

Return a $\mathbf{Z}$-basis matrix of the lattice L.

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absolute_basis — Method
absolute_basis(L::AbstractLat) -> Vector

Return a $\mathbf{Z}$-basis of the lattice L.

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generators — Method
generators(L::AbstractLat; minimal = false) -> Vector{Vector}

Return a set of generators of the lattice L over the base ring of L.

If minimal == true, the number of generators is minimal. Note that computing minimal generators is expensive.

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gram_matrix_of_generators — Method
gram_matrix_of_generators(L::AbstractLat; minimal::Bool = false) -> MatElem

Return the Gram matrix of a generating set of the lattice L.

If minimal == true, then a minimal generating set is used. Note that computing minimal generators is expensive.

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Examples


julia> K, a = rationals_as_number_field();
julia> Kt, t = K["t"];
julia> g = t^2 + 7;
julia> E, b = number_field(g, "b");
julia> D = matrix(E, 4, 4, [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [2, -1, 0, 0]), map(E, [-3, 0, -1, 0]), map(E, [0, 0, 0, -1]), map(E, [b, 0, 0, 0])];
julia> Lherm = hermitian_lattice(E, gens, gram = D);
julia> rank(Lherm), degree(Lherm)(4, 4)
julia> discriminant(Lherm)Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <7, 7>) * [1 0]
(1//2 * <7, 7>) * [0 1]
julia> base_field(Lherm)Relative number field with defining polynomial t^2 + 7
  over number field with defining polynomial x - 1
    over rational field
julia> base_ring(Lherm)Relative maximal order of Relative number field of degree 2 over number field
with pseudo-basis
(1, 1//1 * <1, 1>)
(b + 1, 1//2 * <1, 1>)
julia> fixed_field(Lherm)Number field with defining polynomial x - 1
  over rational field
julia> fixed_ring(Lherm)Maximal order of Number field of degree 1 over QQ
with basis AbsSimpleNumFieldElem[1]
julia> involution(Lherm)Map
  from relative number field of degree 2 over number field
  to relative number field of degree 2 over number field
julia> pseudo_matrix(Lherm)Pseudo-matrix over Relative maximal order of Relative number field of degree 2 over number field
with pseudo-basis
(1, 1//1 * <1, 1>)
(b + 1, 1//2 * <1, 1>)
Fractional ideal with row [1 0 0 0]
Fractional ideal with row [5 1 0 0]
Fractional ideal with row [3 0 1 0]
Fractional ideal with row [0 0 0 1]
julia> pseudo_basis(Lherm)4-element Vector{Tuple{Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}, Hecke.RelNumFieldOrderFractionalIdeal{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal, Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}}}:
 ([1, 0, 0, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <7, 28>) * [1 0]
(1//2 * <1, 1>) * [6 1])
 ([5, 1, 0, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1])
 ([3, 0, 1, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1])
 ([0, 0, 0, 1], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1])
julia> coefficient_ideals(Lherm)4-element Vector{Hecke.RelNumFieldOrderFractionalIdeal{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal, Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}}:
 Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <7, 28>) * [1 0]
(1//2 * <1, 1>) * [6 1]
 Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1]
 Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1]
 Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1]
julia> absolute_basis_matrix(Lherm)[            7               0               0               0]
[1//2*b + 7//2               0               0               0]
[            5               1               0               0]
[5//2*b + 5//2   1//2*b + 1//2               0               0]
[            3               0               1               0]
[3//2*b + 3//2               0   1//2*b + 1//2               0]
[            0               0               0               1]
[            0               0               0   1//2*b + 1//2]
julia> absolute_basis(Lherm)8-element Vector{Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}}:
 [7, 0, 0, 0]
 [1//2*b + 7//2, 0, 0, 0]
 [5, 1, 0, 0]
 [5//2*b + 5//2, 1//2*b + 1//2, 0, 0]
 [3, 0, 1, 0]
 [3//2*b + 3//2, 0, 1//2*b + 1//2, 0]
 [0, 0, 0, 1]
 [0, 0, 0, 1//2*b + 1//2]
julia> generators(Lherm)4-element Vector{Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}}:
 [2, -1, 0, 0]
 [-3, 0, -1, 0]
 [0, 0, 0, -1]
 [b, 0, 0, 0]
julia> gram_matrix_of_generators(Lherm)[  5     -6   0   -2*b]
[ -6     10   0    3*b]
[  0      0   1      0]
[2*b   -3*b   0      7]

Module operations

Let $L$ be a lattice over $E/K$ inside the space $(V, \Phi)$. The dual lattice of $L$ is defined to be the following lattice over $E/K$ in $(V, \Phi)$:

\[ L^{\#} = \left\{ x \in V \mid \Phi(x,L) \subseteq \mathcal O_E \right\}.\]

For any fractional (left) ideal $\mathfrak a$ of $\mathcal O_E$, one can define the lattice $\mathfrak aL$ to be the lattice over $E/K$, in the same space $(V, \Phi)$, obtained by rescaling the coefficient ideals of a pseudo-basis of $L$ by $\mathfrak a$. In another flavour, for any non-zero element $a \in K$, one defines the rescaled lattice $L^a$ to be the lattice over $E/K$ with the same underlying module as $L$ (i.e. the same pseudo-bases) but in space $(V, a\Phi)$.

+ — Method
+(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.

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* — Method
*(a::NumFieldElem, L::AbstractLat) -> AbstractLat

Return the lattice $aL$ inside the ambient space of the lattice L.

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* — Method
*(a::NumFieldOrderIdeal, L::AbstractLat) -> AbstractLat

Return the lattice $aL$ inside the ambient space of the lattice L.

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* — Method
*(a::NumFieldOrderFractionalIdeal, L::AbstractLat) -> AbstractLat

Return the lattice $aL$ inside the ambient space of the lattice L.

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rescale — Method
rescale(L::AbstractLat, a::NumFieldElem) -> AbstractLat

Return the rescaled lattice $L^a$. Note that this has a different ambient space than the lattice L.

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dual — Method
dual(L::AbstractLat) -> AbstractLat

Return the dual lattice of the lattice L.

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intersect — Method
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.

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primitive_closure — Method
primitive_closure(M::AbstractLat, N::AbstractLat) -> AbstractLat

Given two lattices M and N defined over a number field E, with $N \subseteq E\otimes M$, return the primitive closure $M \cap E\otimes N$ of N in M.

One can also use the alias saturate(L, M).

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orthogonal_submodule — Method
orthogonal_submodule(L::AbstractLat, M::AbstractLat) -> AbstractLat

Return the largest submodule of L orthogonal to M.

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Examples


julia> K, a = rationals_as_number_field();
julia> D = matrix(K, 3, 3, [2, 0, 0, 0, 2, 0, 0, 0, 2]);
julia> gens = Vector{AbsSimpleNumFieldElem}[map(K, [1, 1, 0]), map(K, [1, 0, 1]), map(K, [2, 0, 0])];
julia> Lquad = quadratic_lattice(K, gens, gram = D);
julia> D = matrix(K, 3, 3, [2, 0, 0, 0, 2, 0, 0, 0, 2]);
julia> gens = Vector{AbsSimpleNumFieldElem}[map(K, [-35, 25, 0]), map(K, [30, 40, -20]), map(K, [5, 10, -5])];
julia> Lquad2 = quadratic_lattice(K, gens, gram = D);
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 7)[1][1];
julia> pseudo_matrix(Lquad + Lquad2)Pseudo-matrix over Maximal order of Number field of degree 1 over QQ
with basis AbsSimpleNumFieldElem[1]
1//1 * <2, 2> with row [1 0 0]
1//1 * <1, 1> with row [1 1 0]
1//1 * <1, 1> with row [1 0 1]
julia> pseudo_matrix(intersect(Lquad, Lquad2))Pseudo-matrix over Maximal order of Number field of degree 1 over QQ
with basis AbsSimpleNumFieldElem[1]
1//1 * <10, 10> with row [1 0 0]
1//1 * <25, 25> with row [1//5 1 0]
1//1 * <5, 5> with row [0 3 1]
julia> pseudo_matrix(p*Lquad)Pseudo-matrix over Maximal order of Number field of degree 1 over QQ
with basis AbsSimpleNumFieldElem[1]
1//1 * <14, 126> with row [1 0 0]
1//1 * <7, 7> with row [1 1 0]
1//1 * <7, 7> with row [1 0 1]
julia> ambient_space(rescale(Lquad,3*a))Quadratic space of dimension 3
  over number field of degree 1 over QQ
with gram matrix
[6   0   0]
[0   6   0]
[0   0   6]
julia> pseudo_matrix(Lquad)Pseudo-matrix over Maximal order of Number field of degree 1 over QQ
with basis AbsSimpleNumFieldElem[1]
1//1 * <2, 2> with row [1 0 0]
1//1 * <1, 1> with row [1 1 0]
1//1 * <1, 1> with row [1 0 1]

Categorical constructions

Given finite collections of lattices, one can construct their direct sums, which are also direct products in this context. They are also sometimes called biproducts. Depending on the user usage, it is possible to call one of the following functions.

direct_sum — Method
direct_sum(x::Vararg{T}) where T <: AbstractLat -> T, Vector{AbstractSpaceMor}
direct_sum(x::Vector{T}) where T <: AbstractLat -> T, Vector{AbstractSpaceMor}

Given a collection of quadratic or hermitian lattices $L_1, \ldots, L_n$, return their direct sum $L := L_1 \oplus \ldots \oplus L_n$, together with the injections $L_i \to L$ (seen as maps between the corresponding ambient spaces).

For objects of type AbstractLat, 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 $L \to L_i$, one should call direct_product(x). If one wants to obtain L as a biproduct with the injections $L_i \to L$ and the projections $L \to L_i$, one should call biproduct(x).

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direct_sum(g1::QuadSpaceCls, g2::QuadSpaceCls) -> QuadSpaceCls

Return the isometry class of the direct sum of two representatives.

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direct_sum(M::ModuleFP{T}...; task::Symbol = :sum) where T

Given modules $M_1\dots M_n$, say, return the direct sum $\bigoplus_{i=1}^n M_i$.

Additionally, return

  • a vector containing the canonical injections $M_i\to\bigoplus_{i=1}^n M_i$ if task = :sum (default),
  • a vector containing the canonical projections $\bigoplus_{i=1}^n M_i\to M_i$ if task = :prod,
  • two vectors containing the canonical injections and projections, respectively, if task = :both,
  • none of the above maps if task = :none.
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direct_product — Method
direct_product(algebras::StructureConstantAlgebra...; task::Symbol = :sum)
  -> StructureConstantAlgebra, Vector{AbsAlgAssMor}, Vector{AbsAlgAssMor}
direct_product(algebras::Vector{StructureConstantAlgebra}; task::Symbol = :sum)
  -> StructureConstantAlgebra, Vector{AbsAlgAssMor}, Vector{AbsAlgAssMor}

Returns the algebra $A = A_1 \times \cdots \times A_k$. task can be ":sum", ":prod", ":both" or ":none" and determines which canonical maps are computed as well: ":sum" for the injections, ":prod" for the projections.

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direct_product(x::Vararg{T}) where T <: AbstractLat -> T, Vector{AbstractSpaceMor}
direct_product(x::Vector{T}) where T <: AbstractLat -> T, Vector{AbstractSpaceMor}

Given a collection of quadratic or hermitian lattices $L_1, \ldots, L_n$, return their direct product $L := L_1 \times \ldots \times L_n$, together with the projections $L \to L_i$ (seen as maps between the corresponding ambient spaces).

For objects of type AbstractLat, 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 $L_i \to L$, one should call direct_sum(x). If one wants to obtain L as a biproduct with the injections $L_i \to L$ and the projections $L \to L_i$, one should call biproduct(x).

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direct_product(F::FreeMod{T}...; task::Symbol = :prod) where T

Given free modules $F_1\dots F_n$, say, return the direct product $\prod_{i=1}^n F_i$.

Additionally, return

  • a vector containing the canonical projections $\prod_{i=1}^n F_i\to F_i$ if task = :prod (default),
  • a vector containing the canonical injections $F_i\to\prod_{i=1}^n F_i$ if task = :sum,
  • two vectors containing the canonical projections and injections, respectively, if task = :both,
  • none of the above maps if task = :none.
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direct_product(M::ModuleFP{T}...; task::Symbol = :prod) where T

Given modules $M_1\dots M_n$, say, return the direct product $\prod_{i=1}^n M_i$.

Additionally, return

  • a vector containing the canonical projections $\prod_{i=1}^n M_i\to M_i$ if task = :prod (default),
  • a vector containing the canonical injections $M_i\to\prod_{i=1}^n M_i$ if task = :sum,
  • two vectors containing the canonical projections and injections, respectively, if task = :both,
  • none of the above maps if task = :none.
source
biproduct — Method
biproduct(x::Vararg{T}) where T <: AbstractLat -> T, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}
biproduct(x::Vector{T}) where T <: AbstractLat -> T, Vector{AbstractSpaceMor}, Vector{AbstractSpaceMor}

Given a collection of quadratic or hermitian lattices $L_1, \ldots, L_n$, return their biproduct $L := L_1 \oplus \ldots \oplus L_n$, together with the injections $L_i \to L$ and the projections $L \to L_i$ (seen as maps between the corresponding ambient spaces).

For objects of type AbstractLat, 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 $L_i \to L$, one should call direct_sum(x). If one wants to obtain L as a direct product with the projections $L \to L_i$, one should call direct_product(x).

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Invariants

Let $L$ be a lattice over $E/K$, in the space $(V, \Phi)$. We define:

  • the norm $\mathfrak n(L)$ of $L$ to be the ideal of $\mathcal O_K$ generated by the squares $\left\{\Phi(x,x) \mid x \in L \right\}$;
  • the scale $\mathfrak s(L)$ of $L$ to be the set $\Phi(L,L) = \left\{\Phi(x,y) \mid x,y \in L \right\}$;
  • the volume $\mathfrak v(L)$ of $L$ to be the index ideal

\[ \lbrack L^{\#} \colon L \rbrack_{\mathcal O_E} := \langle \left\{ \sigma \mid \sigma \in \text{Hom}_{\mathcal O_E}(L^{\#}, L) \right\} \rangle_{\mathcal O_E}.\]

norm — Method
norm(L::AbstractLat) -> AbsNumFieldOrderFractionalIdeal

Return the norm of the lattice L. This is a fractional ideal of the fixed field of L.

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scale — Method
scale(L::AbstractLat) -> AbsSimpleNumFieldOrderFractionalIdeal

Return the scale of the lattice L.

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volume — Method
volume(L::AbstractLat) -> AbsSimpleNumFieldOrderFractionalIdeal

Return the volume of the lattice L.

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Examples


julia> K, a = rationals_as_number_field();
julia> Kt, t = K["t"];
julia> g = t^2 + 7;
julia> E, b = number_field(g, "b");
julia> D = matrix(E, 4, 4, [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [2, -1, 0, 0]), map(E, [-3, 0, -1, 0]), map(E, [0, 0, 0, -1]), map(E, [b, 0, 0, 0])];
julia> Lherm = hermitian_lattice(E, gens, gram = D);
julia> norm(Lherm)1//1 * <1, 1>
Norm: 1
Minimum: 1
principal generator 1
basis_matrix
[1]
two normal wrt: 2
julia> scale(Lherm)Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1]
julia> volume(Lherm)Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <7, 7>) * [1 0]
(1//2 * <7, 7>) * [0 1]

Predicates

Let $L$ be a lattice over $E/K$. It is said to be integral if its scale is an integral ideal, i.e. it is contained in $\mathcal O_E$. Moreover, if $\mathfrak p$ is a prime ideal in $\mathcal O_K$, then $L$ is said to be modular (resp. locally modular at $\mathfrak p$) if there exists a fractional ideal $\mathfrak a$ of $\mathcal O_E$ (resp. an integer $v$) such that $\mathfrak aL^{\#} = L$ (resp. $\mathfrak p^vL_{\mathfrak p}^{\#} = L_{\mathfrak p}$).

is_integral — Method
is_integral(L::AbstractLat) -> Bool

Return whether the lattice L is integral.

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is_modular — Method
is_modular(L::AbstractLat) -> Bool, AbsSimpleNumFieldOrderFractionalIdeal

Return whether the lattice L is modular. In this case, the second returned value is a fractional ideal $\mathfrak a$ of the base algebra of L such that $\mathfrak a L^\# = L$, where $L^\#$ is the dual of L.

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is_modular — Method
is_modular(L::AbstractLat, p) -> Bool, Int

Return whether the completion $L_{p}$ of the lattice L at the prime ideal or integer p is modular. If it is the case the second returned value is an integer v such that $L_{p}$ is $p^v$-modular.

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is_positive_definite — Method
is_positive_definite(L::AbstractLat) -> Bool

Return whether the rational span of the lattice L is positive definite.

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is_negative_definite — Method
is_negative_definite(L::AbstractLat) -> Bool

Return whether the rational span of the lattice L is negative definite.

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is_definite — Method
is_definite(L::AbstractLat) -> Bool

Return whether the rational span of the lattice L is definite.

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can_scale_totally_positive — Method
can_scale_totally_positive(L::AbstractLat) -> Bool, NumFieldElem

Return whether there is a totally positive rescaled lattice of the lattice L. If so, the second returned value is an element $a$ such that $L^a$ is totally positive.

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Examples


julia> K, a = rationals_as_number_field();
julia> Kt, t = K["t"];
julia> g = t^2 + 7;
julia> E, b = number_field(g, "b");
julia> D = matrix(E, 4, 4, [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [2, -1, 0, 0]), map(E, [-3, 0, -1, 0]), map(E, [0, 0, 0, -1]), map(E, [b, 0, 0, 0])];
julia> Lherm = hermitian_lattice(E, gens, gram = D);
julia> OK = maximal_order(K);
julia> is_integral(Lherm)true
julia> is_modular(Lherm)[1]false
julia> p = prime_decomposition(OK, 7)[1][1];
julia> is_modular(Lherm, p)(false, 0)
julia> is_positive_definite(Lherm)true
julia> can_scale_totally_positive(Lherm)(true, 1)

Local properties

local_basis_matrix — Method
local_basis_matrix(L::AbstractLat, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}; type = :any) -> MatElem

Given a prime ideal p and a lattice L, return a basis matrix of a lattice M such that $M_{p} = L_{p}$. Note that if p is an ideal in the base ring of L, the completions are taken at the minimum of p (which is an ideal in the base ring of the order of p).

  • If type == :submodule, the lattice M will be a sublattice of L.
  • If type == :supermodule, the lattice M will be a superlattice of L.
  • If type == :any, there may not be any containment relation between M and L.
source
jordan_decomposition — Method
jordan_decomposition(L::AbstractLat, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem})
                            -> Vector{MatElem}, Vector{MatElem}, Vector{Int}

Return a Jordan decomposition of the completion of the lattice L at a prime ideal p.

The returned value consists of three lists $(M_i)_i$, $(G_i)_i$ and $(s_i)_i$ of the same length $r$. The completions of the row spans of the matrices $M_i$ yield a Jordan decomposition of $L_{p}$ into modular sublattices $L_i$ with Gram matrices $G_i$ and scale of $p$-adic valuation $s_i$.

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is_isotropic — Method
is_isotropic(L::AbstractLat, p::Union{AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}, InfPlc}) -> Bool

Return whether the completion of the lattice L at the place p is isotropic.

source

Examples


julia> K, a = rationals_as_number_field();
julia> D = matrix(K, 3, 3, [2, 0, 0, 0, 2, 0, 0, 0, 2]);
julia> gens = Vector{AbsSimpleNumFieldElem}[map(K, [1, 1, 0]), map(K, [1, 0, 1]), map(K, [2, 0, 0])];
julia> Lquad = quadratic_lattice(K, gens, gram = D);
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 7)[1][1];
julia> local_basis_matrix(Lquad, p)[1   0   0]
[1   1   0]
[1   0   1]
julia> jordan_decomposition(Lquad, p)(AbstractAlgebra.Generic.MatSpaceElem{AbsSimpleNumFieldElem}[[1 0 0; 0 1 0; 0 0 1]], AbstractAlgebra.Generic.MatSpaceElem{AbsSimpleNumFieldElem}[[2 0 0; 0 2 0; 0 0 2]], [0])
julia> is_isotropic(Lquad, p)true

Automorphisms for definite lattices

Let $L$ and $L'$ be two lattices over the same extension $E/K$, inside their respective ambient spaces $(V, \Phi)$ and $(V', \Phi')$. Similarly to homomorphisms of spaces, we define a homomorphism of lattices from $L$ to $L'$ to be an $\mathcal{O}_E$-module$ homomorphism $f \colon L \to L'$ such that for all $x,y \in L$, one has

\[ \Phi'(f(x), f(y)) = \Phi(x,y).\]

Again, any automorphism of lattices is called an isometry and any monomorphism is called an embedding. We refer to the set of isometries from a lattice $L$ to itself as the automorphism group of $L$.

automorphism_group_order — Method
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.

source
automorphism_group_generators — Method
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.

source

Examples


julia> K, a = rationals_as_number_field();
julia> Kt, t = K["t"];
julia> g = t^2 + 7;
julia> E, b = number_field(g, "b");
julia> D = matrix(K, 3, 3, [2, 0, 0, 0, 2, 0, 0, 0, 2]);
julia> gens = Vector{AbsSimpleNumFieldElem}[map(K, [1, 1, 0]), map(K, [1, 0, 1]), map(K, [2, 0, 0])];
julia> Lquad = quadratic_lattice(K, gens, gram = D);
julia> is_definite(Lquad)true
julia> automorphism_group_order(Lquad)48
julia> automorphism_group_generators(Lquad)6-element Vector{AbstractAlgebra.Generic.MatSpaceElem{AbsSimpleNumFieldElem}}:
 [-1 0 0; 0 -1 0; 0 0 -1]
 [1 0 0; 0 -1 0; 0 0 -1]
 [1 0 0; 0 0 -1; 0 -1 0]
 [0 -1 0; 0 0 -1; 1 0 0]
 [1 0 0; 0 1 0; 0 0 -1]
 [0 1 0; 1 0 0; 0 0 1]

Isometry

is_isometric — Method
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.

source
is_isometric_with_isometry — Method
is_isometric_with_isometry(L::AbstractLat, M::AbstractLat; ambient_representation::Bool = true
                                                           depth::Int = -1, bacher_depth::Int = 0)
                                                          -> (Bool, MatElem)

Return whether the lattices L and M are isometric. If this is the case, the second returned value is an isometry T from L to M.

By default, that isometry is represented with respect to the bases of the ambient spaces, that is, $T V_M T^t = V_L$ where $V_L$ and $V_M$ are the Gram matrices of the ambient spaces of L and M respectively. If ambient_representation == false, then the isometry is represented with respect to the (pseudo-)bases of L and M, that is, $T G_M T^t = G_L$ where $G_M$ and $G_L$ are the Gram matrices of the (pseudo-)bases of L and M respectively.

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.

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is_locally_isometric — Method
is_locally_isometric(L::AbstractLat, M::AbstractLat, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Bool

Return whether the completions of the lattices L and M at the prime ideal p are isometric.

source

Examples


julia> K, a = rationals_as_number_field();
julia> D = matrix(K, 3, 3, [2, 0, 0, 0, 2, 0, 0, 0, 2]);
julia> gens = Vector{AbsSimpleNumFieldElem}[map(K, [1, 1, 0]), map(K, [1, 0, 1]), map(K, [2, 0, 0])];
julia> Lquad = quadratic_lattice(K, gens, gram = D);
julia> D = matrix(K, 3, 3, [2, 0, 0, 0, 2, 0, 0, 0, 2]);
julia> gens = Vector{AbsSimpleNumFieldElem}[map(K, [-35, 25, 0]), map(K, [30, 40, -20]), map(K, [5, 10, -5])];
julia> Lquad2 = quadratic_lattice(K, gens, gram = D);
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 7)[1][1];
julia> is_isometric(Lquad, Lquad2)false
julia> is_locally_isometric(Lquad, Lquad2, p)true

Maximal integral lattices

is_maximal_integral — Method
is_maximal_integral(L::AbstractLat, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Bool, AbstractLat

Given a lattice L and a prime ideal p of the fixed ring $\mathcal O_K$ of L, return whether the completion of L at p has integral norm and that L has no proper overlattice satisfying this property.

If the norm of L is not integral at p, the second output is L by default. Otherwise, either L is maximal at p and the second output is L, or the second output is a lattice M in the ambient space of L whose completion at p is a minimal overlattice of $L_p$ with integral norm.

source
is_maximal_integral — Method
is_maximal_integral(L::AbstractLat) -> Bool, AbstractLat

Given a lattice L, return whether L has integral norm and has no proper overlattice satisfying this property.

If the norm of L is not integral, the second output is L by default. Otherwise, either L is maximal and the second output is L, or the second output is a minimal overlattice M of L with integral norm.

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is_maximal — Method
is_maximal(L::AbstractLat, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> Bool, AbstractLat

Given a lattice L and a prime ideal p in the fixed ring $\mathcal O_K$ of L such that the norm of $L_p$ is integral, return whether L is maximal integral at p.

If L is locally maximal at p, the second output is L, otherwise it is a lattice M in the same ambient space of L whose completion at p has integral norm and is a proper overlattice of $L_p$.

source
maximal_integral_lattice — Method
maximal_integral_lattice(L::AbstractLat, p::AbsNumFieldOrderIdeal{AbsSimpleNumField, AbsSimpleNumFieldElem}) -> AbstractLat

Given a lattice L and a prime ideal p of the fixed ring $\mathcal O_K$ of L such that the norm of $L_p$ is integral, return a lattice M in the ambient space of L which is maximal integral at p and which agrees with L locally at all the places different from p.

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maximal_integral_lattice — Method
maximal_integral_lattice(L::AbstractLat) -> AbstractLat

Given a lattice L with integral norm, return a maximal integral overlattice M of L.

source
maximal_integral_lattice — Method
maximal_integral_lattice(V::AbstractSpace) -> AbstractLat

Given a space V, return a lattice in V with integral norm and which is maximal in V satisfying this property.

source

Examples


julia> K, a = rationals_as_number_field();
julia> Kt, t = K["t"];
julia> g = t^2 + 7;
julia> E, b = number_field(g, "b");
julia> D = matrix(E, 4, 4, [1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1]);
julia> gens = Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}[map(E, [2, -1, 0, 0]), map(E, [-3, 0, -1, 0]), map(E, [0, 0, 0, -1]), map(E, [b, 0, 0, 0])];
julia> Lherm = hermitian_lattice(E, gens, gram = D);
julia> OK = maximal_order(K);
julia> p = prime_decomposition(OK, 7)[1][1];
julia> is_maximal_integral(Lherm, p)(false, Hermitian lattice of rank 4 and degree 4)
julia> is_maximal_integral(Lherm)(false, Hermitian lattice of rank 4 and degree 4)
julia> is_maximal(Lherm, p)(false, Hermitian lattice of rank 4 and degree 4)
julia> pseudo_basis(maximal_integral_lattice(Lherm, p))4-element Vector{Tuple{Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}, Hecke.RelNumFieldOrderFractionalIdeal{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal, Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}}}:
 ([1, 0, 0, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1])
 ([0, 1, 0, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1])
 ([2, 4, 1, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//14 * <1, 1>) * [6 1])
 ([4, 5, 0, 1], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//14 * <1, 1>) * [6 1])
julia> pseudo_basis(maximal_integral_lattice(Lherm))4-element Vector{Tuple{Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}, Hecke.RelNumFieldOrderFractionalIdeal{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal, Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}}}:
 ([1, 0, 0, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1])
 ([0, 1, 0, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1])
 ([2, 4, 1, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//14 * <1, 1>) * [6 1])
 ([4, 5, 0, 1], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//14 * <1, 1>) * [6 1])
julia> pseudo_basis(maximal_integral_lattice(ambient_space(Lherm)))4-element Vector{Tuple{Vector{Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}, Hecke.RelNumFieldOrderFractionalIdeal{AbsSimpleNumFieldElem, AbsSimpleNumFieldOrderFractionalIdeal, Hecke.RelSimpleNumFieldElem{AbsSimpleNumFieldElem}}}}:
 ([1, 0, 0, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1])
 ([0, 1, 0, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//2 * <1, 1>) * [0 1])
 ([3, 2, 1, 0], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//14 * <1, 1>) * [6 1])
 ([5, 3, 0, 1], Fractional ideal of
Relative maximal order with pseudo-basis (1) * 1//1 * <1, 1>, (b + 1) * 1//2 * <1, 1>
with basis pseudo-matrix
(1//1 * <1, 1>) * [1 0]
(1//14 * <1, 1>) * [6 1])