Enumeration of isometries

admissible_triples(C::ZZGenus, p::Integer; pA::Int = -1,
                                           nA::Int = -1,
                                           pB::Int = -1,
                                           nB::Int = -1,
                                           b::Int = 0)
                                           -> Vector{Tuple{ZZGenus, ZZGenus}}

Given a $\mathbb Z$-genus $C$ and a prime number $p$, return all tuples of $\mathbb Z$-genera $(A, B)$ such that $(A, B, C)$ is $p$-admissible and $B$ is of rank divisible by $p-1$.

One can choose the positive signatures for the genera $A$ and $B$ in output respectively by setting pA and pB to the desired values. Similarly with the negative signatures nA and nB. The function returns an error if the choice of these values is inconsistent.

If b is set to 0, we allow in output the trivial pair, i.e. when $B$ is the genus of rank 0 lattices. Otherwise, if b is set to 1, the trivial pair is discarded.

Examples

julia> L = root_lattice(:A,5);

julia> g = genus(L)
Genus symbol for integer lattices
Signatures: (5, 0, 0)
Local symbols:
  Local genus symbol at 2: 1^-4 2^1_7
  Local genus symbol at 3: 1^-4 3^1

julia> admissible_triples(g, 5)
2-element Vector{Tuple{ZZGenus, ZZGenus}}:
 (Genus symbol: II_(5, 0) 2^-1_3 3^1, Genus symbol: II_(0, 0))
 (Genus symbol: II_(1, 0) 2^1_7 3^1 5^1, Genus symbol: II_(4, 0) 5^1)

julia> admissible_triples(g, 2)
8-element Vector{Tuple{ZZGenus, ZZGenus}}:
 (Genus symbol: II_(5, 0) 2^-1_3 3^1, Genus symbol: II_(0, 0))
 (Genus symbol: II_(4, 0) 2^2_6 3^1, Genus symbol: II_(1, 0) 2^1_1)
 (Genus symbol: II_(3, 0) 2^-3_1 3^1, Genus symbol: II_(2, 0) 2^2_2)
 (Genus symbol: II_(3, 0) 2^3_3, Genus symbol: II_(2, 0) 2^-2 3^1)
 (Genus symbol: II_(2, 0) 2^-2 3^1, Genus symbol: II_(3, 0) 2^3_3)
 (Genus symbol: II_(2, 0) 2^2_2, Genus symbol: II_(3, 0) 2^-3_1 3^1)
 (Genus symbol: II_(1, 0) 2^1_1, Genus symbol: II_(4, 0) 2^2_6 3^1)
 (Genus symbol: II_(0, 0), Genus symbol: II_(5, 0) 2^-1_3 3^1)

is_admissible_triple(A::ZZGenus, B::ZZGenus, C::ZZGenus, p::Integer) -> Bool

Given a triple of $\mathbb Z$-genera $(A, B, C)$ and a prime number $p$, such that the rank of $B$ is divisible by $p-1$, return whether $(A, B, C)$ is $p$-admissible.

Examples

A standard example is the following: let $(L, f)$ be a lattice with isometry of prime order $p$, let $F:= L^f$ and $C:= L_f$ be respectively the invariant and coinvariant sublattices of $(L, f)$. Then, the triple of genera $(g(F), g(C), g(L))$ is $p$-admissible.

julia> L = root_lattice(:A,5);

julia> f = matrix(QQ, 5, 5, [1  1  1  1  1;
                             0 -1 -1 -1 -1;
                             0  1  0  0  0;
                             0  0  1  0  0;
                             0  0  0  1  0]);

julia> Lf = integer_lattice_with_isometry(L, f);

julia> F = invariant_lattice(Lf);

julia> C = coinvariant_lattice(Lf);

julia> is_admissible_triple(genus(F), genus(C), genus(Lf), 5)
true

representatives_of_hermitian_type(G::ZZGenus, chi::Union{ZZPolyRingElem, QQPolyRingElem}; first::Bool=false)
representatives_of_hermitian_type(L::ZZLat, chi::Union{ZZPolyRingElem, QQPolyRingElem}; first::Bool=false)
                                                               -> Vector{ZZLatWithIsom}

Given a non-empty genus of integer lattices $G$ and a polynomial $chi$ irreducible over $\mathbb Q$, such that the equation order of the associated number field is maximal, return a list of representatives of isomorphism classes of pairs $(M, g)$ consisting of a lattice $M$ in $G$ and $g \in O(M)$ is an isometry of minimal polynomial $chi$.

One can also provide a representative $L$ of $G$ instead.

If first is set to true, only return the first representative computed.

representatives_of_hermitian_type(G::ZZGenus, m::Int; first::Bool=false)
representatives_of_hermitian_type(L::ZZLat, m::Int; first::Bool=false)
                                                   -> Vector{ZZLatWithIsom}

Given a non-empty genus of integer lattices $G$, return a list of representatives of isomorphism classes of pairs $(M, g)$ consisting of a lattice $M$ in $G$ and $g \in O(M)$ is an isometry of minimal polynomial $\Phi_m(X)$, the $m-$th cyclotomic polynomial.

If $m = 1,2$, this goes back to enumerate $G$ as a genus of integer lattices.

One can also provide a representative $L$ of $G$ instead.

If first is set to true, only return the first representative computed.

enumerate_classes_of_lattices_with_isometry(L::ZZLat, order::IntegerUnion)
                                                        -> Vector{ZZLatWithIsom}
enumerate_classes_of_lattices_with_isometry(G::ZZGenus, order::IntegerUnion)
                                                        -> Vector{ZZLocalGenus}

Given an even integer lattice $L$, return representatives of isomorphism classes of lattice with isometry $(M ,g)$ where $M$ is in the genus of $L$, and $g$ has order order. Alternatively, one can input a given genus symbol $G$ for even integer lattices as an input - the function first computes a representative of $G$.

Note that currently we support only orders which admit at most 2 prime divisors.

enumerate_prime_power_isometries(L::ZZLat, q::IntegerUnion, vq::IntegerUnion) -> Vector{ZZLatWithIsom}
enumerate_prime_power_isometries(G::ZZGenus, q::IntegerUnion, vq::IntegerUnion) -> Vector{ZZLatWithIsom}

Given a genus $G$ of even integer lattices, or a representative $L$ of $G$, return representatives of isomorphism classes of lattices with isometry $(M, g)$ where $M$ is a lattice in $G$ and $g$ has order $q^vq$.

$q$ must be a prime number and $vq$ must be a positive integer.

representatives_of_hermitian_type(Lf::ZZLatWithIsom, m::Int = 1)
                                                   -> Vector{ZZLatWithIsom}

Given a lattice with isometry $(L, f)$ of finite hermitian type (i.e. the minimal polynomial of $f$ is irreducible cyclotomic) and a positive integer $m$, return a set of representatives of isomorphism classes of lattices with isometry $(M, g)$ of finite hermitian type such that the type of $(M, g^m)$ is equal to the type of $(L, f)$.

Note that in this case, the isometries $g$'s are of order $nm$.

Examples

julia> L = root_lattice(:A,2);

julia> Lf = integer_lattice_with_isometry(L);

julia> reps = representatives_of_hermitian_type(Lf, 6)
1-element Vector{ZZLatWithIsom}:
 Integer lattice with isometry of finite order 6

julia> is_of_hermitian_type(reps[1])
true

splitting_of_hermitian_prime_power(Lf::ZZLatWithIsom, p::Int) -> Vector{ZZLatWithIsom}

Given an even lattice with isometry $(L, f)$ of hermitian type with $f$ of order $q^e$ for some prime number $q$, and given another prime number $p \neq q$, return a set of representatives of the isomorphism classes of lattices with isometry $(M, g)$ such that the type of $(M, g^p)$ is equal to the type of $(L, f)$.

Note that $e$ can be 0.

Examples

julia> L = root_lattice(:A,2);

julia> f = matrix(QQ, 2, 2, [0 1; -1 -1]);

julia> Lf = integer_lattice_with_isometry(L, f);

julia> is_of_hermitian_type(Lf)
true

julia> reps = splitting_of_hermitian_prime_power(Lf, 2)
2-element Vector{ZZLatWithIsom}:
 Integer lattice with isometry of finite order 6
 Integer lattice with isometry of finite order 3

julia> all(is_of_hermitian_type, reps)
true

julia> is_of_same_type(Lf, reps[1]^2)
true

julia> is_of_same_type(Lf, reps[2]^2)
true

splitting_of_prime_power(Lf::ZZLatWithIsom, p::Int, b::Int = 0)
                                                   -> Vector{ZZLatWithIsom}

Given an even lattice with isometry $(L, f)$ with $f$ of order $q^e$ for some prime number $q$, a prime number $p \neq q$ and an integer $b = 0, 1$, return a set of representatives of the isomorphism classes of lattices with isometry $(M, g)$ such that the type of $(M, g^p)$ is equal to the type of $(L, f)$.

If b == 1, return only the lattices with isometry $(M, g)$ where $g$ has order $pq^e$.

Note that $e$ can be 0.

Examples

julia> L = root_lattice(:A,2);

julia> Lf = integer_lattice_with_isometry(L);

julia> splitting_of_prime_power(Lf, 2)
4-element Vector{ZZLatWithIsom}:
 Integer lattice with isometry of finite order 2
 Integer lattice with isometry of finite order 2
 Integer lattice with isometry of finite order 2
 Integer lattice with isometry of finite order 1

julia> splitting_of_prime_power(Lf, 3, 1)
1-element Vector{ZZLatWithIsom}:
 Integer lattice with isometry of finite order 3

splitting_of_pure_mixed_prime_power(Lf::ZZLatWithIsom, p::Int)
                                             -> Vector{ZZLatWithIsom}

Given an even lattice with isometry $(L, f)$ and a prime number $p$, such that $\prod_{i=0}^e\Phi_{p^dq^i}(f)$ is trivial for some $d > 0$ and $e \geq 0$, return a set of representatives of the isomorphism classes of lattices with isometry $(M, g)$ such that the type of $(M, g^p)$ is equal to the type of $(L, f)$.

Note that $e$ can be 0, while $d$ has to be positive.

splitting_of_mixed_prime_power(Lf::ZZLatWithIsom, p::Int, b::Int = 1)
                                      -> Vector{ZZLatWithIsom}

Given an even lattice with isometry $(L, f)$ and a prime number $p$ such that $f$ has order $p^dq^e$ for some prime number $q \neq p$, return a set of representatives of the isomorphism classes of lattices with isometry $(M, g)$ such that the type of $(M, g^p)$ is equal to the type of $(L, f)$.

If b == 1, return only the lattices with isometry $(M, g)$ where $g$ has order $p^{d+1}q^e$.

Note that $d$ and $e$ can be both 0.

Examples

julia> L = root_lattice(:E,7);

julia> f = matrix(QQ, 7, 7, [ 1  1  2  1  0  0  1;
                             -1 -2 -3 -2 -1 -1 -1;
                              0  1  2  1  1  1  1;
                              0  0 -1 -1 -1 -1 -1;
                              1  1  2  2  2  1  1;
                              0  0 -1 -1 -1  0  0;
                              0  0  0  1  0  0  0]);

julia> Lf = integer_lattice_with_isometry(L, f)
Integer lattice of rank 7 and degree 7
  with isometry of finite order 6
  given by
  [ 1    1    2    1    0    0    1]
  [-1   -2   -3   -2   -1   -1   -1]
  [ 0    1    2    1    1    1    1]
  [ 0    0   -1   -1   -1   -1   -1]
  [ 1    1    2    2    2    1    1]
  [ 0    0   -1   -1   -1    0    0]
  [ 0    0    0    1    0    0    0]

julia> reps = splitting_of_mixed_prime_power(Lf, 2)
2-element Vector{ZZLatWithIsom}:
 Integer lattice with isometry of finite order 12
 Integer lattice with isometry of finite order 12

julia> all(LL -> is_of_same_type(Lf, LL^2), reps)
true