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The genus of an exact lattice has a distinct type SymGen
which holds a representative lattice, and the local data defining
the genus. Each genus consists of 2n spinor genera, for some
integer n, typically 1. The spinor genera share the same
type SymGen. Unlike the genus, the spinor genus is not
determined solely by the local data of the genus, so the cached
representative is necessary to define the spinor class.
Equality testing of genera is fast, since this requires only
a comparison of the canonical local information. It is also
possible to enumerate representatives of all equivalences classes
in a genus or spinor genus. This is done by a process of exploration
of the p-neighbour graph, for an appropriate prime p.
The neighbouring functions can be applied to individual lattices to
find p-neighbours or the closure under the p-neighbour process.
Functions for computing and comparing the local p-adic equivalence
classes of lattices, mediated by the type SymGenLoc.
Subsections
Genus(G) : SymGen -> SymGen
Given an exact lattice L or a spinor genus G this function
returns the genus of L. If given a genus the function returns
G itself.
Given an exact lattice L, returns the spinor genus of L.
Given a genus G, returns the sequence of spinor genera.
If G is a spinor genus, then this function returns the sequence
consisting of G itself.
Returns a representative lattice for the genus symbol G.
Returns true if and only if G is a spinor genus. This is the
negation of IsGenus(G).
Returns true if and only if G is a genus. This is the negation
of IsSpinorGenus(G).
Returns the determinant of the genus symbol G.
Returns the sequence of p-adic genera of the genus symbol G.
Returns a representative lattice for the genus symbol G.
Given two genus symbols, return true if and only if they represent
the same genus. This computation is fast for genera, but currently
for spinor genera invokes a call to Representatives.
The number of isometry classes in the genus or spinor genus G.
Enumeration of isometry classes is done by an explicit call to
Representatives, so that #G is an expensive computation.
Return the spinor characters of the genus symbol G as a sequence of Dirichlet characters
whose kernels intersect exactly in the group of automorphous
numbers. Consult Conway and Sloane [JC98]
for precise definitions and significance of the spinor kernel
and automorphous numbers.
Return the spinor generators of the genus symbol G as a sequence of primes which generate
the group of spinor norms. The primes generate a group dual
to that generated by the spinor characters.
AutomorphousClasses(G,p) : SymGen, RngIntElt -> RngIntElt
A set of integer representatives of the p-adic square classes
in the image of the spinor norm of the lattice L (respectively the genus symbol G).
Returns true if and only if p is coprime to 2 and the determinant,
and p is the norm of an element of the spinor kernel of G.
Return the prime p for which G represents the p-adic genus.
Returns a canonical representative lattice of the p-adic genus G,
with Gram matrix in Jordan form. For odd p the Jordan form is
diagonalized.
This function
returns a canonical p-adic representative of the determinant of the p-adic
genus G.
The determinant is well-defined only up to squares.
Return the dimension of the p-adic
genus G.
Given local genus symbols G1 and G2, return true if and only if they
have the same prime and the same canonical Jordan form.
Neighbor(L, v, p) : Lat, LatElt, RngIntElt -> Lat
Let L be an integral lattice, p a prime which does not divide
( Determinant)(L) and v a vector in L - pL with
(v, v) ∈p2 Z.
The p-neighbour of L with respect to v is the lattice generated
by Lv and p - 1 v, where Lv := { x ∈L | (x, v) ∈p Z }.
See [Kne57] for the original definition and [SP91]
for a generalization of the neighbouring method.
Neighbors(L, p) : Lat, RngIntElt -> Lat
For an integral lattice L and prime p, returns the sequence of
p-neighbours of L.
NeighborClosure(L, p) : Lat, RngIntElt -> Lat
Bound: RngIntElt Default: 2^{24}
Depth: RngIntElt Default:
For an integral lattice L and prime p, returns the sequence of
lattices obtained by transitive closure of the p-neighbours of L.
Note that neighbours with respect to two vectors v1, v2 whose
images in L/pL lie in the same projective orbit of ( Aut)(L)
on L/pL are isometric. Therefore only projective orbit
representatives of the action of ( Aut)(L) on L/pL are used.
The large number of orbits restricts the complexity of this
algorithm, hence the function gives an error if p^(( Rank)(L))
is greater than Bound, by default set to 224.
The system chooses the value for the Depth parameter by default,
for the calls to AutomorphismGroup and IsIsometric (see the
section on automorphism group and isometry testing).
The default choice is ( Min)( ⌊(( Rank)(L) - 5)/2 ⌋, 4 ).
This can be overruled by setting Depth := d.
SpinorRepresentatives(L) : Lat -> [ Lat ]
Representatives(G) : SymGen -> [ Lat ]
Bound: RngIntElt Default: 2^{24}
Depth: RngIntElt Default:
For an exact lattice L with genus or spinor genus G, this
function enumerates the isometry classes in G by constructing
the p-neighbour closure (up to isometry). This construction used
using an appropriate prime or primes p not dividing the determinant
of L. For the genus, sufficiently many primes p are chosen to
generate the full image, modulo the spinor kernel, of each character
defining the spinor kernel. The parameters are exactly as for the
NeighbourClosure function.
For a genus or spinor genus G, this function determines the
adjacency matrix of the p-neighbour graph on the representative
classes for G. The integer p must be prime, and if G is a
spinor genus, then an error ensues if p is not an automorphous
number for G.
We construct the root lattice E8 (the unique even unimodular lattice of
dimension 8) as a 2-neighbour of the 8-dimensional standard lattice.
> Z8 := StandardLattice(8);
> v := Z8 ! [1,1,1,1,1,1,1,1];
> E8 := Neighbour(Z8, v, 2);
> E8;
Lattice of rank 8 and degree 8
Basis:
( 2 0 0 0 0 0 0 2)
( 2 0 0 0 0 0 0 -2)
( 1 1 -1 1 1 -1 1 1)
( 1 1 -1 -1 -1 -1 -1 -1)
( 1 -1 -1 -1 -1 -1 1 -1)
( 0 0 2 0 0 0 0 2)
( 0 0 0 0 2 0 0 2)
( 0 0 0 0 0 2 0 2)
Basis denominator: 2
The so-obtained lattice is in fact identical to the one returned by the standard
construction.
> L := Lattice("E", 8);
> L;
Lattice of rank 8 and degree 8
Basis:
( 4 0 0 0 0 0 0 0)
(-2 2 0 0 0 0 0 0)
( 0 -2 2 0 0 0 0 0)
( 0 0 -2 2 0 0 0 0)
( 0 0 0 -2 2 0 0 0)
( 0 0 0 0 -2 2 0 0)
( 0 0 0 0 0 -2 2 0)
( 1 1 1 1 1 1 1 1)
Basis Denominator: 2
> E8 eq L;
true
In this example we enumerate representatives for the genus of the
Coxeter-Todd lattice, performing the major steps manually.
The whole computation can be done simply by calling the
GenusRepresentatives function but the example illustrates
how the function actually works.
We use a combination of the automorphism group, isometry and neighbouring
functions. The idea is that the neighbouring graph spans the full genus
which therefore can be computed by successively generating neighbours
and checking them for isometry with already known ones. The automorphism
group comes into play, since neighbours with respect to vectors in the
same projective orbit under the automorphism group are isometric.
> L := CoordinateLattice(Lattice("Kappa", 12));
> G := AutomorphismGroup(L);
> G2 := ChangeRing(G, GF(2));
> O := LineOrbits(G2);
> [ Norm(L!Rep(o).1) : o in O ];
[ 4, 8, 10 ]
Hence only the first and second orbits give rise to a 2-neighbour.
To obtain an even neighbour, the second vector has to be adjusted
by an element of 2 * L such that it has norm divisible by 8.
> v1 := L ! Rep(O[1]).1;
> v1 +:= 2 * Rep({ u : u in Basis(L) | (v1,u) mod 2 eq 1 });
> v2 := L ! Rep(O[2]).1;
> Norm(v1), Norm(v2);
16 8
> L1 := Neighbour(L, v1, 2);
> L2 := Neighbour(L, v2, 2);
> bool := IsIsometric(L, L1); bool;
true
> bool := IsIsometric(L, L2); bool;
false
So we obtain only one non-isometric even neighbour of L.
To obtain the full genus we can now proceed with L2 in the
same way, and do this with the following function EvenGenus.
Note that this function is simply one component of the function
GenusRepresentatives.
> function EvenGenus(L)
> // Start with the lattice L
> Lambda := [ CoordinateLattice(LLL(L)) ];
> cand := 1;
> while cand le #Lambda do
> L := Lambda[cand];
> G := ChangeRing( AutomorphismGroup(L), GF(2) );
> // Get the projective orbits on L/2L
> O := LineOrbits(G);
> for o in O do
> v := L ! Rep(o).1;
> if Norm(v) mod 4 eq 0 then
> // Adjust the vector such that its norm is divisible by 8
> if not Norm(v) mod 8 eq 0 then
> v +:= 2 * Rep({ u : u in Basis(L) | (v,u) mod 2 eq 1 });
> end if;
> N := LLL(Neighbour(L, v, 2));
> new := true;
> for i in [1..#Lambda] do
> if IsIsometric(Lambda[i], N) then
> new := false;
> break i;
> end if;
> end for;
> if new then
> Append(~Lambda, CoordinateLattice(N));
> end if;
> end if;
> end for;
> cand +:= 1;
> end while;
> return Lambda;
> end function;
>
> time Lambda := EvenGenus(L);
Time: 9.300
> #Lambda;
10
> [ Minimum(L) : L in Lambda ];
[ 4, 2, 2, 2, 2, 2, 2, 2, 2, 2 ]
> &+[ 1/#AutomorphismGroup(L) : L in Lambda ];
4649359/4213820620800
We see that the genus consists of 10 classes of lattices where only the
Coxeter-Todd lattice has minimum 4 and get the mass of the genus as
4649359/4213820620800.
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