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For definite quaternion orders or ideals one can compute reduced bases
and Gram matrices.
If the base ring of the order or ideal is Z or Fq[X] with q odd,
the Gram matrices can be made unique up to isomorphism. In fact, in
these cases, the isomorphism testing of ideals and orders is based on
this reduction.
Given a quaternion algebra A over any field F not of characteristic
2, this function returns the underlying F-space with inner product
the norm form. A map from A into the structure is returned as second
value.
NormModule(S) : AlgQuatOrd -> ModTupRng, Map
Given a quaternion order S over Z or Fq[X] (with q odd),
this function returns the underlying module over its base ring, with
inner product respect to the norm. A map from O into the structure
is returned as second value.
GramMatrix(I) : AlgQuatOrdIdl -> AlgMatElt
The Gram matrix of the quaternion order S or ideal I over Z
or Fq[X] with respect to the norm on the basis for S.
ReducedGramMatrix(S) : AlgQuatOrdIdl[RngInt] -> AlgMatElt
Given an order or ideal S over Z in a definite quaternion algebra,
this function returns the Gram matrix G of the corresponding lattice.
The quaternion ideal machinery makes use of a Minkowski basis reduction
algorithm which returns a unique normalized reduced Gram matrix G
for any definite quaternion ideal. This forms the core of the isomorphism
testing for quaternion ideals.
ReducedBasis(S) : AlgQuatOrdIdl[RngInt] -> SeqEnum
Given an order or ideal S over Z in a definite quaternion algebra,
this function returns some basis B of S the Gram matrix G of the
corresponding lattice associated with S. Note that while there exists
a unique Minkowski-reduced Gram matrix G, the basis B is not unique.
Recall that Minkowski basis reduction is used which returns a unique
normalized reduced Gram matrix G for any definite quaternion ideal.
This forms the core of the isomorphism testing for quaternion ideals.
We illustrate this by applying it to representatives of the set of left
ideal classes of an order.
> A := QuaternionOrder(19,2);
> ideals := LeftIdealClasses(A);
> #ideals;
5
> [ (1/Norm(I))*ReducedGramMatrix(I) : I in ideals ];
[
[ 2 0 1 1]
[ 0 2 1 1]
[ 1 1 20 1]
[ 1 1 1 20],
[6 0 1 3]
[0 6 3 1]
[1 3 8 1]
[3 1 1 8],
[6 0 1 3]
[0 6 3 1]
[1 3 8 1]
[3 1 1 8],
[ 4 0 1 -1]
[ 0 4 1 1]
[ 1 1 10 0]
[-1 1 0 10],
[ 4 0 1 -1]
[ 0 4 1 1]
[ 1 1 10 0]
[-1 1 0 10]
]
ReducedGramMatrix(S) : AlgQuatOrdIdl[RngUPol] -> AlgMatElt, SeqEnum
ReducedBasis(S) : AlgQuatOrd[RngUPol] -> SeqEnum
ReducedBasis(S) : AlgQuatOrdIdl[RngUPol] -> SeqEnum
Canonical: BoolElt Default: false
Given an order or ideal S over Fq[X] in a quaternion algebra A,
this function returns a Gram matrix and/or a basis of S whose Gram matrix
is in dominant diagonal form (see the function DominantDiagonalForm
in Section Automorphism Group and Isometry Testing over Fq[t]).
The Gram matrix will not be unique unless A is definite and
Canonical is set to true.
ReducedBasis(I) : AlgAssVOrdIdl[RngOrd] -> [AlgAssVOrdElt]
Returns a "reduced" basis for the order O or the ideal I over some
number ring. If O or I arise from a definite quaternion algebra,
then this basis is LLL-reduced with respect to the norm form; otherwise,
the basis is reduced with respect to a Minkowski-like embedding
(see [KV10, Section 4]).
OptimisedRepresentation(O) : AlgAssVOrd -> AlgQuat, Map
Given an order O contained in a quaternion algebra A over Q or
a number field F, this function returns a new quaternion algebra A'
such that A' = ((a, b)/F) where a and b are small (with respect to O),
and, as second return value, an isomorphism A to A'.
OptimisedRepresentation(A) : AlgQuat -> AlgQuat, Map
Given a quaternion algebra A over Q or a number field F,
this function returns a new quaternion algebra A' such that A' = ((a, b)/F)
where a and b are small. An isomorphism A to A' is returned
as second value.
Enumerate(O, B) : AlgQuatOrd[RngInt], RngIntElt -> [AlgQuatOrdElt]
Enumerate(O, A, B) : AlgQuatOrdIdl[RngInt], RngElt, RngElt -> [AlgQuatElt]
Enumerate(O, B) : AlgQuatOrd[RngInt], RngElt -> [AlgQuatElt]
The sequence of all elements x (up to sign) in the definite quaternion
order O or ideal I over Z such that the reduced norm of x
lies in the interval [A, ... B] or [0, ... B], respectively.
Enumerate(O, B) : AlgAssVOrd[RngOrd], RngElt -> [AlgAssVOrdElt]
Enumerate(O, B) : AlgAssVOrd[RngOrd], [RngElt] -> [AlgAssVOrdElt]
Enumerate(I, B) : AlgAssVOrdIdl[RngOrd], [RngElt] -> [AlgAssVOrdElt]
The sequence of elements x (up to sign) in the definite quaternion order O or
ideal I over a number ring such that the absolute trace of the norm of x
lies in the interval [A, ... B] or [0, ... B], respectively.
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