Predicates of Orders

Let \(O\) be a quaternion order with base ring \({\mathbb{Z}}\), \({\mathbb{F}}_q[X]\) with \(q\) odd, or a number ring. Then Magma can test the following predicates.

IsMaximal(O): AlgAssVOrd -> BoolElt: Returns true if and only if the order \(O\) is maximal.

IspMaximal(O, p): AlgAssVOrd, RngOrdIdl -> BoolElt
IspMaximal(O, p): AlgQuatOrd, RngElt -> BoolElt: Returns true if and only if the order \(O\) is maximal at the prime or prime ideal \(p\).

IsEichler(O): AlgAssVOrd -> BoolElt, AlgAssVOrd, AlgAssVOrd

MaximalOrders: BoolElt                    Default: false

Returns true if and only if the order \(O\) is Eichler, that is an intersection of two (not necessarily distinct) maximal orders. The function calls the EichlerInvariant intrinsic explained below.

If the optional argument MaximalOrders is set to true, the algorithm also returns two maximal orders such that \(O\) is their intersection.

IsEichler(O, p): AlgAssVOrd, RngOrdIdl -> BoolElt, AlgAssVOrd, AlgAssVOrd

IsEichler(O, p): AlgQuatOrd, RngElt -> BoolElt, AlgQuatOrd, AlgQuatOrd

MaximalOrders: BoolElt                    Default: false

Returns true if and only if the completion of the order \(O\) at the prime (ideal) \(p\) is Eichler.

If the optional argument MaximalOrders is set to true, the algorithm also returns two \(p\)-maximal orders such that \(O\) is their intersection.

EichlerInvariant(O, p): AlgAssVOrd, RngOrdIdl -> RngIntElt
EichlerInvariant(O, p): AlgQuatOrd, RngElt -> RngIntElt: Returns the local Eichler invariant of \(O\) at some prime (ideal) \(p\) which divides the discriminant of \(O\). Let \(R\) be the base ring of \(O\) and let \(J\) be the Jacobson radical of the \(R/p\)-algebra \(O/pO\). If \(J\) has dimension \(3\) then the Eichler invariant is defined to be \(0\). Otherwise the quotient of \(O/pO\) by \(J\) is either isomorphic to a direct sum of two copies of \(R/p\) or a quadratic field extension of \(R/p\). In the first case the Eichler invariant is \(1\), in the latter it is \(-1\).

IsHereditary(O): AlgAssVOrd -> BoolElt: Returns true if and only if the order \(O\) is a hereditary order in a quaternion algebra \(A\). That is, every lattice in \(A\) of full rank such that \(O\) is contained in its left order is a projective left \(O\)-module. The hereditary orders are precisely those with squarefree discriminant.

IsHereditary(O, p): AlgAssVOrd, RngOrdIdl -> BoolElt
IsHereditary(O, p): AlgQuatOrd, RngElt -> BoolElt: Returns true if and only if the completion of the order \(O\) at the prime (ideal) \(p\) is hereditary.

IsGorenstein(O): AlgAssVOrd -> BoolElt, .: Returns true if and only if the order \(O\) is a Gorenstein order. That is, the dual of \(O\) with respect to the trace bilinear form is a projective \(O\)-module. The second return value is the Brandt invariant of \(O\) as in the GorensteinClosure intrinsic.

IsGorenstein(O, p): AlgAssVOrd, RngOrdIdl -> BoolElt, RngIntElt
IsGorenstein(O, p): AlgQuatOrd, RngElt -> BoolElt, RngIntElt: Returns true if and only if the completion of the order \(O\) at the prime (ideal) \(p\) is Gorenstein. The second return value is the valuation of the Brandt invariant of \(O\) at \(p\).

IsBass(O): AlgAssVOrd -> BoolElt: Returns true if and only if the order \(O\) is a Bass order, i.e. every order which contains \(O\) is Gorenstein.

IsBass(O, p): AlgAssVOrd, RngOrdIdl -> BoolElt
IsBass(O, p): AlgQuatOrd, RngElt -> BoolElt: Returns true if and only if the completion of the order \(O\) at the prime (ideal) \(p\) is Bass.

IsSameType(O1, O2): AlgAssVOrd, AlgAssVOrd -> BoolElt: Returns true if and only if the two quaternion orders \(O1\) and \(O2\) are of the same type which means that they are locally isomorphic. The orders must be over the ring of integers or a number ring.

Predicates of Orders#

Predicates of Orders