Units and Unit Groups

Let \(S\) be a definite quaternion order over \({\mathbb{Z}}\), \({\mathbb{F}}_q[X]\) with \(q\) odd, or a number ring. In the first two cases, the unit group of \(S\) is finite and can be read off any reduced Gram matrix of \(S\). If the base ring of \(S\) is some number ring \(R\), then an explicit description of the finite quotient \(S^*/R^*\) is given in [Vig76].

NormOneGroup(S): AlgAssVOrd -> GrpPerm, Map

NormOneGroup(S): AlgQuatOrd -> GrpAb, Map

ModScalars: BoolElt                    Default: false

Returns a group \(G\) isomorphic to the group \(S_1\) of elements in \(S\) with reduced norm \(1\) (unless ModScalars is set, in which case \(G\) is isomorphic to \(S_1\) modulo \(\{\pm 1\}\)).

The second object returned is a map from \(G\) to \(S\) expressing the isomorphism.

Units(S): AlgQuatOrd -> SeqEnum

Units(S): AlgAssVOrd -> SeqEnum

This intrinsic computes the set of units \(S^*\) for the definite order \(S\). When the base ring of \(S\) is \(Z\) or \({\mathbb{F}}_q[X]\), the returned sequence contains all units of \(S\); when the base field is a number field, the returned sequence contains representatives modulo the unit group of the base ring (since the unit group is infinite in general).

MultiplicativeGroup(S): AlgQuatOrd[RngInt] -> GrpPerm, Map

UnitGroup(S): AlgQuatOrd[RngInt] -> GrpPerm, Map

MultiplicativeGroup(S): AlgQuatOrd[RngUPol] -> GrpAb, Map

UnitGroup(S): AlgQuatOrd[RngUPol] -> GrpAb, Map

MultiplicativeGroup(S): AlgAssVOrd[RngOrd] -> GrpPerm, Map

UnitGroup(S): AlgAssVOrd[RngOrd] -> GrpPerm, Map

This intrinsic computes the unit group \(S^*\) of the definite quaternion order \(S\). The function returns an abstract group \(G\), and a map from \(G\) to \(S\). When the base ring of \(S\) is \(Z\) or \({\mathbb{F}}_q[X]\), \(G\) represents the full group of units of \(S\); when the base field is a number field, \(G\) represents the unit group of \(S\) modulo the unit group of the base ring (since the unit group is infinite in general).

Example: Unit Groups over the Rationals (ex-410b55)

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The following example illustrates the unit group computation for an order in a definite quaternion algebra over \({\mathbb{Q}}\).

> A := QuaternionAlgebra< RationalField() | -1, -1 >;
> S1 := MaximalOrder(A);
> S2 := QuaternionOrder(A,2);
> G1, h1 := UnitGroup(S1);
> #G1;
24
> [ A | h1(g) : g in G1 ];
[ 1, -1, -j, -k, i, -1/2 + 1/2*i - 1/2*j - 1/2*k, 1/2 + 1/2*i - 1/2*j + 1/2*k,
-1/2 - 1/2*i - 1/2*j + 1/2*k, -1/2 + 1/2*i + 1/2*j + 1/2*k, 1/2 - 1/2*i - 1/2*j
- 1/2*k, 1/2 + 1/2*i + 1/2*j - 1/2*k, 1/2 - 1/2*i + 1/2*j + 1/2*k, -1/2 - 1/2*i
+ 1/2*j - 1/2*k, 1/2 - 1/2*i + 1/2*j - 1/2*k, -1/2 + 1/2*i + 1/2*j - 1/2*k, 1/2
+ 1/2*i - 1/2*j - 1/2*k, -1/2 - 1/2*i - 1/2*j - 1/2*k, 1/2 + 1/2*i + 1/2*j +
1/2*k, -1/2 - 1/2*i + 1/2*j + 1/2*k, -1/2 + 1/2*i - 1/2*j + 1/2*k, 1/2 - 1/2*i -
1/2*j + 1/2*k, k, -i, j ]
> G2, h2 := UnitGroup(S2);
> #G2;
8
> [ A | h2(g) : g in G2 ];
[ 1, -1, -j, j, k, -k, -i, i ]

The unit groups of orders in indefinite quaternion algebras \(A\) are infinite arithmetic groups, which are twisted analogues of the groups \({\operatorname{SL}}_2({\mathbb{Z}})\) and their families of subgroups. These are studied in relation to their actions on the upper half complex plane, via an embedding in \({\operatorname{GL}}_2({\mathbb{R}})\) provided by some isomorphism \(A\otimes{\mathbb{R}}\cong M_2({\mathbb{R}})\).

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Example: Unit Groups over a Number Ring (ex-b816c0)

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Now we exhibit unit group computations over a number ring.

> P<x> := PolynomialRing(Rationals());
> F := NumberField(x^3-3*x-1);
> Z_F := MaximalOrder(F);
> Foo := InfinitePlaces(F);

We use SetSeed since the following line makes random choices.

> SetSeed(1);
> A := QuaternionAlgebra(ideal<Z_F | 2>, Foo);
> IsDefinite(A);
true
> O := MaximalOrder(A);
> U, h := UnitGroup(O);
> U;
Permutation group U acting on a set of cardinality 12
Order = 12 = 2^2 * 3
    Id(U)
    (1, 2, 4)(3, 6, 7)(5, 9, 10)(8, 12, 11)
    (1, 3)(2, 5)(4, 8)(6, 11)(7, 9)(10, 12)
> #Units(O);
12

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