Picard Group

Let \(R\) be an order in an étale algebra \(A\) over \({{\Bbb Q}}\) with maximal order \({\cal O}_A\). We say that two fractional \(R\)-ideals \(I\) and \(J\) are isomorphic if there exists a unit \(a\in A\) such that \(I=aJ\). Observe that this happens if and only if \(I\) and \(J\) are isomorphic as \(R\)-modules. We refer to the isomorphism class \([I]\) of \(I\) as its ideal class.

The set of ideal classes of invertible fractional \(R\)-ideals forms a group under the operation induced by ideal multiplication. This group is called the Picard group of \(R\) and denoted \({\rm Pic}(R)\).

If \(K_1\times \ldots \times K_n\) are the components of \(A\) then \({\rm Pic}({\cal O}_A) = \prod_{i=1}^n{\rm Cl}({\cal O}_{K_i})\), where \({\rm Cl}({\cal O}_{K_i})\) is the class group of the number field \(K_i\). Also, the unit group \({\cal O}_A^\times\) of \({\cal O}_A\) satisfies \({\cal O}_A^\times = {\cal O}_{K_1}^\times \times \ldots \times {\cal O}_{K_n}^\times\).

The Picard group and the unit group of the order \(R\) can be computed using the well-known exact sequence:

\[1 \to R^\times \to {\cal O}_A^\times \to {\left( {\cal O}_A/{\frak f} \right)^\times\over\left( R/{\frak f} \right)^\times} \to {\rm Pic}(R)\to {\rm Pic}({\cal O}_A) \to 1,\]

where the first, second and third map are the natural maps, the fourth is induced by \(a \mapsto (a{\cal O}_A \cap R)\) and the last one is induced by the extension map \(I \mapsto I{\cal O}_A\).

ResidueRingUnits(S, I): AlgEtQOrd, AlgEtQIdl -> GrpAb, Map

Returns the group \((S/I)^*\) and a map \((S/I)^* \rightarrow S\). The order \(S\) is required to be maximal.

ResidueRingUnits(I): AlgEtQIdl -> GrpAb, Map

Given a fractional \(S\)-ideal \(I\), returns the group \((S/I)^*\) and a map \((S/I)^* \to S\) giving representatives. Implemented when \(S\) is maximal.

ResidueRingUnitsSubgroupGenerators(F): AlgEtQIdl -> SeqEnum[AlgEtQElt]

Given a fractional \(S\)-ideal \(F\), returns generators of \((S/F)^*\).

IsPrincipal(I1): AlgEtQIdl -> BoolElt, AlgAssElt

GRH: BoolElt                    Default: false

Return if the argument is a principal ideal; if so the function returns also the generator. The optional parameter GRH decides whether the bound for the IsPrincipal test should be conditional. The default value is false.

PicardGroup(S): AlgEtQOrd -> GrpAb, Map

GRH: BoolElt                    Default: false

Return the PicardGroup of the order \(S\), which is not required to be maximal, and a map from the Picard group to a set of representatives of the ideal classes. The optional parameter GRH decides the bound for the computations of the Class group and Unit group of the maximal order. The default value is false.

ExtensionHomPicardGroups(S, T): AlgEtQOrd, AlgEtQOrd -> Map

GRH: BoolElt                    Default: false

Given orders \(S\) and \(T\) such that \(S \subseteq T\), returns the surjective extension map \({\rm Pic}(S) \to {\rm Pic}(T)\). The parameter GRH is passed to PicardGroup.

UnitGroup(S): AlgEtQOrd -> GrpAb, Map

GRH: BoolElt                    Default: false

Return the unit group of a order in a étale algebra. The optional parameter GRH decides the bound for the computation of the unit group of the maximal order. The default value is false.

Example: Picard And Units (ex-034565)

> _<x> := PolynomialRing(Integers());
> A := EtaleAlgebra((x^4+16)*(x^4+81));
> E := EquationOrder(A);
> P, phi := PicardGroup(E);
> AbelianInvariants(P);
[ 2, 24, 24, 24, 24 ]
> U, psi := UnitGroup(E);
> TorsionInvariants(U);
[ 2 ]
> TorsionFreeRank(U);
2

Run in calculatorCopy input

IsIsomorphic(I, J): AlgEtQIdl, AlgEtQIdl -> BoolElt, AlgAssElt

GRH: BoolElt                    Default: false

Checks if \(I=x \cdot J\), for some \(x\). If so, also \(x\) is returned. The optional parameter GRH decides whether the bound for the IsPrincipal test should be conditional. The default value is false.