Constructing Artin Representations

ArtinRepresentations(K): FldNum -> SeqEnum

f                 : RngUPolElt                    Default: 
Ramification      : BoolElt                       Default: false
FactorDiscriminant: BoolElt                       Default: false
p0                : RngIntElt                     Default: 

Compute all irreducible Artin representations that factor through the normal closure \(F\) of the number field \(K\).

If \(K\) is given as a relative extension \(E/F\) (that is, BaseField(K) is not \({\mathbb{Q}}\)), the representations returned are those of \({\operatorname{Gal}}(\bar F/F)\) that factor through the normal closure of \(E\) over \(F\), and LSeries returns the corresponding Artin \(L\)-series over \(F\). To obtain the representations of \({\operatorname{Gal}}(\bar{\mathbb{Q}}/{\mathbb{Q}})\) instead, apply ArtinRepresentations to AbsoluteField(K).

Such relative representations support the operations that derive one Artin representation from another: sums, differences, and tensor products of representations attached to the same field, Decomposition and RationalDecomposition return relative representations over the same base field, and Kernel returns the smallest extension of the base field through which the representation factors. IsRamified, IsWildlyRamified, Minimize, OptimizedRepresentation, DirichletCharacter, and HeckeCharacter are not implemented over a base field other than \(\Q\), and raise an error for a relative representation rather than answering over \(\Q\).

The Galois group \(G={\operatorname{Gal}}(F/K)\) whose representations are constructed is represented as a permutation group on the roots of f, which must be a monic irreducible polynomial with integer coefficients that defines \(K\). By default this is the defining polynomial of \(K\) represented as an extension of \({\mathbb{Q}}\). (It is possible to specify any monic integral polynomial whose splitting field is \(F\), even a reducible one, but PermutationCharacter(K) and the Dedekind \(\zeta\)-function of \(K\) will not work correctly.)

The Ramification parameter specifies whether to pre-compute the inertia groups at all ramified primes and the conductors of all representations.

The parameter FactorDiscriminant determines whether to factorize the discriminant of f completely, even if it appears to contain large prime factors. The factorization is used to determine which primes ramify in \(F/K\), which is necessary to compute the conductors. If the factorization is incomplete, Magma assumes that the primes in the unfactored part of the discriminant are unramified. One may specify

FactorDiscriminant:=

<TrialLimit,PollardRhoLimit,ECMLimit,MPQSLimit,Proof> and these 5 parameters are passed to the Factorization function; the default behaviour (false) is the same as <10000,65535,10,0,false>. When the factorization is incomplete, Magma will print “(?)” following the conductor values, when asked to print an Artin representation.

Finally, p0 specifies which \(p\)-adic field to use for the roots of f, in particular in Galois group computations. It must be chosen so that GaloisGroup(f:Prime:=p0) is successful. By default it is chosen by the Galois group computation.

K !! ch: FldNum, AlgChtrElt -> ArtRep

K !! ch: FldNum, SeqEnum -> ArtRep

Writing \(F\) for the normal closure of \(K/{\mathbb{Q}}\), this function converts an abstract group character of \({\operatorname{Gal}}(F/{\mathbb{Q}})\) or the sequence of its trace values into an Artin representation.

PermutationCharacter(K): FldNum -> ArtRep

Construct the permutation representation \(A\) of the absolute Galois group of \({\mathbb{Q}}\) on the embeddings of \(K\) into \({\mathbb{C}}\). This is an Artin representation of \({\operatorname{Gal}}(F/{\mathbb{Q}})\) of dimension \([K:{\mathbb{Q}}]\), where \(F\) is the normal closure of \(K\), and it is the same as the permutation representation of \({\operatorname{Gal}}(F/{\mathbb{Q}})\) on the cosets of \({\operatorname{Gal}}(F/K)\).

Determinant(A): ArtRep -> ArtRep

Construct the determinant of a given Artin representation. The result is given as a 1-dimensional Artin representation attached to the same field.

ChangeField(A, K): ArtRep, FldNum -> ArtRep, BoolElt

K !! A: FldNum, ArtRep -> ArtRep, BoolElt

MinPrimes: RngIntElt                    Default: 20

Given an Artin representation (attached to some number field) that is known to factor through the Galois closure of \(K\), attempts to recognize it as such. Returns “the resulting Artin representation attached to \(K\)”, true if successful, and 0, false if it proves that there is no such representation. The parameter MinPrimes specifies the number of additional primes for which to compare traces of Frobenius elements.

Example: Artin Representations of a Quadratic Field (ex-81e6bd)

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A quadratic field \(K\) has two irreducible Artin representations the factor through \({\operatorname{Gal}}(K/{\mathbb{Q}})\), the trivial one and the quadratic character of \(K\):

> K<i> := QuadraticField(-1);
> triv, sign := Explode(ArtinRepresentations(K));
> sign;
Artin representation C2: (1,-1) of Q(sqrt(-1))

An alternative way to define them is directly by their character:

> triv,sign:Magma;
QuadraticField(-1) !! [1,1]
QuadraticField(-1) !! [1,-1]

The regular representation of \({\operatorname{Gal}}(K/{\mathbb{Q}})\) is their sum:

> PermutationCharacter(K);
Artin representation C2: (2,0) of Q(sqrt(-1))
> $1 eq triv+sign;
true

Next, let \(L=K(\sqrt{-2-i})\), presented as a relative extension of \(K\). As \(L\) is defined over \(K\), ArtinRepresentations returns the representations of \({\operatorname{Gal}}(\bar K/K)\), here the trivial character and the quadratic character of \({\operatorname{Gal}}(L/K)\):

> P<x> := PolynomialRing(K);
> L := ext<K | x^2 + i + 2>;
> reps := ArtinRepresentations(L);
> [Dimension(A): A in reps];
[ 1, 1 ]
> BaseField(reps[2]) eq K;
true

The normal closure \(F\) of \(L\) over \({\mathbb{Q}}\) has \({\operatorname{Gal}}(F/{\mathbb{Q}})=D_4\), the dihedral group of order 8. Passing to the absolute field instead gives the Artin representations of \({\operatorname{Gal}}(\bar{\mathbb{Q}}/{\mathbb{Q}})\):

> Labs := AbsoluteField(L);
> GroupName(GaloisGroup(Labs));
D4
> [Dimension(A): A in ArtinRepresentations(Labs)];
[ 1, 1, 1, 1, 2 ]

We use ChangeField to lift Artin representations from \({\operatorname{Gal}}(K/{\mathbb{Q}})\) to \({\operatorname{Gal}}(F/{\mathbb{Q}})\), and check that it is still the same as an Artin representation.

> A := ChangeField(sign,L);
> A;
Artin representation D4: (1,1,-1,1,-1) of ext<Q(sqrt(-1))|x^2+i+2>
> A eq sign;
true

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