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The base field K of the Galois representation A: Gal(bar K/K)toGLm(C).
Dimension(A) : GalRep -> RngIntElt
Degree (=dimension) m of a Galois representation A: Gal(bar K/K)toGLm(C)
> K:=pAdicField(3,20);
> R<x>:=PolynomialRing(K);
> F:=ext<K|x^3-3>;
> list:=GaloisRepresentations(F,K);
> forall{A: A in list | BaseField(A) eq K};
true
> [Degree(A): A in list];
[ 1, 1, 2 ]
GaloisGroup(A) : GalRep -> GrpPerm
Finite Galois group Gal(F/K) that computes the finite part of a
Galois representation, where F is Field(A) and K is BaseField(A).
> K:=pAdicField(2,20);
> R<x>:=PolynomialRing(K);
> list1:=[PrincipalCharacter(K),CyclotomicCharacter(K),SP(K,3)];
> [Degree(A): A in list1];
[ 1, 1, 3 ]
> [GroupName(Group(A)): A in list1];
[ C1, C1, C1 ]
> list2:=GaloisRepresentations(x^4-2);
> [Degree(A): A in list2];
[ 1, 1, 1, 1, 2 ]
> [GroupName(Group(A)): A in list2];
[ D4, D4, D4, D4, D4 ]
> list1[1] eq list2[1];
true
An arithmetic Frobenius element of Group(A) for a Galois
representation A.
Take K=Q 2 and F=Q 2(ζ 5), a degree 4 unramified extension of K.
A Frobenius element Frob∈Gal(F/K) is characterized by the property that
Frob(x) ≡ x qmod m F. In this example, q=2 (size of the residue
field of K).
> K:=pAdicField(2,20);
> R<x>:=PolynomialRing(K);
> A:=GaloisRepresentations(x^4+x^3+x^2+x+1)[4]; A;
1-dim unramified Galois representation (1,-1,-zeta(4)_4,zeta(4)_4)
with G=C4, I=C1 over Q2[20]
> frob:=FrobeniusElement(A); frob;
(1, 2, 4, 3)
> F<u>:=Field(A);
> Valuation(Automorphism(A,frob)(u)-u^2) gt 0;
true
Character of the finite part of a Galois representation A.
For this to be well-defined, A must have only one component
(and not several components with different characters).
> K:=pAdicField(2,20);
> R<x>:=PolynomialRing(K);
> A1:=PrincipalCharacter(K);
> Character(A1);
( 1 )
> A2:=CyclotomicCharacter(K);
> Character(A2);
( 1 )
> A3:=PermutationCharacter(ext<K|3>,K);
> Character(A3);
( 3, 0, 0 )
> A1+A2+A3;
5-dim unramified Galois representation Unr(1-3/2*x+1/2*x^2) + (3,0,0)
with G=C3, I=C1 over Q2[20]
Given a Galois representation A, return the
p-adic field F such that the finite part of A factors through
Gal(F/K), K = BaseField(A).
For a Galois representation A over K, returns a
polynomial over K whose splitting field is F=Field(A).
The Galois group Gal(F/K) is represented as a permutation group
Group(A) on the roots of this polynomial.
In this example K=Q 5 and F/K is a dihedral extension of degree 12,
represented as a splitting field of a degree 6 polynomial.
> K:=pAdicField(5,20);
> E:=EllipticCurve([K|0,5]);
> A:=GaloisRepresentation(E); A;
2-dim Galois representation Unr(sqrt(5)*i)*(2,-2,0,0,-1,1) with G=D6, I=C6,
conductor 5^2 over Q5[20]
> Field(A);
Totally ramified extension defined by the polynomial x^6 - 5
over Unramified extension defined by the polynomial x^2 + 4*x + 2
over 5-adic field mod 5^20
> DefiningPolynomial(A);
x^6 + O(5^20)*x^5 + O(5^20)*x^4 + O(5^20)*x^3 + O(5^20)*x^2 + O(5^20)*x - 5 +
O(5^20)
The automorphism of Field(A)/BaseField(A) given by g.
In this example K=Q 5 and F/K is a dihedral extension of degree 12,
represented as a splitting field of the polynomial x 6 - 5.
> R<x>:=PolynomialRing(Rationals());
> K:=NumberField(x^6-5);
> a:=ArtinRepresentations(K)[6];
> A:=GaloisRepresentation(a,5); A;
2-dim Galois representation (2,-2,0,0,-1,1) with G=D6, I=C6, conductor 5^2
over Q5[40]
> F:=Field(A); F;
Totally ramified extension defined by the polynomial x^6 - 5
over Unramified extension defined by the polynomial x^2 + 4*x + 2
over 5-adic field mod 5^40
> DefiningPolynomial(A);
x^6 - 5
The 12 elements σ∈Gal(F/K) isomorphic to D 6 act on π=root 6 of 5
by multiplying it by 6th roots of unity, with σ(π)=π for 2
of them, and v(σ(π) - π)=1 for the other 10.
> pi:=UniformizingElement(F);
> autF:=[* Automorphism(A,g): g in Group(A) *];
> [Valuation(sigma(pi)-pi): sigma in autF];
[ 241, 1, 1, 1, 1, 1, 240, 1, 1, 1, 1, 1 ]
R: Rng Default:
Euler factor (=local polynomial) of a Galois representation A
over a p-adic field K. It is defined by
P(T) = det (1 - FrobK - 1T|AIK),
and has degree Dimension(A) if and only if A is unramified.
The coefficient ring of P (rational/complex/cyclotomic field) may be specified
with the optional parameter R.
> G<chi>:=DirichletGroup(5);
> A:=GaloisRepresentation(chi,2);
> EulerFactor(A);
x + 1
> A:=GaloisRepresentation(chi,5);
> EulerFactor(A);
1
Return true if A is the Galois representation 0.
Return true if A is the trivial 1-dimensional Galois representation.
Returns the list of tuples < χi,ni,ρi >,
where A is the direct sum over i of
twists by (SP)(ni) by unramified representations with Euler
factor χi, and a finite Weil representation given by a character
ρi of Group(A).
> R<x>:=PolynomialRing(ComplexField()); // prettier print for complex polys
> K:=pAdicField(2,20);
> S:=SP(K,2);
> S; Factorization(S);
2-dim Galois representation SP(2) over Q2[20]
[*
<-x + 1, 2, ( 1 )>
*]
> A:=Semisimplification(S);
> A; Factorization(A);
2-dim unramified Galois representation Unr(1-3/2*x+1/2*x^2) over Q2[20]
[*
<1/2*x^2 - 3/2*x + 1, 1, ( 1 )>
*]
> [Factorization(I)[1]: I in Decomposition(A)];
[ <-x + 1, 1, ( 1 )>, <-1/2*x + 1, 1, ( 1 )> ]
For a Galois representation A over a p-adic field K this is the image
of inertia IK⊂Gal(bar K/K) under the semisimplification of A.
Equivalently, if
A = ψ tensor SP(n) tensor R,
as in ParaNotation and Printing, with ψ unramified and R a representation of
a finite Galois group Gal(F/K), this is the image of the inertia
subgroup of Gal(F/K) under R. If F is chosen to be minimal possible
(so that R is faithful), then InertiaGroup(A) simply is
the inertia subgroup of Gal(F/K).
Take K=Q 3 and F=K(ζ 6, root 6of 3), a D 6-extension of K.
For each of the 6 irreducible representations of Gal(F/K) we compute
their inertia (=ramification) groups:
> K:=pAdicField(3,20);
> R<x>:=PolynomialRing(K);
> list:=GaloisRepresentations(x^6-3);
> [GroupName(InertiaGroup(A)): A in list];
[ C1, C2, C1, C2, S3, S3 ]
The nth (lower) ramification subgroup of InertiaGroup(A).
> K:=pAdicField(2,20);
> R<x>:=PolynomialRing(K);
> list:=GaloisRepresentations(x^8-2);
> a:=list[#list]; a;
2-dim Galois representation (2,-2,0,0,0,zeta(8)_8^3+zeta(8)_8,
-zeta(8)_8^3-zeta(8)_8) with G=SD16, I=SD16, conductor 2^10 over Q2[20]
> [GroupName(InertiaGroup(a,n)): n in [1..17]];
[ SD16, SD16, C8, C8, C4, C4, C4, C4, C2, C2, C2, C2, C2, C2, C2, C2, C1 ]
Return true if a Galois representation is unramified.
> K:=pAdicField(2,20);
> IsUnramified(CyclotomicCharacter(K));
true
> IsUnramified(SP(K,2));
false
> IsUnramified(Semisimplification(SP(K,2)));
true
Return true if a Galois representation is ramified.
> K:=pAdicField(2,20);
> IsRamified(CyclotomicCharacter(K));
false
> IsRamified(SP(K,2));
true
> IsRamified(Semisimplification(SP(K,2)));
false
Return true if a Galois representation A over a p-adic field K is tamely ramified.
Equivalently, InertiaGroup(A) has order prime to p
(and is then automatically cyclic).
Return true if a Galois representation A over a p-adic field K is wildly ramified,
i.e. not tamely ramified.
Equivalently, InertiaGroup(A) has non-trivial p-Sylow.
Galois representations attached to elliptic curves are always tamely ramified
when p≥5, but may be wildly ramified when p=2 or 3.
> E:=EllipticCurve("75a1");
> A5:=GaloisRepresentation(E,5); A5;
2-dim Galois representation Unr(sqrt(5)*i)*(2,0,-1) with G=S3, I=C3, conductor
5^2 over Q5[40]
> IsWildlyRamified(A5);
false
> E:=EllipticCurve("256a1");
> A2:=GaloisRepresentation(E,2); A2;
2-dim Galois representation Unr(sqrt(2))*(2,-2,0,0,0) with G=D4, I=C4,
conductor 2^8 over Q2[40]
> IsWildlyRamified(A2);
true
Inertia invariants of a Galois representation A. This is an unramified Galois
representation.
> K:=pAdicField(5,20);
> E:=BaseChange(EllipticCurve("15a1"),K);
> A:=GaloisRepresentation(E); A;
2-dim Galois representation Unr(5)*SP(2) over Q5[20]
> I:=InertiaInvariants(A); I;
1-dim trivial Galois representation 1 over Q5[20]
> Dimension(A),Dimension(I);
2 1
Conductor exponent of a Galois representation.
Conductor of a Galois representation.
> K:=pAdicField(2,40);
> E:=BaseChange(EllipticCurve("256a1"),K);
> A:=GaloisRepresentation(E); A;
2-dim Galois representation Unr(sqrt(2))*(2,-2,0,0,0) with G=D4, I=C4,
conductor 2^8 over Q2[40]
> ConductorExponent(A);
8
> Conductor(A);
2^8 + O(2^48)
> Conductor(E); // same, by definition
2^8 + O(2^48)
Epsilon-factor ε(A) of a Galois representation over a p-adic field.
Currently only implemented in a few basic cases, and returns 0 otherwise.
See also Example H58E52 in ParaExample: Local and Global Epsilon Factors for Dirichlet Characters.
Root number ε(A)/|ε(A)|
of a Galois representation.
Currently only implemented in a few basic cases, and returns 0 otherwise.
See also Example H58E52 in ParaExample: Local and Global Epsilon Factors for Dirichlet Characters.
> E:=EllipticCurve("98a1");
> A:=GaloisRepresentation(E,7); A;
2-dim Galois representation Unr(7)*SP(2)*(1,-1) with G=C2, I=C2, conductor 7^2
over Q7[40]
> RootNumber(A);
-1
> RootNumber(E,7); // same
-1
Return true if a Galois representation A is irreducible.
We take the polynomial x 8 - 6 with Galois group C 8:C 22 over Q,
its irreducible 4-dimensional Artin representation A,
and compute whether its local components over Q 2, Q 3,
Q 5 and Q 7 are irreducible.
> R<x>:=PolynomialRing(Rationals());
> K:=NumberField(x^8-6);
> GroupName(GaloisGroup(K));
C8:C2^2
> assert exists(A){A: A in ArtinRepresentations(K) | Degree(A) eq 4};
> A;
Artin representation C8:C2^2: (4,-4,0,0,0,0,0,0,0,0,0) of ext<Q|x^8-6>
> [IsIrreducible(GaloisRepresentation(A,p)): p in PrimesUpTo(10)];
[ true, false, false, false ]
Return true if a Galois representation A is indecomposable
(for semisimple representations, i.e. Weil representations, same as irreducible).
Return true if a Galois representation A is semisimple, i.e. a Weil representation.
Semisimplification of a Galois representation A.
Decompose A into indecomposable (for semisimple representations
same as irreducible) consituents and return them as a sequence,
possibly with repetitions.
> K:=pAdicField(2,20);
> S:=SP(K,2);
> IsIndecomposable(S);
true
> IsIrreducible(S);
false
> IsSemisimple(S);
false
> Decomposition(S);
[ 2-dim Galois representation SP(2) over Q2[20] ]
> Decomposition(Semisimplification(S));
[
1-dim trivial Galois representation 1 over Q2[20],
1-dim unramified Galois representation Unr(1/2) over Q2[20]
]
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