Galois representation 0 over a p-adic field K. It is 0-dimensional, and 0 + A=A, 0 tensor A=0 for every Galois representation A.
> K:=pAdicField(3,20); > zero:=ZeroRepresentation(K); > zero; Galois representation 0 with G=C1, I=C1 over Q3[20] > zero + CyclotomicCharacter(K) eq CyclotomicCharacter(K); true > zero*CyclotomicCharacter(K) eq zero; true
Principal character 1 of the absolute Galois group of K, as a Galois representation. It is a 1-dimensional unramified representation, same as UnramifiedCharacter(K,1). Thus 1 tensor A=1 for every Galois representation A.
> K:=pAdicField(3,20); > one:=PrincipalCharacter(K); > one; 1-dim trivial Galois representation 1 over Q3[20] > F:=ext<K|4>; > A1,A2,A3,A4:=Explode(GaloisRepresentations(F,K)); > A1 eq one; true
Cyclotomic character over K. It is an unramified character (trivial on inertia) and takes the value q, the size of the residue field of K, on any Frobenius element.
> K:=pAdicField(3,20); > chi:=CyclotomicCharacter(K); > chi,EulerFactor(chi); 1-dim unramified Galois representation Unr(1/3) over Q3[20] -1/3*x + 1 > chi^3,EulerFactor(chi^3); 1-dim unramified Galois representation Unr(1/27) over Q3[20] -1/27*x + 1
Galois representation over K given by an unramified character that sends the arithmetic Frobenius element FrobK |-> c - 1 (and, so, the geometric Frobenius element FrobK - 1 |-> c.) The parameter c must be a non-zero complex number.
> K:=pAdicField(3,20); > assert UnramifiedCharacter(K,1) eq PrincipalCharacter(K); > assert UnramifiedCharacter(K,1/3) eq CyclotomicCharacter(K); > C<i>:=ComplexField(); > UnramifiedCharacter(K,2+i); 1-dim unramified Galois representation Unr(2+i) over Q3[20]
Unique unramified Galois representation ρ over K with Euler factor det(1 - FrobK - 1|ρ)=(CharPoly).
> K:=pAdicField(37,20); > R<x>:=PolynomialRing(Rationals()); > rho:=UnramifiedRepresentation(K,(1-37*x)*(1-3*x)); > rho; 2-dim unramified Galois representation Unr(1-40*x+111*x^2) over Q37[20] > rho eq CyclotomicCharacter(K)^(-1)+UnramifiedCharacter(K,3); true
Unramified Galois representation over K of dimension dim, with Euler factor CharPoly computed up to and inclusive degree dimcomputed.
Consider the hyperelliptic curve C: y2=x5 + x + 1 over the p-adic field Q10007.
> _<x>:=PolynomialRing(Rationals()); > p:=10007; > K:=pAdicField(p,20); > _<X>:=PolynomialRing(K); > C:=HyperellipticCurve(X^5+X+1); C; Hyperelliptic Curve defined by y^2 = x^5 + O(10007^20)*x^4 + O(10007^20)*x^3 + O(10007^20)*x^2 + x + 1 + O(10007^20) over pAdicField(10007, 20)
The Galois representation A associated to H1(C) is unramified, of dimension 4, and could be defined by
hfilUnramifiedRepresentation(K,1-ap*x+bp*x^2-p*ap+p^2);hfil hfil
if we find ap and bp by counting points of C over Fp and Fp2. The coefficient ap can be computed very quickly:
> k:=ResidueClassField(Integers(K)); > _<X>:=PolynomialRing(k); > Ck:=HyperellipticCurve(X^5+X+1); > ap:=p+1-#Ck; ap; -21
However, bp would take a long time. If we are only interested in working with A up to degree 1 (e.g. to compute L-series of C/Q with <108 terms), there is no reason to compute it. Instead, we can define an unramified Galois representation of degree 4, which is known to be computed only up to degree 1:
> A:=UnramifiedRepresentation(K,4,1,1-ap*x); > A; 4-dim unramified Galois representation Unr(1+21*x+O(x^2)) over Q10007[20]
One can still take direct sums, and tensor products of such representations with (possibly ramified) Galois representations, and the Euler factors will still be correct up to degree 1:
> A*A; 16-dim unramified Galois representation Unr(1-441*x+O(x^2)) over Q10007[20] > EulerFactor(A*A+1/CyclotomicCharacter(K)); -10448*x + 1
The n-dimensional indecomposable Galois representation SP(n) over a p-adic field K; see ParaNotation and Printing for its description.
> K:=pAdicField(3,20); > SP(K,1) eq PrincipalCharacter(K); true > rho:=SP(K,2); rho; 2-dim Galois representation SP(2) over Q3[20] > Degree(rho); 2 > Semisimplification(rho); 2-dim unramified Galois representation Unr(1-4/3*x+1/3*x^2) over Q3[20] > $1 eq PrincipalCharacter(K)+CyclotomicCharacter(K); true > InertiaInvariants(rho); 1-dim unramified Galois representation Unr(1/3) over Q3[20] > EulerFactor(rho); -1/3*x + 1
Unramified twist ψ tensor (SP(n)) over a p-adic field K, with ψ specified by its Euler factor f.
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(Rationals()); > SP(K,1-x^2,2); 4-dim Galois representation Unr(1-x^2)*SP(2) over Q2[20] > $1*$1; // Tensor product with itself 16-dim Galois representation Unr(1-1/2*x^2+1/16*x^4) + Unr(1-2*x^2+x^4)*SP(3) over Q2[20]
For a p-adic extension F/K, compute all irreducible Galois representations that factor through the (Galois closure of) F/K.
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(K); > F:=ext<K|x^8+2>; > list:=GaloisRepresentations(F,K); > list; [ 1-dim trivial Galois representation 1 over Q2[20], 1-dim Galois representation (1,1,-1,-1,1,1,1) with G=D8, I=D8, conductor 2^2 over Q2[20], 1-dim Galois representation (1,1,-1,1,1,-1,-1) with G=D8, I=D8, conductor 2^3 over Q2[20], 1-dim Galois representation (1,1,1,-1,1,-1,-1) with G=D8, I=D8, conductor 2^3 over Q2[20], 2-dim Galois representation (2,2,0,0,-2,0,0) with G=D8, I=D8, conductor 2^8 over Q2[20], 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=D8, I=D8, conductor 2^10 over Q2[20], 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=D8, I=D8, conductor 2^10 over Q2[20] ]
The first 5 characters are not faithful, and we can descend them to smaller quotients of Gal(F/K).
> min:=[Minimize(rho): rho in list | not IsFaithful(Character(rho))]; > min; [ 1-dim trivial Galois representation 1 over Q2[20], 1-dim Galois representation (1,-1) with G=C2, I=C2, conductor 2^2 over Q2[20], 1-dim Galois representation (1,-1) with G=C2, I=C2, conductor 2^3 over Q2[20], 1-dim Galois representation (1,-1) with G=C2, I=C2, conductor 2^3 over Q2[20], 2-dim Galois representation (2,-2,0,0,0) with G=D4, I=D4, conductor 2^8 over Q2[20] ]
For a polynomial f over a p-adic field K and splitting field F, returns irreducible representations of Gal(F/K).
> K:=pAdicField(2,20); > R:=PolynomialRing(K); > GaloisRepresentations(R!CyclotomicPolynomial(8)); [ 1-dim trivial Galois representation 1 over Q2[20], 1-dim Galois representation (1,-1,1,-1) with G=C2^2, I=C2^2, conductor 2^3 over Q2[20], 1-dim Galois representation (1,1,-1,-1) with G=C2^2, I=C2^2, conductor 2^2 over Q2[20], 1-dim Galois representation (1,-1,-1,1) with G=C2^2, I=C2^2, conductor 2^3 over Q2[20] ]
For a p-adic extension F/K, compute C[Gal(bar(K)/K)/Gal(bar(K)/F)] as a Galois representation over K of degree [F:K].
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(K); > F:=ext<K|x^3-2>; > PermutationCharacter(F,K); 3-dim Galois representation (3,1,0) with G=S3, I=C3, conductor 2^2 over Q2[20] > $1 - PrincipalCharacter(K); 2-dim Galois representation (2,0,-1) with G=S3, I=C3, conductor 2^2 over Q2[20]
Change a Galois representation by a finite representation with character χ, which must be a character of Group(A), or a list of values that determine such a character.
> K:=pAdicField(2,20); > R<x>:=PolynomialRing(K); > F:=ext<K|x^3-2>; > rho:=PermutationCharacter(F,K); > rho!![1,1,1]; 1-dim trivial Galois representation 1 over Q2[20] > rho!![6,0,0]; 6-dim Galois representation (6,0,0) with G=S3, I=C3, conductor 2^4 over Q2[20] > rho!![0,0,0]; // but not [-1,0,0] - may not be virtual Galois representation 0 with G=S3, I=C3 over Q2[20]
Precision: RngIntElt Default: 40
Local Galois representation at p of a Dirichlet character χ.
Local components of a Dirichet character χ of order 6 at p=2, 3, 7.
> G<chi>:=FullDirichletGroup(7); > GaloisRepresentation(chi,2); 1-dim unramified Galois representation (1,-zeta(3)_3-1,zeta(3)_3) with G=C3, I=C1 over Q2[40] > GaloisRepresentation(chi,3); 1-dim unramified Galois representation (1,-1,-zeta(3)_3-1,zeta(3)_3,-zeta(3)_3, zeta(3)_3+1) with G=C6, I=C1 over Q3[40] > GaloisRepresentation(chi,7); 1-dim Galois representation (1,-1,-zeta(3)_3-1,zeta(3)_3,-zeta(3)_3,zeta(3)_3+1) with G=C6, I=C6, conductor 7^1 over Q7[40]
By our convention, the character χ, the associated Artin representation, and the Galois representations associated to them all have the same Euler factors.
> loc1:=EulerFactor(chi,2); > loc2:=EulerFactor(ArtinRepresentation(chi),2); > loc3:=EulerFactor(GaloisRepresentation(chi,2)); > loc4:=EulerFactor(GaloisRepresentation(ArtinRepresentation(chi),2)); > [PolynomialRing(ComplexField(5))| loc1,loc2,loc3,loc4]; [ (0.50000 - 0.86603*$.1)*$.1 + 1.0000, (0.50000 - 0.86603*$.1)*$.1 + 1.0000, (0.50000 - 0.86603*$.1)*$.1 + 1.0000, (0.50000 - 0.86603*$.1)*$.1 + 1.0000 ]
Precision: RngIntElt Default: 40
Minimize: BoolElt Default: false
Local Galois representation at p of an Artin representation A. This is the representation of a decomposition group at p of Gal(bar Q/Q) on the dual vector space of A. (The reason for the dual is that, by our convention, the global and the local Euler factors agree; see ParaConventions and Example H57E14.) If Minimize is true, choose the field through which it factors to be as small as possible (automatic for faithful representations).
> R<x>:=PolynomialRing(Rationals()); > K:=NumberField(x^7-7*x-3); > GroupName(GaloisGroup(K)); PSL(2,7) > A:=ArtinRepresentations(K)[5]; > GaloisRepresentation(A,2); 7-dim unramified Galois representation (7,0,0,0,0,0,0) with G=C7, I=C1 over Q2[40] > GaloisRepresentation(A,3); 7-dim Galois representation (7,-1,1) with G=S3, I=S3, conductor 3^8 over Q3[40] > GaloisRepresentation(A,5); 7-dim unramified Galois representation (7,0,0,0,0,0,0) with G=C7, I=C1 over Q5[40] > GaloisRepresentation(A,7); 7-dim Galois representation (7,1,1,0,0) with G=C7:C3, I=C7:C3, conductor 7^8 over Q7[40] > Conductor(A) eq 3^8*7^8; true
Minimize: BoolElt Default: true
Local Galois representation of (the first l-adic étale cohomology group of) an elliptic curve over a p-adic field. If Minimize is true (default), choose the field through which it factors to be as small as possible.
> K:=pAdicField(5,20); > E:=EllipticCurve([K|0,5]); > E; Elliptic Curve defined by y^2 = x^3 + O(5^20)*x + (5 + O(5^21)) over pAdicField(5, 20) > loc:=LocalInformation(E); loc; <5 + O(5^21), 2, 2, 1, II, true>
Its Galois representation is an unramified twist of a representation with finite image that factors through the dihedral extension Q5(ζ6, (root 6of 5)) of Q5.
> 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
Precision: RngIntElt Default: 40
Minimize: BoolElt Default: true
Local Galois representation of (the first l-adic étale cohomology group of) an elliptic curve over Q at p. If Minimize is true (default), choose the field through which it factors to be as small as possible.
> E:=EllipticCurve("20a1");
> GaloisRepresentation(E,3);
2-dim unramified Galois representation Unr(1+2*x+3*x^2) over Q3[40]
> GaloisRepresentation(E,5);
2-dim Galois representation Unr(-5)*SP(2) over Q5[40]
> GaloisRepresentation(E,2);
2-dim Galois representation Unr(sqrt(2)*i)*(2,0,-1) with G=S3, I=C3, conductor
2^2 over Q2[40]
> EulerFactor($3),EulerFactor($2),EulerFactor($1);
3*x^2 + 2*x + 1
x + 1
1
Precision: RngIntElt Default: 40
Minimize: BoolElt Default: true
Local Galois representation of (the first l-adic étale cohomology group of) an elliptic curve E over a number field F at a given prime ideal P. If Minimize is true (default), choose the field through which it factors to be as small as possible.
> K:=CyclotomicField(5);
> E:=BaseChange(EllipticCurve("75a1"),K);
> P:=Ideal(Decomposition(K,5)[1,1]);
> GaloisRepresentation(E,P);
2-dim Galois representation Unr(sqrt(5)*i)*(2,0,-1) with G=S3, I=C3, conductor
pi^2 over ext<Q5[10]|x^4-15*x^3-40*x^2-90*x-45>
Degree: RngIntElt Default: ∞
Minimize: BoolElt Default: false
Galois representation associated to (H1 of) a hyperelliptic curve C over a p-adic field. Degree specifies that Euler factors of unramified pieces should only be computed up to that degree. (See Example H57E7.) Setting Minimize:=true forces the representation to be minimized. (See Minimize.)
> K:=pAdicField(23,20); > R<x>:=PolynomialRing(K); > C:=HyperellipticCurve(-x,x^3+x^2+1); // genus 2, conductor 23^2 > A:=GaloisRepresentation(C); A; 4-dim Galois representation Unr(1-46*x+529*x^2)*SP(2) over Q23[20]
If F/K is a finite extension, then the base change of A to F is the same as the Galois representation of C/F:
> F:=ext<K|2>; > BaseChange(A,F); 4-dim Galois representation Unr(1-1058*x+279841*x^2)*SP(2) over ext<Q23[20]|2> > GaloisRepresentation(BaseChange(C,F)); 4-dim Galois representation Unr(1-1058*x+279841*x^2)*SP(2) over ext<Q23[20]|2>
Degree: RngIntElt Default: ∞
Minimize: BoolElt Default: false
Galois representation associated to (H1 of) a hyperelliptic curve C/Q at p. Degree specifies that Euler factors of unramified pieces should only be computed up to that degree. (See Example H57E7.) Setting Minimize:=true forces the representation to be minimized. (See Minimize.)
> R<x>:=PolynomialRing(Rationals()); > C:=HyperellipticCurve((x^2+5)*(x+1)*(x+2)*(x+3)); > GaloisRepresentation(C,5); // bad reduction 4-dim Galois representation Unr(1+2*x+5*x^2) + Unr(5)*SP(2) over Q5[20] > GaloisRepresentation(C,11); // good reduction 4-dim unramified Galois representation Unr(1-2*x+6*x^2-22*x^3+121*x^4) over Q11[5] > GaloisRepresentation(C,997: Degree:=1); // don't count pts over GF(997^2) 4-dim unramified Galois representation Unr(1+26*x+O(x^2)) over Q997[5]
Degree: RngIntElt Default: ∞
Minimize: BoolElt Default: false
Galois representation associated to (H1 of) a hyperelliptic curve C over a number field at a prime ideal P. Degree specifies that Euler factors of unramified pieces should only be computed up to that degree. (See Example H57E7.) Setting Minimize:=true forces the representation to be minimized. (See Minimize.)
> K<zeta>:=CyclotomicField(11); > R<x>:=PolynomialRing(K); > C:=HyperellipticCurve(x^9+x^2+(zeta-1)); > P:=Ideal(Decomposition(K,11)[1,1]); > GaloisRepresentation(C,P); 8-dim Galois representation Unr(1-44*x^3+1331*x^6) + Unr(11)*SP(2) over ext<Q11[2]|x^10+22*x^9+55*x^8+44*x^7-33*x^6-22*x^5-22*x^4-33*x^3+44*x^2+ 55*x+11>
Precision: RngIntElt Default: 40
Local Galois representation at p of a modular form f. Currently only implemented when p2 does not divide the level.
> f:=Newforms("5k4")[1,1];
> GaloisRepresentation(f,3);
2-dim unramified Galois representation Unr(1-2*x+27*x^2) over Q3[40]
> GaloisRepresentation(f,5);
2-dim Galois representation Unr(-25)*SP(2) over Q5[40]