Creating Galois Representations

Contents

ZeroRepresentation(K) : FldPad -> GalRep
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.

Example GalRep_galrep-zerorepresentation (H58E2)

> 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
PrincipalCharacter(K) : FldPad -> GalRep
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.

Example GalRep_galrep-principalcharacter (H58E3)

Take K=Q3, and F/K the unramified extension of degree 4, so that G=Gal(F/K) isomorphic to C4. The 4 irreducible representations of G can be viewed as Galois representations, and the first one of these is the principal character (for any group). 

> 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
CyclotomicCharacter(K) : FldPad -> GalRep
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.

Example GalRep_galrep-cyclotomiccharacter (H58E4)

> 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
UnramifiedCharacter(K,c) : FldPad, Any -> GalRep
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.

Example GalRep_galrep-unramifiedcharacter (H58E5)

> 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]
UnramifiedRepresentation(K,CharPoly) : FldPad,RngUPolElt -> GalRep
Unique unramified Galois representation ρ over K with Euler factor det(1 - FrobK - 1|ρ)=(CharPoly).

Example GalRep_galrep-unramifiedrepresentation (H58E6)

> 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
UnramifiedRepresentation(K,dim,dimcomputed,CharPoly) : FldPad,RngIntElt,RngIntElt,RngUPolElt -> GalRep
Unramified Galois representation over K of dimension dim, with Euler factor CharPoly computed up to and inclusive degree dimcomputed.

Example GalRep_galrep-notfullycomputed (H58E7)

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
SP(K,n) : FldPad,RngIntElt -> GalRep
The n-dimensional indecomposable Galois representation SP(n) over a p-adic field K; see ParaNotation and Printing for its description.

Example GalRep_galrep-sp (H58E8)

> 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
SP(K,f,n) : FldPad,RngUPolElt,RngIntElt -> GalRep
Unramified twist ψ tensor (SP(n)) over a p-adic field K, with ψ specified by its Euler factor f.

Example GalRep_galrep-sp-1 (H58E9)

> 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]

Representations from Finite Extensions

GaloisRepresentations(F,K) : FldPad,FldPad -> SeqEnum[GalRep]
For a p-adic extension F/K, compute all irreducible Galois representations that factor through the (Galois closure of) F/K.

Example GalRep_galrep-galoisrepresentations-1 (H58E10)

We take F to be a degree 16 dihedral extension of K=Q2, and compute the irreducible characters of Gal(F/K), viewed as Galois representations over 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]
]
GaloisRepresentations(f) : RngUPolEltFldPad -> SeqEnum[GalRep]
For a polynomial f over a p-adic field K and splitting field F, returns irreducible representations of Gal(F/K).

Example GalRep_galrep-galoisrepresentations (H58E11)

We construct 4 one-dimensional characters of Q2(ζ8)/Q2. 

> 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]
]
PermutationCharacter(F,K) : FldPad,FldPad -> GalRep
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].

Example GalRep_galrep-permutationcharacter (H58E12)

Take K=Q2 and F=Q2(root 3 of 2). Then PermutationCharacter(F,K) is a 3-dimensional representation which is the trivial representation plus a 2-dimensional irreducible one. 

> 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]
A !! chi : GalRep, AlgChtrElt -> GalRep
A !! chi : GalRep, SeqEnum -> GalRep
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.

Example GalRep_galrep-smash (H58E13)

Take K=Q2 and F=Q2(root 3 of 2), so that G=Gal(F/K) isomorphic to S3. Using !! we can start with any Galois representation whose finite part comes from this Galois group, and replace it by any other character of G. 

> 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]

Local Representations of Global Objects

GaloisRepresentation(chi,p) : GrpDrchElt,RngIntElt -> GalRep
    Precision: RngIntElt                Default: 40
Local Galois representation at p of a Dirichlet character χ.

Example GalRep_galrep-dirichlet (H58E14)

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
]
GaloisRepresentation(A,p) : ArtRep,RngIntElt -> GalRep
    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 H58E14.) If Minimize is true, choose the field through which it factors to be as small as possible (automatic for faithful representations).

Example GalRep_galrep-artin (H58E15)

Local components of an Artin representation. We take the Trinks' polynomial x7 - 7x - 3 with Galois group PSL(2, 7) over Q, one of its 7-dimensional representations A of conductor 3878, and compute its local components over Q2, Q3, Q5 and Q7. 

> 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
GaloisRepresentation(E) : CrvEll -> GalRep
    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.

Example GalRep_galrep-E (H58E16)

Take an elliptic curve E/Q5, with additive (potentially good) reduction of type II. 

> 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
GaloisRepresentation(E,p) : CrvEll,RngIntElt -> GalRep
    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.

Example GalRep_galrep-E-p (H58E17)

We take the elliptic curve 20a1 over Q and compute its local Galois representation at a prime p=3 of good reduction, p=5 of (non-split) multiplicative reduction and p=2 of additive reduction. 

> 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
GaloisRepresentation(E,P) : CrvEll,RngOrdIdl -> GalRep
    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.

Example GalRep_galrep-E-P (H58E18)

> 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>
GaloisRepresentation(C) : CrvHyp[FldPad] -> GalRep
    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 H58E7.) Setting Minimize:=true forces the representation to be minimized. (See Minimize.)

Example GalRep_galrep-crvhyp-Qp (H58E19)

We take a curve C over K=Q23 of conductor 232 and compute its Galois representation. 

> 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>
GaloisRepresentation(C,p) : CrvHyp[FldRat], RngIntElt -> GalRep
    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 H58E7.) Setting Minimize:=true forces the representation to be minimized. (See Minimize.)

Example GalRep_galrep-crvhyp-Q (H58E20)

> 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]
GaloisRepresentation(C,P) : CrvHyp[FldNum],RngOrdIdl -> GalRep
    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 H58E7.) Setting Minimize:=true forces the representation to be minimized. (See Minimize.)

Example GalRep_galrep-crvhyp-nf (H58E21)

We take a curve of genus 4 over Q(ζ11) and compute its Galois representation at a unique prime P above 11. 

> 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>
GaloisRepresentation(f,p) : ModFrmElt,RngIntElt -> GalRep
    Precision: RngIntElt                Default: 40
Local Galois representation at p of a modular form f. Currently only implemented when p2 does not divide the level.

Example GalRep_galrep-f-p (H58E22)

We take a rational modular form of weight 4 and level 5, and compute its Galois representations at p=3 (unramified principal series) and p=5 (Steinberg). 

> 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]
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