Arithmetic

A1 + A2 : GalRep,GalRep -> GalRep
Direct sum of two Galois representations, both defined over the same p-adic field K.

Example GalRep_galrep-sum (H57E41)

> K:=pAdicField(2,20);
> SP(K,1)+SP(K,2)+SP(K,3);
6-dim Galois representation 1 + SP(2) + SP(3) over Q2[20]
A1 - A2 : GalRep,GalRep -> BoolElt
Assuming A2 is a Galois subrepresentation of A1, compute A1 - A2.

Example GalRep_galrep-difference (H57E42)

We take K=Q2, F its unique degree 3 unramified extension, and compute the regular representation Gal(F/K) minus the trivial representation, as a Galois representation over K. 
> K:=pAdicField(2,20);
> F:=ext<K|3>;
> PermutationCharacter(F,K)-PrincipalCharacter(K);
2-dim unramified Galois representation (2,-1,-1) with G=C3, I=C1 over Q2[20]
A1 * A2 : GalRep,GalRep -> GalRep
Tensor product of two Galois representations, both defined over the same p-adic field K.

Example GalRep_galrep-product (H57E43)

We take K=Q2 and compute SP(K,3) tensor SP(K,3); this is basically the Clebsch-Gordan decomposition. 
> K:=pAdicField(2,20);
> SP(K,3)*SP(K,3);
9-dim Galois representation Unr(1/4) + Unr(1/2)*SP(3) + SP(5) over Q2[20]
A1 / A2 : GalRep,GalRep -> GalRep
A1 / A2 : RngIntElt,GalRep -> GalRep
Tensor A1 with A2 - 1, for 1-dimensional A2. (A1 may also be the integer 0 or the integer 1.)

A ^ n : GalRep,Any -> GalRep
Tensor power of a Galois representation. The power n should be a non-negative integer for a general representation, but may be negative for 1-dimensional representations, and an arbitrary complex number for (powers of) the cyclotomic character.

Example GalRep_galrep-power (H57E44)

> K:=pAdicField(2,20);
> CyclotomicCharacter(K)^(1/2);
1-dim unramified Galois representation Unr(1/2*sqrt(2)) over Q2[20]
> SP(K,2)^2 / CyclotomicCharacter(K);
4-dim Galois representation 1 + Unr(2)*SP(3) over Q2[20]
A1 eq A2 : GalRep,GalRep -> BoolElt
Return true if the two Galois representations are equal.

Example GalRep_galrep-eq (H57E45)

> K:=pAdicField(2,20);
> w:=CyclotomicCharacter(K);
> A:=Semisimplification(SP(K,2));
> A eq w^0+w;
true
Determinant(A) : GalRep -> GalRep
Determinant of a Galois representation (a 1-dimensional Galois representation).

Example GalRep_galrep-determinant (H57E46)

> 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]
> Determinant(A) eq CyclotomicCharacter(K)^(-1);
true
TateTwist(A,n) : GalRep,RngIntElt -> GalRep
Tate twist A(n) of a Galois representation. So A(n)=A tensor w tensor n where w is the cyclotomic character.

Example GalRep_galrep-tatetwist (H57E47)

> K:=pAdicField(5,20);
> R<x>:=PolynomialRing(K);
> A:=GaloisRepresentations(x^2-2)[2];
> A;
1-dim unramified Galois representation (1,-1) with G=C2, I=C1 over Q5[20]
> TateTwist(A,1);
1-dim unramified Galois representation Unr(1/5)*(1,-1) with G=C2, I=C1
   over Q5[20]
> [EulerFactor(TateTwist(A,n)): n in [-2..2]];
[ 25*x + 1, 5*x + 1, x + 1, 1/5*x + 1, 1/25*x + 1 ]
V2.28, 13 July 2023