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Examples


Example RepLoc_example1 (H139E3)

We consider a newform of weight 5 and level 7, whose local representation at 7 is principal series.

> S := CuspidalSubspace(ModularSymbols(Gamma1(7), 5, 1));
> newforms := NewformDecomposition(S);
> Eigenform(newforms[1], 15);
q + q^2 - 15*q^4 + 49*q^7 - 31*q^8 + 81*q^9 - 206*q^11 + 49*q^14 + O(q^15)
> pi := LocalComponent(newforms[1], 7);
> pi;
Ramified Principal Series Representation of GL(2,Q_7)
> chi := CentralCharacter(pi);
> Conductor(chi);
7
> parameters := PrincipalSeriesParameters(pi);
These are Dirichlet characters on Z/7Z (the trivial character and the character of order 2):

> Conductor(parameters[1]), Order(parameters[1]);
1 1
> Conductor(parameters[2]), Order(parameters[2]);
7 2
The principal series representation πis the induction up to GL2(Q7) of a character of the Borel subgroup inflated from a character of the diagonal group Q7 x x Q7 x . The restriction of this character to Z7 x x Z7 x gives the pair of Dirichlet characters above. We now compute the Galois representation.

> G, map, F, rho := WeilRepresentation(pi);
> F;
Totally ramified extension defined by the polynomial 
x^6 + 7*x^5 + 21*x^4 + 35*x^3 + 35*x^2 + 21*x + 7
over 7-adic ring mod 7^10
> IsAbelian(G);
true
The Weil representation is simply the sum of the two characters above, considered as characters of the Galois group of Q7 via local class field theory.

> IsIrreducible(rho);
false 
GModule of dimension 1 over Cyclotomic Field of order 6 and degree 2
GModule of dimension 1 over Cyclotomic Field of order 6 and degree 2

Example RepLoc_example2 (H139E4)

We consider a supercuspidal representation of conductor 121, associated to a newform of weight 2 and level 121.

> S := CuspidalSubspace(ModularSymbols(Gamma0(121), 2, 1));
> newforms := NewformDecomposition(S);
> newforms;
[
    Modular symbols space for Gamma_0(121) of weight 2 and dimension 1 over Q,
    Modular symbols space for Gamma_0(121) of weight 2 and dimension 1 over Q,
    Modular symbols space for Gamma_0(121) of weight 2 and dimension 1 over Q,
    Modular symbols space for Gamma_0(121) of weight 2 and dimension 1 over Q,
    Modular symbols space for Gamma_0(121) of weight 2 and dimension 2 over Q
]
> Eigenform(newforms[2], 11);
q + q^2 + 2*q^3 - q^4 + q^5 + 2*q^6 - 2*q^7 - 3*q^8 + q^9 + q^10 + O(q^11)
> pi := LocalComponent(newforms[2], 11);
> pi;
Supercuspidal Representation of GL(2,Q_11)
This means the representation of the Weil group associated to pi is irreducible.

> Conductor(pi);
121
> W := CuspidalInducingDatum(pi);
> W;
GModule W of dimension 10 over Rational Field
W is a module over a group which is a quotient of GL2(Z11), namely GL2(Z/(11Z)). The representation πis induced from some extension of W to the open subgroup Q11 x GL2(Z11).

> Group(W); 
MatrixGroup(2, IntegerRing(11)) of order 2^4 * 3 * 5^2 * 11
Generators:
    [2 0]
    [0 1]


    [1 1]
    [0 1]


    [ 0  1]
    [10  0]
> Group(W) eq GL(2, Integers(11));
true
> G, alpha, L, rho := WeilRepresentation(pi);
This gives the Weil representation attached to πup to multiplication by an unramified twist. This consists of a permutation group G, an isomorphism alpha identifying G with the automorphisms of the local field L/Qp, and a 2-dimensional G-module rho.

> G; 
Permutation group G acting on a set of cardinality 6
Order = 6 = 2 * 3
    Id(G)
    (1, 2, 4)(3, 6, 5)
    (1, 3)(2, 5)(4, 6)
> bool := IsIsomorphic(G, DihedralGroup(3));
> bool;
true 
> L;
Totally ramified extension defined by the polynomial x^3 + 11
 over Unramified extension defined by the polynomial x^2 + 7*x + 2
 over 11-adic field mod 11^10

Example RepLoc_example3 (H139E5)

We consider a supercuspidal representation of conductor 33, associated to a newform of weight 4 and level 27.

> S := CuspidalSubspace(ModularSymbols(Gamma0(27), 4, 1));
> newforms := NewformDecomposition(S);
> Eigenform(newforms[1], 14);
q + 3*q^2 + q^4 + 15*q^5 - 25*q^7 - 21*q^8 + 45*q^10 - 15*q^11 + 20*q^13 + O(q^14)
> pi:=LocalComponent(newforms[1], 3); 
> pi;
Supercuspidal Representation of GL(2,Q_3)
> W:=CuspidalInducingDatum(pi);
> W;
GModule W of dimension 2 over Rational Field
> Group(W);
MatrixGroup(2, IntegerRing(9)) of order 2^2 * 3^5
Generators:
    [1 1]
    [0 1]


    [2 0]
    [0 1]


    [1 0]
    [0 2]


    [1 0]
    [3 1]
These matrices generate (topologically) the Iwahori subgroup of GL2(Z3) consisting of matrices which are upper-triangular modulo 3. W is an irreducible two-dimensional G-module. The representation πis induced from some extension of W to the normalizer of the Iwahori in GL2(Q3).

> E, chi:=AdmissiblePair(pi);
> E;
Totally ramified extension defined by the polynomial x^2 - 3
 over 3-adic ring mod 3^10
> E.1^2;
3
> chi(1+E.1);
zeta_3
Note that chi can only be evaluated on units of E, so that chi(E.1) would result in an error.

> G, alpha, L, rho := WeilRepresentation(pi);
> L;
Totally ramified extension defined by the polynomial x^6 - 12*x^4 - 12*x^2 - 3
 over Unramified extension defined by the polynomial x + 2
 over 3-adic field mod 3^10
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Version: V2.19 of Mon Dec 17 14:40:36 EST 2012