Example ModAbVar_Operators-Creation (H148E112)

We compute the main Atkin-Lehner operator and the Hecke operator T₂ on J₀(23).

> A := JZero(23);
> AtkinLehnerOperator(A,23);
Homomorphism W23 from JZero(23) to JZero(23) given on integral homology by:
[-1  0  0  0]
[ 0 -1  0  0]
[ 0  0 -1  0]
[ 0  0  0 -1]
> HeckeOperator(A,2);
Homomorphism T2 from JZero(23) to JZero(23) given on integral homology by:
[ 0  1 -1  0]
[ 0  1 -1  1]
[-1  2 -2  1]
[-1  1  0 -1]

> A := JZero(100); A;
Modular abelian variety JZero(100) of dimension 7 and 
level 2^2*5^2 over Q
> w4 := AtkinLehnerOperator(A,4); 
> Factorization(CharacteristicPolynomial(w4));
[
    <x - 1, 4>,
    <x + 1, 10>
]
> w25 := AtkinLehnerOperator(A,25); 
> Factorization(CharacteristicPolynomial(w25));
[
    <x - 1, 8>,
    <x + 1, 6>
]
> w4*w25 eq AtkinLehnerOperator(A);
true

> A := Js(17);
> B := BaseExtend(A,CyclotomicField(17));
> w := AtkinLehnerOperator(B);
> Factorization(CharacteristicPolynomial(w));
[
    <x - 1, 4>,
    <x + 1, 6>
]

Finally we compute Hecke operators on the quotient of a simple factor of J₀(65) by a finite subgroup.

> A := Decomposition(JZero(65))[2]; A;
Modular abelian variety 65B of dimension 2, level 5*13 and conductor 
5^2*13^2 over Q
> G := nTorsionSubgroup(A,2); G;
Finitely generated subgroup of abelian variety with invariants 
[ 2, 2, 2, 2 ]
> H := Subgroup([G.1]); H;
Finitely generated subgroup of abelian variety with invariants [ 2 ]
> B := A/H; B;
Modular abelian variety of dimension 2 and level 5*13 over Qbar
> T2 := HeckeOperator(B,2); T2;
Homomorphism from modular abelian variety of dimension 2 to 
modular abelian variety of dimension 2 (up to isogeny) on 
integral homology by:
[ -2 1/2   0   0]
[ -2   2   0   0]
[ -2   1  -2   1]
[ -6   1  -1   2]
> FactoredCharacteristicPolynomial(T2);
[
    <x^2 - 3, 2>
]

Invariants

Intrinsics are provided which compute characteristic polynomials, factored characteristic polynomials and minimal polynomials of Hecke operators.

HeckePolynomial(A, n) : ModAbVar, RngIntElt -> RngUPolElt

The characteristic polynomial of the Hecke operator T_n acting on the abelian variety A.

FactoredHeckePolynomial(A, n) : ModAbVar, RngIntElt -> RngUPolElt

The factored characteristic polynomial of the Hecke operator T_n acting on the abelian variety A. This can be faster than first computing T_n, then computing the characteristic polynomial, and factoring, because we can take into account information about the decomposition of A, in order to avoid factoring.

MinimalHeckePolynomial(A, n) : ModAbVar, RngIntElt -> RngUPolElt

The minimal polynomial of the Hecke operator T_n acting on the abelian variety A.

> FactoredHeckePolynomial(JZero(65),2);
[
    <x + 1, 2>,
    <x^2 - 3, 2>,
    <x^2 + 2*x - 1, 2>
]
> HeckePolynomial(JZero(65),2);
x^10 + 6*x^9 + 5*x^8 - 32*x^7 - 62*x^6 + 28*x^5 + 130*x^4 + 
    48*x^3 - 51*x^2 - 18*x + 9
> MinimalHeckePolynomial(JZero(65),2);
x^5 + 3*x^4 - 2*x^3 - 10*x^2 - 3*x + 3