These commands compute endomorphisms induced by the Atkin-Lehner and Hecke operators on modular abelian varieties. The Atkin-Lehner involution Wq is defined for each positive integer q that exactly divides the level (and is divisible by the conductor of any relevant character).
The Atkin-Lehner operator Wq of index q induced on the abelian variety A by virtue of A being modular. In general Wq need not be a morphism except in the category of abelian varieties up to isogeny so this intrinsic also returns an integer d such that d * Wq is an endomorphism of A, and when Wq doesn't leave A invariant, returns d=0. If the ambient modular symbols space of A contains a space with character of conductor r, then currently an error occurs unless r divides q.
The morphism (or morphism tensor Q) on (or from) the abelian variety A induced by the Atkin-Lehner operator.
The Hecke operator Tn of index n induced on the abelian variety A by virtue of its morphism to a modular symbols abelian variety. In general Tn need not be a morphism. Also, if A is contained in e.g., J0(N), then the Tn on J0(N) need not even leave A invariant. In that case this command composes Tn with a map back to A to obtain an endomorphism of A. For the exact Hecke operators induced by their action on J0(N), say, use the RestrictEndomorphism command.
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]Next we compute w4 and w25 on J100, and note that their product equals w100.
> 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
Next we compute W25 acting on J1(25).
> 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>
]
Intrinsics are provided which compute characteristic polynomials, factored characteristic polynomials and minimal polynomials of Hecke operators.
The characteristic polynomial of the Hecke operator Tn acting on the abelian variety A.
The factored characteristic polynomial of the Hecke operator Tn acting on the abelian variety A. This can be faster than first computing Tn, then computing the characteristic polynomial, and factoring, because we can take into account information about the decomposition of A, in order to avoid factoring.
The minimal polynomial of the Hecke operator Tn 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