Hecke and Atkin-Lehner Operators

Creation

These commands compute endomorphisms induced by the Atkin-Lehner and Hecke operators on modular abelian varieties. The Atkin-Lehner involution \(W_q\) is defined for each positive integer \(q\) that exactly divides the level (and is divisible by the conductor of any relevant character).

AtkinLehnerOperator(A, q): ModAbVar, RngIntElt -> MapModAbVar, RngIntElt

The Atkin-Lehner operator \(W_q\) of index \(q\) induced on the abelian variety \(A\) by virtue of \(A\) being modular. In general \(W_q\) 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 \(d*W_q\) is an endomorphism of \(A\), and when \(W_q\) 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 \(q\).

AtkinLehnerOperator(A): ModAbVar -> MapModAbVar

The morphism (or morphism tensor Q) on (or from) the abelian variety \(A\) induced by the Atkin-Lehner operator.

HeckeOperator(A, n): ModAbVar, RngIntElt -> MapModAbVar

The Hecke operator \(T_n\) of index \(n\) induced on the abelian variety \(A\) by virtue of its morphism to a modular symbols abelian variety. In general \(T_n\) need not be a morphism. Also, if \(A\) is contained in e.g., \(J_0(N)\), then the \(T_n\) on \(J_0(N)\) need not even leave \(A\) invariant. In that case this command composes \(T_n\) with a map back to \(A\) to obtain an endomorphism of \(A\). For the exact Hecke operators induced by their action on \(J_0(N)\), say, use the RestrictEndomorphism command.

Example: Operators Creation (ex-6db02a)

We compute the main Atkin-Lehner operator and the Hecke operator \(T_2\) on \(J_0(23)\).

> 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]

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Next we compute \(w_4\) and \(w_{25}\) on \(J_{100}\), and note that their product equals \(w_{100}\).

> 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

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Next we compute \(W_{25}\) acting on \(J_1(25)\).

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

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Finally we compute Hecke operators on the quotient of a simple factor of \(J_0(65)\) 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>
]

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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 \(A\).

Example: Operators Invariants (ex-605974)

> 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

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