Example ModAbVar_Homology-Creation (H145E44)

The homology of the elliptic curve J₀(14) is of dimension 2.

> A := JZero(14); A;
Modular abelian variety JZero(14) of dimension 1 and level 2*7 over Q
> Homology(A);
Modular abelian variety homology space of dimension 2

If, for efficiency purposes, we work in the +1 quotient, then we are only working with half the homology, so the dimension of the homology is 1.

> Homology(JZero(14,2 : Sign := +1));
Modular abelian variety homology space of dimension 1

Example ModAbVar_Homology-Invariants (H145E45)

> Dimension(Homology(JZero(100)));
14
> Dimension(Homology(JZero(100,2 : Sign := +1)));
7
> Dimension(Homology(JZero(100,2 : Sign := -1)));
7

Example ModAbVar_additional8 (H145E46)

The integral, rational and real homology of a modular abelian variety A is H₁(A, R), where R=Z, R, Q, respectively. This homology is just a free module over R of dimension twice dim(A) (unless the sign of A is ∓ 1, in which case the homology is of dimension dim(A)).

> J := JZero(22); J;
Modular abelian variety JZero(22) of dimension 2 and level 2*11 over Q
> IntegralHomology(J);
Standard Lattice of rank 4 and degree 4
> RationalHomology(J);
Full Vector space of degree 4 over Rational Field
> RealHomology(J);
Full Vector space of degree 4 over Real Field
> J := JZero(22, 2, +1); J;
Modular abelian variety JZero(22) of dimension 2 and level 2*11 over Q
with sign 1
> RationalHomology(J);
Full Vector space of degree 2 over Rational Field

Example ModAbVar_Homology-Functors_to_Categories_of_Lattices_and_Vector_Spaces (H145E47)

The following code demonstrates the above commands applied to the abelian surface attached to the space of cusp forms of level 37 with quadratic character.

> G<eps> := DirichletGroup(37);
> Order(eps);
2
> A := ModularAbelianVariety(eps); A;
Modular abelian variety of dimension 2 and level 37 over Q
> Decomposition(A);
[
    Modular abelian variety image(37A[18]) of dimension 2, level
    37 and conductor 37^2 over Q
]
> IntegralHomology(A);
Standard Lattice of rank 4 and degree 4
> H := Homology(A); H;
Modular abelian variety homology space of dimension 4
> Lattice(H);
Standard Lattice of rank 4 and degree 4
> RationalHomology(A);
Full Vector space of degree 4 over Rational Field
> RealHomology(A);
Full Vector space of degree 4 over Real Field
> RealVectorSpace(H);
Full Vector space of degree 4 over Real Field
> VectorSpace(H);
Full Vector space of degree 4 over Rational Field

Example ModAbVar_Homology-Functors_to_Categories_of_Lattices_and_Vector_Spaces2 (H145E48)

The code below illustrates that the lattice need not just be the free lattice on dim(H) generators. For efficiency reasons many internal computations only require knowing H₁(A, Q), which is sometimes all that gets computed, and the lattice returned by this command is H₁(A, Z) written with respect to a basis for H₁(A, Q). Thus Lattice should be viewed as giving an integral structure of H₁(A, Q).

> J := JZero(37);
> A := J(1);
> Lattice(Homology(A));
Lattice of rank 2 and degree 2
Basis:
( 1  1)
( 1 -1)
> IntegralHomology(JOne(17));
Lattice of rank 10 and degree 10
Basis:
( 0  0  1  0  0  0  1  0  0  0)
( 0  0  1  0  0  0  0  0 -1  0)
( 0  0  0  1  0  0  0  1  0  0)
( 0  0  0  1  0  0  0 -1  0  0)
( 0  0  0  0  1  0  0  0 -1  0)
( 0  0  0  0  0  1  0  0  0  1)
( 0  0  0  0  0  1  0  0  0 -1)
( 0  0  0  0  0  0  1  0 -1  0)
( 0  2  1  0  0  0  0  0  0  0)
( 2 -1  0  0  0  0  1 -1  0  1)

Example ModAbVar_Homology-Modular_Structure (H145E49)

In this example, we create the product J₀(23) x J₀(11), and find that its homology is attached modular symbols, since both J₀(23) and J₀(11) are attached to modular symbols. We then display the corresponding spaces of modular symbols.

> A := JZero(23) * JZero(11);
> H := Homology(A); H;
Modular abelian variety homology space of dimension 6
> IsAttachedToModularSymbols(H);
true
> ModularSymbols(H);
[
    Modular symbols space for Gamma_0(23) of weight 2 and
    dimension 4 over Rational Field,
    Modular symbols space for Gamma_0(11) of weight 2 and
    dimension 2 over Rational Field
]

Example ModAbVar_homology1 (H145E50)

The following example illustrates each of the above intrinsics for the homology of J₀(23).

> A := JZero(23); A;
Modular abelian variety JZero(23) of dimension 2 and level 23 over Q
> H := Homology(A); H;
Modular abelian variety homology space of dimension 4

Notice that homology has its own type:

> Type(H);
ModAbVarHomol

In this case, the homology is attached to a space of modular symbols, since J₀(23).

> IsAttachedToModularSymbols(H);
true
> ModularSymbols(H);
[
    Modular symbols space for Gamma_0(23) of weight 2 and
    dimension 4 over Rational Field
]
> Dimension(H);
4
> Lattice(H);
Standard Lattice of rank 4 and degree 4
> RealVectorSpace(H);
Full Vector space of degree 4 over Real Field
> VectorSpace(H);
Full Vector space of degree 4 over Rational Field

We can also obtain various homologies of A more directly.

> IntegralHomology(A);
Standard Lattice of rank 4 and degree 4
> RationalHomology(A);
Full Vector space of degree 4 over Rational Field
> RealHomology(A);
Full Vector space of degree 4 over Real Field

Example ModAbVar_additional20 (H145E51)

The following example illustrates that a factor of J₀(37) is not presented as something associated to a space of modular symbols. Also notice that the two lattices are different.

> J := JZero(37);
> D := Decomposition(J);
> H1 := Homology(D[1]);  H2 := Homology(D[2]);
> IsAttachedToModularSymbols(H1);
false
> Lattice(H1);
Lattice of rank 2 and degree 2
Basis:
( 1  1)
( 1 -1)
> Lattice(H2);
Standard Lattice of rank 2 and degree 2