The Intersection Pairing

Magma can compute the intersection pairing

\[H_1(X_0(N),{\mathbb{Q}}) \times H_1(X_0(N),{\mathbb{Q}}) \rightarrow {\mathbb{Q}}\]

on the homology of the modular curve \(X_0(N)\). The algorithm that we implemented is essentially the one given in [Merel, 1993]. (Warning: There is a typo in Proposition 4 of [Merel, 1993]; \(W_i\) should be replaced by \(W_i^{\varepsilon_i}\).)

IntersectionPairing(x, y): ModSymElt, ModSymElt -> FldRatElt

The intersection pairing of the homology classes corresponding to the weight-\(2\) cuspidal modular symbols \(x\) and \(y\). The symbols \(x\) and \(y\) must have the same parent, which must have trivial character and not be a \(+1\) or \(-1\) quotient.

Example: Intersection Pairing (ex-3889b3)

In this example, we illustrate several basic properties of the intersection pairing on \(H_1(X_0(37),{\mathbb{Z}})\). First, let H37 be the space of modular symbols that corresponds to \(H_1(X_0(37),{\mathbb{Z}})\), and compute a basis for H37.

> M37 := ModularSymbols(37,2);
> H37 := CuspidalSubspace(M37);
> Z := IntegralBasis(H37); Z;
[
    {-1/29, 0},
    {-1/22, 0},
    {-1/12, 0},
    {-1/18, 0}
]

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Now we compute some intersection numbers.

> IntersectionPairing(Z[1],Z[2]);
-1
> IntersectionPairing(Z[3],Z[4]);
0

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The intersection pairing is perfect and skew-symmetric, so the matrix that defines it is skew-symmetric and has determinant \(\pm 1\) (in fact, it has determinant \(+1\)).

> A := MatrixAlgebra(RationalField(),4);
> I := A![IntersectionPairing(x,y) : x in Z, y in Z]; I;
[ 0  1  0  1]
[-1  0  1  1]
[ 0 -1  0  0]
[-1 -1  0  0]
> I + Transpose(I) eq 0;
true
> Determinant(I);
1

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The Hecke operators are compatible with the intersection pairing in the sense that \((T_n x, y) = (x, T_n y).\)

> T2 := HeckeOperator(M37,2);
> IntersectionPairing(Z[1]*T2,Z[2]);
1
> IntersectionPairing(Z[1],Z[2]*T2);
1

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It is note the case \((T_n x, T_n y) = (x, y)\) for all \(n\), \(x\), and \(y\).

> IntersectionPairing(Z[1]*T2,Z[2]*T2);
-2

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The existence of the intersection pairing implies that \(H_1(X_0(N),{\mathbb{Z}})\) is isomorphic, as a module over the Hecke algebra, to its linear dual \({\operatorname{Hom}}(H_1(X_0(N),{\mathbb{Z}}),{\mathbb{Z}})\).