[Next][Prev] [Right] [Left] [Up] [Index] [Root] 
The Intersection Pairing






Magma
can compute the intersection pairing 
 H_1(X_0(N), Q) x H_1(X_0(N), Q) -> Q
on the homology of the modular curve X_0(N).
The algorithm that we implemented is essentially the one given in
[Mer93].
(Warning: There is a typo in Proposition 4 
of [Mer93]; W_i should be 
replaced by W_i^(eps_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 ModSym_IntersectionPairing (H88E18)

In this example, we illustrate several basic properties of
the intersection pairing on H_1(X_0(37), Z).  
First, let H37 be the space of modular
symbols that corresponds to H_1(X_0(37), 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}
]


Now we compute some intersection numbers.





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




The intersection pairing is perfect and skew-symmetric,
so the matrix that defines it is skew-symmetric and has determinant +- 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


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


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




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



 [Next][Prev] [Right] [Left] [Up] [Index] [Root]