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The Monodromy Pairing










The monodromy pairing is a nondegenerate (imperfect) Hecke-equivariant
integer-valued pairing on the module of supersingular points.  This
pairing is diagonal with respect to the basis of enhanced
supersingular elliptic curves, and the pairing of an enhanced curve
with itself is half the number of automorphisms of that enhanced
curve.



MonodromyPairing(P, Q) : ModSSElt, ModSSElt -> RngIntElt



The monodromy pairing of P and Q.



MonodromyWeights(M) : ModSS -> SeqEnum



The diagonal entries that define the monodromy pairing on the
ambient space of M.





Example ModSS_Monodromy pairing (H92E9)

For prime level p = 11 mod 12, the elliptic curves of 
j-invariants 0 and 12^3, with extra automorphisms, are 
supersingular.  This gives rise to a nontrivial monodromy pairing 
on the supersingular module, which we demonstrate for p = 23
--- note that j = 12^3 = 3 mod 23.





> M := SupersingularModule(23);
> MonodromyWeights(M);
[ 2, 1, 3 ]
> Basis(M);
[
    (3, 3),
    (19, 19),
    (0, 0)
]
> S := CuspidalSubspace(M);
> P1 := M!Basis(S)[1];
> P2 := M!Basis(S)[2];
> P1;
(3, 3) - (0, 0)
> P2;
(19, 19) - (0, 0)
> E := EisensteinSubspace(M);
> P0 := M!Basis(E)[1];
> P0;
3*(3, 3) + 6*(19, 19) + 2*(0, 0)
> MonodromyPairing(P0,P1);
0
> MonodromyPairing(P0,P2);
0
> Matrix(2,[ MonodromyPairing(P,Q) : P, Q in [P1,P2] ]);
[5 3]
[3 4]
> Determinant(1);
11


For an application the monodromy pairing to the computation 
of the order of component groups of J_0(Np), we refer to 
Kohel - Stein cite(kohel - stein - ants4).  In this example the 
modular Jacobian J_0(23) is an abelian surface, whose 
component group at 23 is of order 11 - - - the determinant 
of this monodromy pairing giving precisely this invariant. --- the determinant 
of this monodromy pairing giving precisely this invariant.

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