The standard arithmetic operations + and - and left scalar
multiplication are defined for elements of
supersingular modules. Also one can add and intersect
two submodules of an ambient supersingular module.
First we illustrate some arithmetic on elements.
> M := SupersingularModule(11);
> P := M.1; P;
(1, 1)
> Q := M.2; Q;
(0, 0)
> P + Q;
(1, 1) + (0, 0)
> P - Q;
(1, 1) - (0, 0)
> 3*P;
3*(1, 1)
Next we illustrate some arithmetic on submodules.
> E := EisensteinSubspace(M);
> S := CuspidalSubspace(M);
> V := E + S;
> V;
Supersingular module associated to X_0(1)/GF(11) of dimension 2
> Basis(V);
[
(1, 1) + 4*(0, 0),
5*(0, 0)
]
The index of E + S in M is of interest since it is related
to congruences between Eisenstein series and cusp forms.
Upon converting each of E and S to an
RSpace,
we find that the index is 5.
> RSpace(M)/RSpace(V);
Full Quotient RSpace of degree 1 over Integer Ring
Column moduli:
[ 5 ]
The intersection of E and S is the zero module.
> W := E meet S; W;
Supersingular module associated to X_0(1)/GF(11) of dimension 0
> Basis(W);
[]
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