Arithmetic#

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.

P + Q: ModSSElt, ModSSElt -> ModSSElt#
P - Q: ModSSElt, ModSSElt -> ModSSElt#
a * P: RngElt, ModSSElt -> ModSSElt#
M1 + M2: ModSS, ModSS -> ModSS#: The submodule generated by all sums of elements in the supersingular modules \(M_1\) and \(M_2\).

M1 meet M2: ModSS, ModSS -> ModSS#: The intersection of the supersingular modules \(M_1\) and \(M_2\).

Example: Arithmetic (ex-a36e45)#

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)

Run in calculator

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)
]

Run in calculator

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 ]

Run in calculator

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);
[]

Run in calculator