# 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.

```magma
> 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.

```magma
> 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 $5$.

```magma
> 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.

```magma
> W := E meet S; W;
Supersingular module associated to X_0(1)/GF(11) of dimension 0
> Basis(W);
[]

```