Associated Spaces

BrandtModule(M): ModSS -> ModBrdt

The Brandt module associated to the supersingular module \(M\).

ModularSymbols(M : parameters): ModSS -> ModSym

Proof: BoolElt                    Default: true

The space of modular symbols corresponding to the supersingular module \(M\).

ModularSymbols(M, sign : parameters): ModSS, RngIntElt -> ModSym

Proof: BoolElt                    Default: true

The \(+1\) or \(-1\) quotient of the space of modular symbols corresponding to the supersingular module \(M\).

RSpace(M): ModSS -> ModTupRng, Map

The \({\mathbb{Z}}\)-module \(V\) underlying the supersingular module \(M\) along with an invertible map \(V \rightarrow M\).

Example: Associated (ex-16e86e)

We compute the Brandt module and modular symbols spaces associated to the supersingular module for \(p=3\), \(N=11\), and verify that \(T_2\) has the same characteristic polynomial on each.

> M := SupersingularModule(3,11);
> B := BrandtModule(M); B;
Brandt module of level (3,11), dimension 2, and degree 2 over
Integer Ring
> MS := ModularSymbols(M); MS;
Modular symbols space for Gamma_0(33) of weight 2 and dimension 4
over Rational Field
> Factorization(CharacteristicPolynomial(HeckeOperator(B, 2)));
[
    <$.1 - 3, 1>,
    <$.1 - 1, 1>
]
> Factorization(CharacteristicPolynomial(HeckeOperator(MS, 2)));
[
    <$.1 - 3, 2>,
    <$.1 - 1, 2>
]

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There is an associated Brandt module even if the underlying computations on \(M\) are done using the Mestre-Oesterle graph method.

> M := SupersingularModule(11);
> UsesMestre(M);
true
> B := BrandtModule(M); B;   // takes a while
Brandt module of level (11,1), dimension 2, and degree 2 over
Integer Ring

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