An ambient supersingular points module is specified by giving the prime p and the level N.
Brandt: BoolElt Default: false
The module M of supersingular points on X_0(N)/F_p, where N = 1 if not provided, p is prime, and N is a positive integer coprime to p. Equivalently, this is the free abelian group on the isomorphism classes of supersingular elliptic curves in characteristic p enhanced with level N structure. If N = 1, then the basis of supersingular points are given by the supersingular j-invariants in F_(p^2).
> SupersingularModule(11); Supersingular module associated to X_0(1)/GF(11) of dimension 2 > CuspidalSubspace($1); Module of supersingular points on X_0(1)/GF(11) of dimension 1So we verify that the supersingular cuspidal space of level 11 is of dimension 1 --- precisely that associated to the isogeny class of the elliptic curve of level 11. In the full space of modular forms or modular symbols of level 33, there are be two copies of this oldspace of lower level.
> M33 := ModularSymbols(33,2,1); > S33 := CuspidalSubspace(M33); > S33; Modular symbols space of level 33, weight 2, and dimension 3 > M11x3 := SupersingularModule(11,3); > S11x3 := CuspidalSubspace(M11x3); > S11x3; Module of supersingular points on X_0(3)/GF(11) of dimension 3 > NewSubspace(S33,11); Modular symbols space of level 33, weight 2, and dimension 3 > M3x11 := SupersingularModule(3,11); > S3x11 := CuspidalSubspace(M3x11); > S3x11; Module of supersingular points on X_0(11)/GF(3) of dimension 1 > NewSubspace(S33,3); Modular symbols space of level 33, weight 2, and dimension 1The supersingular module construction selects out the p-new part of the corresponding Hecke module, with the advantage of not first computing a larger dimensional ambient space.
The ith basis element of M.
The coercion of x into M.
> X := SupersingularModule(11,3); > P := X.1; > P; (5, 5) > Eltseq(P); [ 1, 0, 0, 0 ] > X![ 1, 0, 0, 0 ]; (5, 5)The module associated to p=3, N=11 is computed using Brandt matrices (since X_0(11) has genus >0), so elements are printed differently.
> Z := SupersingularModule(3,11); Z; Supersingular module associated to X_0(11)/GF(3) of dimension 2 > P := Z.1; > P; [E1] > Eltseq(P); [ 1, 0 ] > Z![1,0]; [E1]
The cuspidal submodule X of M. Thus X is the submodule of divisors of degree 0 on the supersingular points. It is "cuspidal" in the sense that X tensor Q is isomorphic as a Hecke module to the space S_2(Gamma_0(Np);Q)^(p - new) of p-new cuspforms with Fourier coefficients in Q.
The Eisenstein submodule of M, i.e., the orthogonal complement of the cuspidal subspace of M with respect to the monodromy pairing.
The orthogonal complement of M in the ambient space with respect to the monodromy pairing.
The kernel of I on M. This is the subspace of M obtained by intersecting the kernels of the operators f_i(T_(p_i)), where I is a sequence [< p_1, f_1(x) >, ..., < p_n, f_n(x) >] of pairs consisting of a prime number and a polynomial.
Decomposition of M with respect to the Hecke operators T_1, T_2, ..., T_n. (Warning: The algorithm is currently very simple and no real caching is done.)
> M := SupersingularModule(11); Basis(M);
[
(1, 1),
(0, 0)
]
> S := CuspidalSubspace(M);
> E := EisensteinSubspace(M);
> Basis(S);
[
(1, 1) - (0, 0)
]
> Basis(E);
[
3*(1, 1) + 2*(0, 0)
]
Next we test the orthogonal complement.
> Basis(OrthogonalComplement(E));
[
(1, 1) - (0, 0)
]
> Basis(OrthogonalComplement(S));
[
3*(1, 1) + 2*(0, 0)
]
> S eq OrthogonalComplement(E);
true
The Hecke operator T_2 acts as -2 on the
cuspidal subspace. We can "cut out" the subspace
of M with this property as follows.
> R<x> := PolynomialRing(Integers());
> I := [<2, x + 2>];
> K := Kernel(I,M);
> Basis(K);
[
(1, 1) - (0, 0)
]
We can also directly as for a complete decomposition of M into
submodules for the first few Hecke operators.
> Decomposition(M,5);
[
Supersingular module associated to X_0(1)/GF(11) of dimension 1,
Supersingular module associated to X_0(1)/GF(11) of dimension 1
]