Subspaces

The following functions compute the cuspidal, Eisenstein, and new subspaces, along with the complement of a subspace.

CuspidalSubspace(M) : ModSym -> ModSym

The cuspidal subspace of the space of modular symbols M. This is the kernel of BoundaryMap(M).

IsCuspidal(M) : ModSym -> BoolElt

Returns true if and only if the space of modular symbols M is contained in the cuspidal subspace of the ambient space.

EisensteinSubspace(M) : ModSym -> ModSym

The Eisenstein subspace of the space of modular symbols M. This is the complement in M of the cuspidal subspace of M.

IsEisenstein(M) : ModSym -> BoolElt

Returns true if and only if the space of modular symbols M is contained in the Eisenstein subspace of the ambient space.

NewSubspace(M) : ModSym-> ModSym

The new subspace of the space of modular symbols M. This is the intersection of NewSubspace(M,p) as p varies over all prime divisors of the level of M. Note that M is required to be cuspidal.

IsNew(M) : ModSym -> BoolElt

Returns true if and only if the space of modular symbols M is contained in the new cuspidal subspace of the ambient space.

NewSubspace(M, p) : ModSym, RngIntElt -> ModSym

The p-new subspace of the space of modular symbols M. This is the kernel of the degeneracy map from M to the space of modular symbols of level equal to the level of M divided by p and character the restriction of the character of M. If the character of M does not restrict, then NewSubspace(M,p) is equal to M. Note that M is required to be cuspidal.

Kernel(I, M) : [Tup], ModSym -> ModSym

The kernel of I on the space of modular symbols M. Let T_p denote the pth Hecke operator (see Section Hecke and Atkin-Lehner Operators). This is the subspace of M obtained by intersecting the kernels of the operators f_n(T_pn), where I is a sequence [< p₁, f₁(x) >, ..., < p_n, f_n(x) >] of pairs consisting of a prime number and a polynomial. Only primes p_i which do not divide the level of M are used.

Complement(M) : ModSym -> ModSym

The space of modular symbols complementary to the space of modular symbols M in the ambient space of M. Thus the ambient space of M is equal to the direct sum of M and Complement(M).

BoundaryMap(M) : ModSym -> ModMatFldElt

A matrix that represents the boundary map from the space of modular symbols M to the vector space whose basis consists of the weight k cusps. (Note: At present there is no intrinsic that lists these cusps.)

> M := ModularSymbols(11,2); M;
Full modular symbols space for Gamma_0(11) of weight 2 and dimension 3 
over Rational Field
> IsCuspidal(M);
false
> C := CuspidalSubspace(M); C;
Modular symbols space for Gamma_0(11) of weight 2 and dimension 2 over
Rational Field
> IsCuspidal(C);
true

> IsEisenstein(C);
false
> E := EisensteinSubspace(M); E;
Modular symbols space for Gamma_0(11) of weight 2 and dimension 1 over
Rational Field
> IsEisenstein(E);
true
> E + C eq M;
true

> E eq Complement(C);
true
> C eq Complement(E);
true

> M := ModularSymbols("37B"); M;
Modular symbols space for Gamma_0(37) of weight 2 and dimension 2 over
Rational Field
> BoundaryMap(M);
[0 0]
[0 0]
> A := AmbientSpace(M);
> BoundaryMap(A);
[ 0  0]
[ 0  0]
[ 0  0]
[ 0  0]
[ 1 -1]

> Basis(VectorSpace(EisensteinSubspace(A)));
[
    (0 0 0 1 3)
]