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 Tp 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 fn(Tpn), where I is a sequence [< p1, f1(x) >, ..., < pn, fn(x) >] of pairs consisting of a prime number and a polynomial. Only primes pi 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.)

Example ModSym_Subspaces (H142E11)

First we compute the cuspidal subspace of the space of modular symbols for Γ0(11). 
> 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
Next we compute the Eisenstein subspace. 
> 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
The Eisenstein subspace is the complement of the cuspidal subspace, and conversely. 
> E eq Complement(C);
true
> C eq Complement(E);
true

Example ModSym_BoundaryMap (H142E12)

> 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]
Observe that the Eisenstein subspace of A is not in the kernel of the boundary map. 
> Basis(VectorSpace(EisensteinSubspace(A)));
[
    (0 0 0 1 3)
]
V2.28, 13 July 2023