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 \(p\)th Hecke operator (see Section Operators). This is the subspace of \(M\) obtained by intersecting the kernels of the operators \(f_n(T_{p_n})\), where \(I\) is a sequence \([\langle p_1, f_1(x)\rangle,...,\langle p_n,f_n(x)\rangle]\) 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.)

Example: Subspaces (ex-a39393)#

First we compute the cuspidal subspace of the space of modular symbols for \(\Gamma_0(11)\).

> 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

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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

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The Eisenstein subspace is the complement of the cuspidal subspace, and conversely.

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

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Example: Boundary Map (ex-2e2403)#

> 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]

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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)
]

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