Decomposition

The functions Decomposition and NewformDecomposition express a space of modular symbols as a direct sum of Hecke-stable subspaces.

In the intrinsics below, the Proof parameter affects the internal characteristic polynomial computations. If Proof is set to false and this causes a characteristic polynomial computation to fail, then the sum of the dimensions of the spaces returned by Decomposition will be less than the dimension of M. Thus setting Proof equal to false is usually safe.

Decomposition(M, bound : parameters): ModSym, RngIntElt -> SeqEnum

Proof: BoolElt                    Default: true

The decomposition of the space of modular symbols \(M\) with respect to the Hecke operators \(T_p\) with \(p\) coprime to the level of \(M\) and \(p \le\) bound. If bound is too small, the constituents of the decomposition are not guaranteed to be “irreducible”, in the sense that they can not be decomposed further into kernels and images of Hecke operators \(T_p\) with \(p\) prime to the level of \(M\). When Decomposition is called, the result is cached, so each successive call results in a possibly more refined decomposition.

Important Note: In some cases NewformDecomposition is significantly faster than Decomposition.

NewformDecomposition(M : parameters): ModSym -> SeqEnum

Proof: BoolElt                    Default: true

Unsorted decomposition of the space of modular symbols \(M\) into factors corresponding to the Galois conjugacy classes of newforms of level some divisor of the level of \(M\). We require that IsCuspidal(M) is true.

AssociatedNewSpace(M): ModSym -> ModSym

The space of modular symbols corresponding to the Galois-conjugacy class of newforms associated to the space of modular symbols \(M\). The level of the newforms is allowed to be a proper divisor of the level of \(M\). The space \(M\) must have been created using NewformDecomposition.

SortDecomposition(D): [ModSym] -> SeqEnum

Sort the sequence \(D\) of spaces of modular symbols with respect to the lt comparison operator.

IsIrreducible(M): ModSym -> BoolElt

Returns true if and only if Decomposition(M) has cardinality \(1\).

M1 lt M2: ModSym, ModSym -> BoolElt

The ordering on spaces of modular symbols is determined as follows:

(1)

This rule applies only if NewformDecomposition was used to construct both of \(M_1\) and \(M_2\): If Level(AssociatedNewSpace(M1)) is not equal to that of \(M2\) then the \(M_i\) with larger associated level is first.

(2)

The smaller dimension is first.

(3)

The following applies when the weight is \(2\) and the character is trivial: Order by \(W_q\) eigenvalues, starting with the smallest \(p\mid N\), with the eigenvalue \(+1\) being less than the eigenvalue \(-1\).

(4)

Order by \({\operatorname{abs}}({\rm trace}(a_p))\), with \(p\) not dividing the level, and with positive trace being smaller in the event that the two absolute values are equal.

Rule (3) is included so that our ordering extends the one used in (most of!) [Cremona, 1997].

Example: Decomposition (ex-25e8b4)

First, we compute the decomposition of the space of modular symbols of weight \(2\) and level \(37\).

> M := ModularSymbols(37,2); M;
Full modular symbols space for Gamma_0(37) of weight 2 and dimension 5
over Rational Field
> D := Decomposition(M,2); D;
[
     Modular symbols space for Gamma_0(37) of weight 2 and dimension 1
     over Rational Field,
     Modular symbols space for Gamma_0(37) of weight 2 and dimension 2
     over Rational Field,
     Modular symbols space for Gamma_0(37) of weight 2 and dimension 2
     over Rational Field
]
> IsIrreducible(D[2]);
true
> C := CuspidalSubspace(M); C;
Modular symbols space for Gamma_0(37) of weight 2 and dimension 4 over
Rational Field
> N := NewformDecomposition(C); N;
[
     Modular symbols space for Gamma_0(37) of weight 2 and dimension 2
     over Rational Field,
     Modular symbols space for Gamma_0(37) of weight 2 and dimension 2
     over Rational Field
]

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Next, we use NewformDecomposition to decompose a space having plentiful old subspaces.

> M := ModularSymbols(90,2); M;
Full modular symbols space for Gamma_0(90) of weight 2 and dimension
37 over Rational Field
> D := Decomposition(M,11); D;
[
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 11
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 4
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 2
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 2
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 4
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 8
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 6
     over Rational Field
]
> C := CuspidalSubspace(M); C;
Modular symbols space for Gamma_0(90) of weight 2 and dimension 22
over Rational Field
> N := NewformDecomposition(C); N;
[
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 2
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 2
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 2
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 4
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 4
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 8
     over Rational Field
]

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The above decomposition uses all of the Hecke operator; it suggests that the decomposition D is not as fine as possible. Indeed, D[7] breaks up further:

> Decomposition(D[7],11);
[
    Modular symbols space for Gamma_0(90) of weight 2 and dimension 6
    over Rational Field
]
> Decomposition(D[7],19);
[
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 4
     over Rational Field,
     Modular symbols space for Gamma_0(90) of weight 2 and dimension 2
     over Rational Field
]

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The function AssociatedNewSpace allows us to see where each of these subspace comes from. By definition they each arise by taking images under the degeneracy maps from a single Galois-conjugacy class of newforms of some level dividing \(90\).

> [Level(AssociatedNewSpace(A)) : A in N];
[ 90, 90, 90, 45, 30, 15 ]
> A := N[4];
> qEigenform(AssociatedNewSpace(A),7);
q + q^2 - q^4 - q^5 + O(q^7)
> qExpansionBasis(A,7);
[
    q - 2*q^4 - q^5 + O(q^7),
    q^2 + q^4 + O(q^7)
]

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