41.4 Subspace of ambient spaces of modular symbols

Module: sage.modular.modsym.subspace

Subspace of ambient spaces of modular symbols

Class: ModularSymbolsSubspace

class ModularSymbolsSubspace
Subspace of ambient space of modular symbols
ModularSymbolsSubspace( self, ambient_hecke_module, submodule, [dual_free_module=None], [check=False])

Input:

ambient_hecke_module
- the ambient space of modular symbols in which we're constructing a submodule
submodule
- the underlying free module of the submodule
dual_free_module
- underlying free module of the dual of the submodule (optional)
check
- (default: False) whether to check that the submodule is invariant under all Hecke operators T_p.

sage: M = ModularSymbols(15,4) ; S = M.cuspidal_submodule() # indirect doctest
sage: S
Modular Symbols subspace of dimension 8 of Modular Symbols space of
dimension 12 for Gamma_0(15) of weight 4 with sign 0 over Rational Field
sage: S == loads(dumps(S))
True
sage: M = ModularSymbols(1,24)
sage: A = M.ambient_hecke_module()
sage: B = A.submodule([ x.element() for x in M.cuspidal_submodule().gens() ])
sage: S = sage.modular.modsym.subspace.ModularSymbolsSubspace(A, B.free_module())
sage: S
Modular Symbols subspace of dimension 4 of Modular Symbols space of
dimension 5 for Gamma_0(1) of weight 24 with sign 0 over Rational Field
sage: S == loads(dumps(S))
True

Functions: boundary_map,$ \,$ cuspidal_submodule,$ \,$ dual_star_involution_matrix,$ \,$ eisenstein_subspace,$ \,$ factorization,$ \,$ hecke_bound,$ \,$ is_cuspidal,$ \,$ is_eisenstein,$ \,$ star_involution

boundary_map( self)

The boundary map to the corresponding space of boundary modular symbols. (This is the restriction of the map on the ambient space.)

sage: M = ModularSymbols(1, 24, sign=1) ; M
Modular Symbols space of dimension 3 for Gamma_0(1) of weight 24 with sign
1 over Rational Field
sage: M.basis()
([X^18*Y^4,(0,0)], [X^20*Y^2,(0,0)], [X^22,(0,0)])
sage: M.cuspidal_submodule().basis()
([X^18*Y^4,(0,0)], [X^20*Y^2,(0,0)])
sage: M.eisenstein_submodule().basis()
([X^18*Y^4,(0,0)] + 166747/324330*[X^20*Y^2,(0,0)] +
236364091/6742820700*[X^22,(0,0)],)
sage: M.boundary_map()
Hecke module morphism boundary map defined by the matrix
[ 0]
[ 0]
[-1]
Domain: Modular Symbols space of dimension 3 for Gamma_0(1) of weight ...
Codomain: Space of Boundary Modular Symbols for Congruence Subgroup
Gamma0(1) ...
sage: M.cuspidal_subspace().boundary_map()
Hecke module morphism defined by the matrix
[0]
[0]
Domain: Modular Symbols subspace of dimension 2 of Modular Symbols space
...
Codomain: Space of Boundary Modular Symbols for Congruence Subgroup
Gamma0(1) ...
sage: M.eisenstein_submodule().boundary_map()
Hecke module morphism defined by the matrix
[-236364091/6742820700]
Domain: Modular Symbols subspace of dimension 1 of Modular Symbols space
...
Codomain: Space of Boundary Modular Symbols for Congruence Subgroup
Gamma0(1) ...

cuspidal_submodule( self)

Return the cuspidal subspace of this subspace of modular symbols.

sage: S = ModularSymbols(42,4).cuspidal_submodule() ; S
Modular Symbols subspace of dimension 40 of Modular Symbols space of
dimension 48 for Gamma_0(42) of weight 4 with sign 0 over Rational Field
sage: S.is_cuspidal()
True
sage: S.cuspidal_submodule()
Modular Symbols subspace of dimension 40 of Modular Symbols space of
dimension 48 for Gamma_0(42) of weight 4 with sign 0 over Rational Field

The cuspidal submodule of the cuspidal submodule is just itself:

sage: S.cuspidal_submodule() is S
True
sage: S.cuspidal_submodule() == S
True

An example where we abuse the _set_is_cuspidal function:

sage: M = ModularSymbols(389)
sage: S = M.eisenstein_submodule()
sage: S._set_is_cuspidal(True)
sage: S.cuspidal_submodule()
Modular Symbols subspace of dimension 1 of Modular Symbols space of
dimension 65 for Gamma_0(389) of weight 2 with sign 0 over Rational Field

dual_star_involution_matrix( self)

Return the matrix of the dual star involution, which is induced by complex conjugation on the linear dual of modular symbols.

sage: S = ModularSymbols(6,4) ; S.dual_star_involution_matrix()
[ 1  0  0  0  0  0]
[ 0  1  0  0  0  0]
[ 0 -2  1  2  0  0]
[ 0  2  0 -1  0  0]
[ 0 -2  0  2  1  0]
[ 0  2  0 -2  0  1]
sage: S.star_involution().matrix().transpose() == S.dual_star_involution_matrix()
True

eisenstein_subspace( self)

Return the Eisenstein subspace of this space of modular symbols.

sage: ModularSymbols(24,4).eisenstein_subspace()
Modular Symbols subspace of dimension 8 of Modular Symbols space of
dimension 24 for Gamma_0(24) of weight 4 with sign 0 over Rational Field
sage: ModularSymbols(20,2).cuspidal_subspace().eisenstein_subspace()
Modular Symbols subspace of dimension 0 of Modular Symbols space of
dimension 7 for Gamma_0(20) of weight 2 with sign 0 over Rational Field

factorization( self)

Returns a list of pairs $ (S,e)$ where $ S$ is simple spaces of modular symbols and self is isomorphic to the direct sum of the $ S^e$ as a module over the anemic Hecke algebra adjoin the star involution.

The cuspidal $ S$ are all simple, but the Eisenstein factors need not be simple.

The factors are sorted by dimension - don't depend on much more for now.

ASSUMPTION: self is a module over the anemic Hecke algebra.

Note that if the sign is 1 then the cuspidal factors occur twice, one with each star eigenvalue.

sage: M = ModularSymbols(11)
sage: D = M.factorization(); D
(Modular Symbols subspace of dimension 1 of Modular Symbols space of
dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field) * 
(Modular Symbols subspace of dimension 1 of Modular Symbols space of
dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field) * 
(Modular Symbols subspace of dimension 1 of Modular Symbols space of
dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field)
sage: [A.T(2).matrix() for A, _ in D]
[[-2], [3], [-2]]
sage: [A.star_eigenvalues() for A, _ in D]
[[-1], [1], [1]]

In this example there is one old factor squared.

sage: M = ModularSymbols(22,sign=1)
sage: S = M.cuspidal_submodule()
sage: S.factorization()
(Modular Symbols subspace of dimension 1 of Modular Symbols space of
dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field)^2

sage: M = ModularSymbols(Gamma0(22), 2, sign=1)
sage: M1 = M.decomposition()[1]
sage: M1.factorization()
Modular Symbols subspace of dimension 3 of Modular Symbols space of
dimension 5 for Gamma_0(22) of weight 2 with sign 1 over Rational Field

hecke_bound( self)

Compute the Hecke bound for self; that is, a number n such that the T_m for m <= n generate the Hecke algebra.

sage: M = ModularSymbols(24,8)
sage: M.hecke_bound()
53
sage: M.cuspidal_submodule().hecke_bound()
32
sage: M.eisenstein_submodule().hecke_bound()
53

is_cuspidal( self)

Return True if self is cuspidal.

sage: ModularSymbols(42,4).cuspidal_submodule().is_cuspidal()
True
sage: ModularSymbols(12,6).eisenstein_submodule().is_cuspidal()
False

is_eisenstein( self)

Return True if self is an Eisenstein subspace.

sage: ModularSymbols(22,6).cuspidal_submodule().is_eisenstein()
False
sage: ModularSymbols(22,6).eisenstein_submodule().is_eisenstein()
True

star_involution( self)

Return the star involution on self, which is induced by complex conjugation on modular symbols.

sage: M = ModularSymbols(1,24)
sage: M.star_involution()
Hecke module morphism Star involution on Modular Symbols space of dimension
5 for Gamma_0(1) of weight 24 with sign 0 over Rational Field defined by
the matrix
[ 1  0  0  0  0]
[ 0 -1  0  0  0]
[ 0  0  1  0  0]
[ 0  0  0 -1  0]
[ 0  0  0  0  1]
Domain: Modular Symbols space of dimension 5 for Gamma_0(1) of weight ...
Codomain: Modular Symbols space of dimension 5 for Gamma_0(1) of weight ...
sage: M.cuspidal_subspace().star_involution()
Hecke module morphism defined by the matrix
[ 1  0  0  0]
[ 0 -1  0  0]
[ 0  0  1  0]
[ 0  0  0 -1]
Domain: Modular Symbols subspace of dimension 4 of Modular Symbols space
...
Codomain: Modular Symbols subspace of dimension 4 of Modular Symbols space
...
sage: M.plus_submodule().star_involution()
Hecke module morphism defined by the matrix
[1 0 0]
[0 1 0]
[0 0 1]
Domain: Modular Symbols subspace of dimension 3 of Modular Symbols space
...
Codomain: Modular Symbols subspace of dimension 3 of Modular Symbols space
...
sage: M.minus_submodule().star_involution()
Hecke module morphism defined by the matrix
[-1  0]
[ 0 -1]
Domain: Modular Symbols subspace of dimension 2 of Modular Symbols space
...
Codomain: Modular Symbols subspace of dimension 2 of Modular Symbols space
...

Special Functions: __init__,$ \,$ _compute_sign_subspace,$ \,$ _repr_,$ \,$ _set_is_cuspidal

_compute_sign_subspace( self, sign, [compute_dual=True])

Return the subspace of self that is fixed under the star involution.

Input:

sign
- int (either -1 or +1)
compute_dual
- bool (default: True) also compute dual subspace. This are useful for many algorithms.
Output: subspace of modular symbols

sage: S = ModularSymbols(100,2).cuspidal_submodule() ; S
Modular Symbols subspace of dimension 14 of Modular Symbols space of
dimension 31 for Gamma_0(100) of weight 2 with sign 0 over Rational Field
sage: S._compute_sign_subspace(1)
Modular Symbols subspace of dimension 7 of Modular Symbols space of
dimension 31 for Gamma_0(100) of weight 2 with sign 0 over Rational Field
sage: S._compute_sign_subspace(-1)
Modular Symbols subspace of dimension 7 of Modular Symbols space of
dimension 31 for Gamma_0(100) of weight 2 with sign 0 over Rational Field
sage: S._compute_sign_subspace(-1).sign()
-1

_repr_( self)

Return the string representation of self.

sage: ModularSymbols(24,4).cuspidal_subspace()._repr_()
'Modular Symbols subspace of dimension 16 of Modular Symbols space of
dimension 24 for Gamma_0(24) of weight 4 with sign 0 over Rational Field'

_set_is_cuspidal( self, t)

Used internally to declare that a given submodule is cuspidal.

We abuse this command:

sage: M = ModularSymbols(389)
sage: S = M.eisenstein_submodule()
sage: S._set_is_cuspidal(True)
sage: S.is_cuspidal()
True

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