41.7 Space of boundary modular symbols

Module: sage.modular.modsym.boundary

Space of boundary modular symbols.

Used mainly for computing the cuspidal subspace of modular symbols.

Module-level Functions

is_FreeModuleElement( )

Class: BoundarySpace

class BoundarySpace
Space of boundary symbols for a congruence subgroup of SL_2(Z).

This class is an abstract base class, so only derived classes should be instantiated.

Input:

weight
- int, the weight
group
- congroup.CongruenceGroup, a congruence subgroup.
sign
- int, either -1, 0, or 1
base_ring
- rings.Ring (defaults to the rational numbers)

BoundarySpace( self, [group=Congruence Subgroup Gamma0(1)], [weight=2], [sign=0], [base_ring=Rational Field], [character=None])

Initialize a space of boundary symbols.

Functions: character,$ \,$ element,$ \,$ filename,$ \,$ free_module,$ \,$ gen,$ \,$ group,$ \,$ is_ambient,$ \,$ rank,$ \,$ sign,$ \,$ weight

character( self)

Return the Dirichlet character assocaited to this space of boundary modular symbols.

element( self, x)

Creates and returns an element of self from a modular or sage symbol, if possible. Input:

x
- an object of one of the following types: ModularSymbol, ManinSymbol. We ignore the group, weight, and character of the input modular or sage symbol.
Output:
ModularSymbol
- a modular symbol with parent self.

filename( self)

Returns the filename of self that should be used to store self in the database. Input:

self
- space of modular symbols
Output:
str
- a string

gen( self, [i=0])

Return the i-th generator of this space.

group( self)

Return the congruence subgroup associated to this space of boundary modular symbols.

rank( self)

The rank of the space generated by boundary symbols that have been found so far in the course of computing the boundary map.

WARNING: This number may change as more elements are coerced into this space!! (This is an implementation detail that will likely change.)

sign( self)

Return the sign of the complex conjugation involution on this space of boundary modular symbols.

weight( self)

Return the weight of this space of boundary modular symbols.

Special Functions: __call__,$ \,$ __init__,$ \,$ __len__,$ \,$ _coerce_impl,$ \,$ _coerce_in_manin_symbol,$ \,$ _cusp_index,$ \,$ _known_cusps,$ \,$ _repr_

__call__( self, x)

Coerce x into a boundary symbol space.

If x is a modular symbol (with the same group, weight, character, sign, and base field), this returns the image of that modular symbol under the boundary map.

_known_cusps( self)

Return cusps found so far.

Class: BoundarySpace_wtk_eps

class BoundarySpace_wtk_eps
BoundarySpace_wtk_eps( self, eps, weight, [sign=0])

Space of boundary modular symbols with given weight, character, and sign.

Input:

eps
- dirichlet.DirichletCharacter, the "Nebentypus" character.
weight
- int, the weight >= 2
sign
- int, either -1, 0, or 1

Special Functions: __init__,$ \,$ _coerce_cusp,$ \,$ _cusp_index,$ \,$ _is_equiv,$ \,$ _repr_

_coerce_cusp( self, c)

Coerce symbol into a boundary symbol space.

Class: BoundarySpace_wtk_g0

class BoundarySpace_wtk_g0
Boundary symbols for Gamma_0(N) of integer weight k > 2 over the field F.
BoundarySpace_wtk_g0( self, level, weight, sign, F)

Initialize a space of boundary symbols of weight k for Gamma_0(N), over F.

For weight 2, it is faster to use BoundarySpace_wt2_g0.

Input:

level
- int, the level
weight
- integer weight >= 2.
sign
- int, either -1, 0, or 1
F
- field

Special Functions: __init__,$ \,$ _coerce_cusp,$ \,$ _is_equiv,$ \,$ _repr_

_coerce_cusp( self, c)

Coerce cusp into a boundary symbol space.

Class: BoundarySpace_wtk_g1

class BoundarySpace_wtk_g1
BoundarySpace_wtk_g1( self, level, weight, sign, F)

Initialize a space of boundary modular symbols for Gamma1(N).

Input:

level
- int, the level
weight
- int, the weight >= 2
sign
- int, either -1, 0, or 1
F
- base ring

sage: from sage.modular.modsym.boundary import BoundarySpace_wtk_g1
sage: BoundarySpace_wtk_g1(17, 2, 0, QQ)
Boundary Modular Symbols space for Gamma_1(17) of weight 2 over Rational
Field

Special Functions: __init__,$ \,$ _coerce_cusp,$ \,$ _cusp_index,$ \,$ _is_equiv,$ \,$ _repr_

_coerce_cusp( self, c)

Coerce symbol into a boundary symbol space.

Class: BoundarySpace_wtk_gamma_h

class BoundarySpace_wtk_gamma_h
BoundarySpace_wtk_gamma_h( self, group, weight, sign, F)

Initialize a space of boundary modular symbols for GammaH(N).

Input:

group
- congruence subgroup Gamma_H(N).
weight
- int, the weight >= 2
sign
- int, either -1, 0, or 1
F
- base ring

sage: from sage.modular.modsym.boundary import BoundarySpace_wtk_gamma_h
sage: BoundarySpace_wtk_gamma_h(GammaH(13,[3]), 2, 0, QQ)
Boundary Modular Symbols space for Congruence Subgroup Gamma_H(13) with H
generated by [3] of weight 2 over Rational Field

Special Functions: __init__,$ \,$ _coerce_cusp,$ \,$ _cusp_index,$ \,$ _is_equiv,$ \,$ _repr_

_coerce_cusp( self, c)

Coerce symbol into a boundary symbol space.

Class: BoundarySpaceElement

class BoundarySpaceElement
A boundary symbol.
BoundarySpaceElement( self, parent, x)

Create a boundary symbol.

Input:

parent
- BoundarySpace; a space of boundary modular symbols
x
- a dict with integer keys and values in the base field of parent.

Functions: coordinate_vector

Special Functions: __init__,$ \,$ __neg__,$ \,$ _add_,$ \,$ _lmul_,$ \,$ _repr_,$ \,$ _rmul_,$ \,$ _sub_

_repr_( self)

Returns a string representation for self for printing purposes.

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