Space of boundary modular symbols

Used mainly for computing the cuspidal subspace of modular symbols. The space of boundary symbols of sign 0 is isomorphic as a Hecke module to the dual of the space of Eisenstein series, but this does not give a useful method of computing Eisenstein series, since there is no easy way to extract the constant terms.

We represent boundary modular symbols as a sum of Manin symbols of the form \([P, u/v]\), where \(u/v\) is a cusp for our group \(G\). The group of boundary modular symbols naturally embeds into a vector space \(B_k(G)\) (see Stein, section 8.4, or Merel, section 1.4, where this space is called \(\CC[\Gamma \backslash \QQ]_k\), for a definition), which is a finite dimensional \(\QQ\) vector space of dimension equal to the number of cusps for \(G\). The embedding takes \([P, u/v]\) to \(P(u,v)\cdot [(u,v)]\). We represent the basis vectors by pairs \([(u,v)]\) with u, v coprime. On \(B_k(G)\), we have the relations

\[[\gamma \cdot (u,v)] = [(u,v)]\]

for all \(\gamma \in G\) and

\[[(\lambda u, \lambda v)] = \operatorname{sign}(\lambda)^k [(u,v)]\]

for all \(\lambda \in \QQ^\times\).

It’s possible for these relations to kill a class, i.e., for a pair \([(u,v)]\) to be 0. For example, when \(N=4\) and \(k=3\) then \((-1,-2)\) is equivalent mod \(\Gamma_1(4)\) to \((1,2)\) since \(2=-2 \bmod 4\) and \(1=-1 \bmod 2\). But since \(k\) is odd, \([(-1,-2)]\) is also equivalent to \(-[(1,2)]\). Thus this symbol is equivalent to its negative, hence 0 (notice that this wouldn’t be the case in characteristic 2). This happens for any irregular cusp when the weight is odd; there are no irregular cusps on \(\Gamma_1(N)\) except when \(N = 4\), but there can be more on \(\Gamma_H\) groups. See also prop 2.30 of Stein’s Ph.D. thesis.

In addition, in the case that our space is of sign \(\sigma = 1\) or \(-1\), we also have the relation \([(-u,v)] = \sigma \cdot [(u,v)]\). This relation can also combine with the above to kill a cusp class - for instance, take (u,v) = (1,3) for \(\Gamma_1(5)\). Then since the cusp \(\tfrac{1}{3}\) is \(\Gamma_1(5)\)-equivalent to the cusp \(-\tfrac{1}{3}\), we have that \([(1,3)] = [(-1,3)]\). Now, on the minus subspace, we also have that \([(-1,3)] = -[(1,3)]\), which means this class must vanish. Notice that this cannot be used to show that \([(1,0)]\) or \([(0,1)]\) is 0.

Note

Special care must be taken when working with the images of the cusps 0 and \(\infty\) in \(B_k(G)\). For all cusps except 0 and \(\infty\), multiplying the cusp by -1 corresponds to taking \([(u,v)]\) to \([(-u,v)]\) in \(B_k(G)\). This means that \([(u,v)]\) is equivalent to \([(-u,v)]\) whenever \(\tfrac{u}{v}\) is equivalent to \(-\tfrac{u}{v}\), except in the case of 0 and \(\infty\). We have the following conditions for \([(1,0)]\) and \([(0,1)]\):

• \([(0,1)] = \sigma \cdot [(0,1)]\), so \([(0,1)]\) is 0 exactly when \(\sigma = -1\).
• \([(1,0)] = \sigma \cdot [(-1,0)]\) and \([(1,0)] = (-1)^k [(-1,0)]\), so \([(1,0)] = 0\) whenever \(\sigma \ne (-1)^k\).

Note

For all the spaces of boundary symbols below, no work is done to determine the cusps for G at creation time. Instead, cusps are added as they are discovered in the course of computation. As a result, the rank of a space can change as a computation proceeds.

REFERENCES:

• Merel, “Universal Fourier expansions of modular forms.” Springer LNM 1585 (1994), pg. 59-95.
• Stein, “Modular Forms, a computational approach.” AMS (2007).

class sage.modular.modsym.boundary.BoundarySpace(group=Modular Group SL(2, Z), weight=2, sign=0, base_ring=Rational Field, character=None)

Bases: sage.modular.hecke.module.HeckeModule_generic

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 - arithgroup.congroup_generic.CongruenceSubgroup, a congruence subgroup.
• sign - int, either -1, 0, or 1
• base_ring - rings.Ring (defaults to the rational numbers)

EXAMPLES:

sage: B = ModularSymbols(Gamma0(11),2).boundary_space()
sage: isinstance(B, sage.modular.modsym.boundary.BoundarySpace)
True
sage: B == loads(dumps(B))
True

character()

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

EXAMPLES:

sage: ModularSymbols(DirichletGroup(7).0, 6).boundary_space().character()
Dirichlet character modulo 7 of conductor 7 mapping 3 |--> zeta6

free_module()

Return the underlying free module for self.

EXAMPLES:

sage: B = ModularSymbols(Gamma1(7), 5, sign=-1).boundary_space()
sage: B.free_module()
Sparse vector space of dimension 0 over Rational Field
sage: x = B(Cusp(0)) ; y = B(Cusp(1/7)) ; B.free_module()
Sparse vector space of dimension 2 over Rational Field

gen(i=0)

Return the i-th generator of this space.

EXAMPLES:

sage: B = ModularSymbols(Gamma0(24), 4).boundary_space()
sage: B.gen(0)
Traceback (most recent call last):
...
ValueError: only 0 generators known for Space of Boundary Modular Symbols for Congruence Subgroup Gamma0(24) of weight 4 and over Rational Field
sage: B(Cusp(1/3))
[1/3]
sage: B.gen(0)
[1/3]

group()

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

EXAMPLES:

sage: ModularSymbols(GammaH(14,[9]), 2).boundary_space().group()
Congruence Subgroup Gamma_H(14) with H generated by [9]

is_ambient()

Return True if self is a space of boundary symbols associated to an ambient space of modular symbols.

EXAMPLES:

sage: M = ModularSymbols(Gamma1(6), 4)
sage: M.is_ambient()
True
sage: M.boundary_space().is_ambient()
True

rank()

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.)

EXAMPLES:

sage: M = ModularSymbols(Gamma0(72), 2) ; B = M.boundary_space()
sage: B.rank()
0
sage: _ = [ B(x) for x in M.basis() ]
sage: B.rank()
16

sign()

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

EXAMPLES:

sage: ModularSymbols(13,2,sign=-1).boundary_space().sign()
-1

weight()

Return the weight of this space of boundary modular symbols.

EXAMPLES:

sage: ModularSymbols(Gamma1(9), 5).boundary_space().weight()
5

class sage.modular.modsym.boundary.BoundarySpaceElement(parent, x)

Bases: sage.modular.hecke.element.HeckeModuleElement

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.

EXAMPLES:

sage: B = ModularSymbols(Gamma0(32), sign=-1).boundary_space()
sage: B(Cusp(1,8))
[1/8]
sage: B.0
[1/8]
sage: type(B.0)
<class 'sage.modular.modsym.boundary.BoundarySpaceElement'>

coordinate_vector()

Return self as a vector on the QQ-vector space with basis self.parent()._known_cusps().

EXAMPLES:

sage: B = ModularSymbols(18,4,sign=1).boundary_space()
sage: x = B(Cusp(1/2)) ; x
[1/2]
sage: x.coordinate_vector()
(1)
sage: ((18/5)*x).coordinate_vector()
(18/5)
sage: B(Cusp(0))
[0]
sage: x.coordinate_vector()
(1)
sage: x = B(Cusp(1/2)) ; x
[1/2]
sage: x.coordinate_vector()
(1, 0)

class sage.modular.modsym.boundary.BoundarySpace_wtk_eps(eps, weight, sign=0)

Bases: sage.modular.modsym.boundary.BoundarySpace

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

EXAMPLES:

sage: B = ModularSymbols(DirichletGroup(6).0, 4).boundary_space() ; B
Boundary Modular Symbols space of level 6, weight 4, character [-1] and dimension 0 over Rational Field
sage: type(B)
<class 'sage.modular.modsym.boundary.BoundarySpace_wtk_eps_with_category'>
sage: B == loads(dumps(B))
True

class sage.modular.modsym.boundary.BoundarySpace_wtk_g0(level, weight, sign, F)

Bases: sage.modular.modsym.boundary.BoundarySpace

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

INPUT:

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

EXAMPLES:

sage: B = ModularSymbols(Gamma0(2), 5).boundary_space()
sage: type(B)
<class 'sage.modular.modsym.boundary.BoundarySpace_wtk_g0_with_category'>
sage: B == loads(dumps(B))
True

class sage.modular.modsym.boundary.BoundarySpace_wtk_g1(level, weight, sign, F)

Bases: sage.modular.modsym.boundary.BoundarySpace

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

EXAMPLES:

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

class sage.modular.modsym.boundary.BoundarySpace_wtk_gamma_h(group, weight, sign, F)

Bases: sage.modular.modsym.boundary.BoundarySpace

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

EXAMPLES:

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