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""" Submodules of Hecke modules """ #***************************************************************************** # Sage: System for Algebra and Geometry Experimentation # # Copyright (C) 2005 William Stein <wstein@gmail.com> # # Distributed under the terms of the GNU General Public License (GPL) # # This code is distributed in the hope that it will be useful, # but WITHOUT ANY WARRANTY; without even the implied warranty of # MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU # General Public License for more details. # # The full text of the GPL is available at: # # http://www.gnu.org/licenses/ #***************************************************************************** from __future__ import absolute_import
import sage.arith.all as arith import sage.misc.misc as misc from sage.misc.cachefunc import cached_method from sage.structure.richcmp import richcmp_method, richcmp_not_equal, richcmp import sage.modules.all
from . import module
def is_HeckeSubmodule(x): r""" Return True if x is of type HeckeSubmodule.
EXAMPLES::
sage: sage.modular.hecke.submodule.is_HeckeSubmodule(ModularForms(1, 12)) False sage: sage.modular.hecke.submodule.is_HeckeSubmodule(CuspForms(1, 12)) True """
@richcmp_method class HeckeSubmodule(module.HeckeModule_free_module): """ Submodule of a Hecke module. """ def __init__(self, ambient, submodule, dual_free_module=None, check=True): r""" Initialise a submodule of an ambient Hecke module.
INPUT:
- ``ambient`` - an ambient Hecke module
- ``submodule`` - a free module over the base ring which is a submodule of the free module attached to the ambient Hecke module. This should be invariant under all Hecke operators.
- ``dual_free_module`` - the submodule of the dual of the ambient module corresponding to this submodule (or None).
- ``check`` - whether or not to explicitly check that the submodule is Hecke equivariant.
EXAMPLES::
sage: CuspForms(1,60) # indirect doctest Cuspidal subspace of dimension 5 of Modular Forms space of dimension 6 for Modular Group SL(2,Z) of weight 60 over Rational Field
sage: M = ModularForms(4,10) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.submodule(M.basis()[:3]).free_module()) sage: S Rank 3 submodule of a Hecke module of level 4
sage: S == loads(dumps(S)) True """ raise TypeError("ambient must be an ambient Hecke module") raise TypeError("submodule must be a free module") raise ValueError("submodule must be a submodule of the ambient free module")
ambient.base_ring(), ambient.level(), ambient.weight()) raise TypeError("dual_free_module must be a free module") raise ArithmeticError("dual_free_module must have the same rank as submodule")
def _repr_(self): r""" String representation of self.
EXAMPLES::
sage: M = ModularForms(4,10) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.submodule(M.basis()[:3]).free_module()) sage: S._repr_() 'Rank 3 submodule of a Hecke module of level 4' """ self.rank(), self.level())
def __add__(self, other): r""" Sum of self and other (as submodules of a common ambient module).
EXAMPLES::
sage: M = ModularForms(4,10) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.submodule(M.basis()[:3]).free_module()) sage: E = sage.modular.hecke.submodule.HeckeSubmodule(M, M.submodule(M.basis()[3:]).free_module()) sage: S + E # indirect doctest Modular Forms subspace of dimension 6 of Modular Forms space of dimension 6 for Congruence Subgroup Gamma0(4) of weight 10 over Rational Field """ raise TypeError("other (=%s) must be a Hecke module."%other) raise ArithmeticError("Sum only defined for submodules of a common ambient space.") return other # Neither is ambient
def _element_constructor_(self, x, check=True): """ Coerce x into the ambient module and checks that x is in this submodule.
EXAMPLES::
sage: M = ModularSymbols(37) sage: S = M.cuspidal_submodule() sage: M([0,oo]) -(1,0) sage: S([0,oo]) Traceback (most recent call last): ... TypeError: x does not coerce to an element of this Hecke module sage: S([-1/23,0]) (1,23) """
def __richcmp__(self, other, op): """ Compare self to other.
EXAMPLES::
sage: M = ModularSymbols(12,6) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: T = sage.modular.hecke.submodule.HeckeSubmodule(M, M.new_submodule().free_module()) sage: S Rank 14 submodule of a Hecke module of level 12 sage: T Rank 0 submodule of a Hecke module of level 12 sage: S > T True sage: T < S True sage: S == S True """ return richcmp_not_equal(lx, rx, op)
################################ # Semi-Private functions ################################ def _compute_dual_hecke_matrix(self, n): """ Compute the matrix for the nth Hecke operator acting on the dual of self.
EXAMPLES::
sage: M = ModularForms(4,10) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.submodule(M.basis()[:3]).free_module()) sage: S._compute_dual_hecke_matrix(3) [ 0 0 1] [ 0 -156 0] [35568 0 72] sage: CuspForms(4,10).dual_hecke_matrix(3) [ 0 0 1] [ 0 -156 0] [35568 0 72] """
def _compute_hecke_matrix(self, n): r""" Compute the matrix of the nth Hecke operator acting on this space, by calling the corresponding function for the ambient space and restricting. If n is not coprime to the level, we check that the restriction is well-defined.
EXAMPLES::
sage: R.<q> = QQ[[]] sage: M = ModularForms(2, 12) sage: f = M(q^2 - 24*q^4 + O(q^6)) sage: A = M.submodule(M.free_module().span([f.element()]),check=False) sage: sage.modular.hecke.submodule.HeckeSubmodule._compute_hecke_matrix(A, 3) [252] sage: sage.modular.hecke.submodule.HeckeSubmodule._compute_hecke_matrix(A, 4) Traceback (most recent call last): ... ArithmeticError: subspace is not invariant under matrix """
def _compute_diamond_matrix(self, d): r""" EXAMPLES:
sage: f = ModularSymbols(Gamma1(13),2,sign=1).cuspidal_subspace().decomposition()[0] sage: a = f.diamond_bracket_operator(2).matrix() # indirect doctest sage: a.charpoly() x^2 - x + 1 sage: a^12 [1 0] [0 1] """
def _compute_atkin_lehner_matrix(self, d): """ Compute the Atkin-Lehner matrix corresponding to the divisor d of the level of self.
EXAMPLES::
sage: M = ModularSymbols(4,10) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S Rank 6 submodule of a Hecke module of level 4 sage: S._compute_atkin_lehner_matrix(1) [1 0 0 0 0 0] [0 1 0 0 0 0] [0 0 1 0 0 0] [0 0 0 1 0 0] [0 0 0 0 1 0] [0 0 0 0 0 1] """
def _set_dual_free_module(self, V): """ Set the dual free module of self to V. Here V must be a vector space of the same dimension as self, embedded in a space of the same dimension as the ambient space of self.
EXAMPLES::
sage: M = ModularSymbols(4,10) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S._set_dual_free_module(M.cuspidal_submodule().dual_free_module()) sage: S._set_dual_free_module(S) """ raise ArithmeticError("The degree of V must equal the rank of the ambient space.") raise ArithmeticError("The rank of V must equal the rank of self.")
################################ # Public functions ################################
def ambient_hecke_module(self): r""" Return the ambient Hecke module of which this is a submodule.
EXAMPLES::
sage: CuspForms(2, 12).ambient_hecke_module() Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(2) of weight 12 over Rational Field """
def ambient(self): r""" Synonym for ambient_hecke_module.
EXAMPLES::
sage: CuspForms(2, 12).ambient() Modular Forms space of dimension 4 for Congruence Subgroup Gamma0(2) of weight 12 over Rational Field """
@cached_method def complement(self, bound=None): """ Return the largest Hecke-stable complement of this space.
EXAMPLES::
sage: M = ModularSymbols(15, 6).cuspidal_subspace() sage: M.complement() Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 20 for Gamma_0(15) of weight 6 with sign 0 over Rational Field sage: E = EllipticCurve("128a") sage: ME = E.modular_symbol_space() sage: ME.complement() Modular Symbols subspace of dimension 17 of Modular Symbols space of dimension 18 for Gamma_0(128) of weight 2 with sign 1 over Rational Field """
return self.ambient_hecke_module().zero_submodule()
else: anemic = True
# TODO: optimize in some cases by computing image of # complementary factor instead of kernel...? while N % p == 0: p = arith.next_prime(p)
# first attempt to compute the complement failed, we now try # the following naive approach: decompose the ambient space, # decompose self, and sum the pieces of ambient that are not # subspaces of self
# failed miserably
def degeneracy_map(self, level, t=1): """ The t-th degeneracy map from self to the space of ambient modular symbols of the given level. The level of self must be a divisor or multiple of level, and t must be a divisor of the quotient.
INPUT:
- ``level`` - int, the level of the codomain of the map (positive int).
- ``t`` - int, the parameter of the degeneracy map, i.e., the map is related to `f(q)` - `f(q^t)`.
OUTPUT: A linear function from self to the space of modular symbols of given level with the same weight, character, sign, etc., as this space.
EXAMPLES::
sage: D = ModularSymbols(10,4).cuspidal_submodule().decomposition(); D [ Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field, Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field ] sage: d = D[1].degeneracy_map(5); d Hecke module morphism defined by the matrix [ 0 0 -1 1] [ 0 1/2 3/2 -2] [ 0 -1 1 0] [ 0 -3/4 -1/4 1] Domain: Modular Symbols subspace of dimension 4 of Modular Symbols space ... Codomain: Modular Symbols space of dimension 4 for Gamma_0(5) of weight ...
::
sage: d.rank() 2 sage: d.kernel() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 10 for Gamma_0(10) of weight 4 with sign 0 over Rational Field sage: d.image() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 4 for Gamma_0(5) of weight 4 with sign 0 over Rational Field """
@cached_method def dual_free_module(self, bound=None, anemic=True, use_star=True): r""" Compute embedded dual free module if possible. In general this won't be possible, e.g., if this space is not Hecke equivariant, possibly if it is not cuspidal, or if the characteristic is not 0. In all these cases we raise a RuntimeError exception.
If use_star is True (which is the default), we also use the +/- eigenspaces for the star operator to find the dual free module of self. If self does not have a star involution, use_star will automatically be set to False.
EXAMPLES::
sage: M = ModularSymbols(11, 2) sage: M.dual_free_module() Vector space of dimension 3 over Rational Field sage: Mpc = M.plus_submodule().cuspidal_submodule() sage: Mcp = M.cuspidal_submodule().plus_submodule() sage: Mcp.dual_free_module() == Mpc.dual_free_module() True sage: Mpc.dual_free_module() Vector space of degree 3 and dimension 1 over Rational Field Basis matrix: [ 1 5/2 5]
sage: M = ModularSymbols(35,2).cuspidal_submodule() sage: M.dual_free_module(use_star=False) Vector space of degree 9 and dimension 6 over Rational Field Basis matrix: [ 1 0 0 0 -1 0 0 4 -2] [ 0 1 0 0 0 0 0 -1/2 1/2] [ 0 0 1 0 0 0 0 -1/2 1/2] [ 0 0 0 1 -1 0 0 1 0] [ 0 0 0 0 0 1 0 -2 1] [ 0 0 0 0 0 0 1 -2 1]
sage: M = ModularSymbols(40,2) sage: Mmc = M.minus_submodule().cuspidal_submodule() sage: Mcm = M.cuspidal_submodule().minus_submodule() sage: Mcm.dual_free_module() == Mmc.dual_free_module() True sage: Mcm.dual_free_module() Vector space of degree 13 and dimension 3 over Rational Field Basis matrix: [ 0 1 0 0 0 0 1 0 -1 -1 1 -1 0] [ 0 0 1 0 -1 0 -1 0 1 0 0 0 0] [ 0 0 0 0 0 1 1 0 -1 0 0 0 0]
sage: M = ModularSymbols(43).cuspidal_submodule() sage: S = M[0].plus_submodule() + M[1].minus_submodule() sage: S.dual_free_module(use_star=False) Traceback (most recent call last): ... RuntimeError: Computation of complementary space failed (cut down to rank 7, but should have cut down to rank 4). sage: S.dual_free_module().dimension() == S.dimension() True
We test that :trac:`5080` is fixed::
sage: EllipticCurve('128a').congruence_number() 32
"""
# if we know the complement we can read off the dual module
return A.zero_submodule()
# ALGORITHM: Compute the char poly of each Hecke operator on # the submodule, then use it to cut out a submodule of the # dual. If the dimension cuts down to the dimension of self # terminate with success. If it stays larger beyond the Sturm # bound, raise a RuntimeError exception.
# In the case that the sign of self is not 1, we need to use # the star involution as well as the Hecke operators in order # to find the dual of self. # # Note that one needs to comment out the line caching the # result of this computation below in order to get meaningful # timings.
# If the star involution doesn't make sense for self, then we # can't use it.
# If the star involution has both + and - eigenspaces on self, # then we compute the dual on each eigenspace, then put them # together. self.minus_submodule(compute_dual = False).dual_free_module()
# At this point, we know that self is an eigenspace for star. else:
else: # Failed to reduce V to the appropriate dimension else: raise RuntimeError("Computation of embedded dual vector space failed " + \ "(cut down to rank %s, but should have cut down to rank %s)."%(V.rank(), self.rank()))
def free_module(self): """ Return the free module corresponding to self.
EXAMPLES::
sage: M = ModularSymbols(33,2).cuspidal_subspace() ; M Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field sage: M.free_module() Vector space of degree 9 and dimension 6 over Rational Field Basis matrix: [ 0 1 0 0 0 0 0 -1 1] [ 0 0 1 0 0 0 0 -1 1] [ 0 0 0 1 0 0 0 -1 1] [ 0 0 0 0 1 0 0 -1 1] [ 0 0 0 0 0 1 0 -1 1] [ 0 0 0 0 0 0 1 -1 0] """
def module(self): r""" Alias for \code{self.free_module()}.
EXAMPLES::
sage: M = ModularSymbols(17,4).cuspidal_subspace() sage: M.free_module() is M.module() True """
def intersection(self, other): """ Returns the intersection of self and other, which must both lie in a common ambient space of modular symbols.
EXAMPLES::
sage: M = ModularSymbols(43, sign=1) sage: A = M[0] + M[1] sage: B = M[1] + M[2] sage: A.dimension(), B.dimension() (2, 3) sage: C = A.intersection(B); C.dimension() 1
TESTS::
sage: M = ModularSymbols(1,80) sage: M.plus_submodule().cuspidal_submodule().sign() # indirect doctest 1 """ raise ArithmeticError("Intersection only defined for subspaces of"\ + " a common ambient modular symbols space.") return self return other
# Neither is ambient
## if sign is nonzero, the intersection will be, too ## this only makes sense for modular symbols spaces (and hence shouldn't really be in this file)
def is_ambient(self): r""" Return ``True`` if self is an ambient space of modular symbols.
EXAMPLES::
sage: M = ModularSymbols(17,4) sage: M.cuspidal_subspace().is_ambient() False sage: A = M.ambient_hecke_module() sage: S = A.submodule(A.basis()) sage: sage.modular.hecke.submodule.HeckeSubmodule.is_ambient(S) True """
def is_new(self, p=None): """ Returns True if this Hecke module is p-new. If p is None, returns True if it is new.
EXAMPLES::
sage: M = ModularSymbols(1,16) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S.is_new() True """ except KeyError: pass
def is_old(self, p=None): """ Returns True if this Hecke module is p-old. If p is None, returns True if it is old.
EXAMPLES::
sage: M = ModularSymbols(50,2) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.old_submodule().free_module()) sage: S.is_old() True sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.new_submodule().free_module()) sage: S.is_old() False """ except KeyError: pass
def is_submodule(self, V): """ Returns True if and only if self is a submodule of V.
EXAMPLES::
sage: M = ModularSymbols(30,4) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S.is_submodule(M) True sage: SS = sage.modular.hecke.submodule.HeckeSubmodule(M, M.old_submodule().free_module()) sage: S.is_submodule(SS) False """ self.free_module().is_subspace(V.free_module())
def linear_combination_of_basis(self, v): """ Return the linear combination of the basis of ``self`` given by the entries of `v`.
The result can be of different types, and is printed accordingly, depending on the type of submodule.
EXAMPLES::
sage: M = ModularForms(Gamma0(2),12)
sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S.basis() ((1, 0, 0, 0), (0, 1, 0, 0)) sage: S.linear_combination_of_basis([3, 10]) (3, 10, 0, 0)
sage: S = M.cuspidal_submodule() sage: S.basis() [ q + 252*q^3 - 2048*q^4 + 4830*q^5 + O(q^6), q^2 - 24*q^4 + O(q^6) ] sage: S.linear_combination_of_basis([3, 10]) 3*q + 10*q^2 + 756*q^3 - 6384*q^4 + 14490*q^5 + O(q^6)
"""
def new_submodule(self, p=None): """ Return the new or p-new submodule of this space of modular symbols.
EXAMPLES::
sage: M = ModularSymbols(20,4) sage: M.new_submodule() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(20) of weight 4 with sign 0 over Rational Field sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S Rank 12 submodule of a Hecke module of level 20 sage: S.new_submodule() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(20) of weight 4 with sign 0 over Rational Field """
def nonembedded_free_module(self): """ Return the free module corresponding to self as an abstract free module, i.e. not as an embedded vector space.
EXAMPLES::
sage: M = ModularSymbols(12,6) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S Rank 14 submodule of a Hecke module of level 12 sage: S.nonembedded_free_module() Vector space of dimension 14 over Rational Field """
def old_submodule(self, p=None): """ Return the old or p-old submodule of this space of modular symbols.
EXAMPLES: We compute the old and new submodules of `\mathbf{S}_2(\Gamma_0(33))`.
::
sage: M = ModularSymbols(33); S = M.cuspidal_submodule(); S Modular Symbols subspace of dimension 6 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field sage: S.old_submodule() Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field sage: S.new_submodule() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 9 for Gamma_0(33) of weight 2 with sign 0 over Rational Field """ return self
self.__is_old[p] = True return self
def rank(self): r""" Return the rank of self as a free module over the base ring.
EXAMPLES::
sage: ModularSymbols(6, 4).cuspidal_subspace().rank() 2 sage: ModularSymbols(6, 4).cuspidal_subspace().dimension() 2 """
def submodule(self, M, Mdual=None, check=True): """ Construct a submodule of self from the free module M, which must be a subspace of self.
EXAMPLES::
sage: M = ModularSymbols(18,4) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: S[0] Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(18) of weight 4 with sign 0 over Rational Field sage: S.submodule(S[0].free_module()) Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 18 for Gamma_0(18) of weight 4 with sign 0 over Rational Field """ else: M = V.span(M)
raise TypeError("M (=%s) must be a submodule of the free module (=%s) associated to this module."%(M, self.free_module()))
def submodule_from_nonembedded_module(self, V, Vdual=None, check=True): """ Construct a submodule of self from V. Here V should be a subspace of a vector space whose dimension is the same as that of self.
INPUT:
- ``V`` - submodule of ambient free module of the same rank as the rank of self.
- ``check`` - whether to check that V is Hecke equivariant.
OUTPUT: Hecke submodule of self
EXAMPLES::
sage: M = ModularSymbols(37,2) sage: S = sage.modular.hecke.submodule.HeckeSubmodule(M, M.cuspidal_submodule().free_module()) sage: V = (QQ**4).subspace([[1,-1,0,1/2],[0,0,1,-1/2]]) sage: S.submodule_from_nonembedded_module(V) Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field """ # We encode the operation of taking the linear combinations of # the basis of E given by the basis of V as a single matrix # multiplication, since matrix multiplication is (presumed to be) # so fast, and their are asymptotically fast algorithms.
def 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.
EXAMPLES::
sage: M = ModularSymbols(24,8) sage: M.hecke_bound() 53 sage: M.cuspidal_submodule().hecke_bound() 32 sage: M.eisenstein_submodule().hecke_bound() 53 """ else: |