Coverage for local/lib/python2.7/site-packages/sage/modular/abvar/cuspidal_subgroup.py : 68%

""" 

Cuspidal subgroups of modular abelian varieties 

AUTHORS: 

- William Stein (2007-03, 2008-02) 

EXAMPLES: We compute the cuspidal subgroup of `J_1(13)`:: 

sage: A = J1(13) 

sage: C = A.cuspidal_subgroup(); C 

Finite subgroup with invariants [19, 19] over QQ of Abelian variety J1(13) of dimension 2 

sage: C.gens() 

[[(1/19, 0, 0, 9/19)], [(0, 1/19, 1/19, 18/19)]] 

sage: C.order() 

361 

sage: C.invariants() 

[19, 19] 

We compute the cuspidal subgroup of `J_0(54)`:: 

sage: A = J0(54) 

sage: C = A.cuspidal_subgroup(); C 

Finite subgroup with invariants [3, 3, 3, 3, 3, 9] over QQ of Abelian variety J0(54) of dimension 4 

sage: C.gens() 

[[(1/3, 0, 0, 0, 0, 1/3, 0, 2/3)], [(0, 1/3, 0, 0, 0, 2/3, 0, 1/3)], [(0, 0, 1/9, 1/9, 1/9, 1/9, 1/9, 2/9)], [(0, 0, 0, 1/3, 0, 1/3, 0, 0)], [(0, 0, 0, 0, 1/3, 1/3, 0, 1/3)], [(0, 0, 0, 0, 0, 0, 1/3, 2/3)]] 

sage: C.order() 

2187 

sage: C.invariants() 

[3, 3, 3, 3, 3, 9] 

We compute the subgroup of the cuspidal subgroup generated by 

rational cusps. 

:: 

sage: C = J0(54).rational_cusp_subgroup(); C 

Finite subgroup with invariants [3, 3, 9] over QQ of Abelian variety J0(54) of dimension 4 

sage: C.gens() 

[[(1/3, 0, 0, 1/3, 2/3, 1/3, 0, 1/3)], [(0, 0, 1/9, 1/9, 7/9, 7/9, 1/9, 8/9)], [(0, 0, 0, 0, 0, 0, 1/3, 2/3)]] 

sage: C.order() 

81 

sage: C.invariants() 

[3, 3, 9] 

This might not give us the exact rational torsion subgroup, since 

it might be bigger than order `81`:: 

sage: J0(54).rational_torsion_subgroup().multiple_of_order() 

243 

TESTS:: 

sage: C = J0(54).cuspidal_subgroup() 

sage: loads(dumps(C)) == C 

True 

sage: D = J0(54).rational_cusp_subgroup() 

sage: loads(dumps(D)) == D 

True 

""" 

from __future__ import absolute_import 

#***************************************************************************** 

# Copyright (C) 2007 William Stein <wstein@gmail.com> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

from .finite_subgroup import FiniteSubgroup 

from sage.rings.all import infinity, QQ, ZZ 

from sage.matrix.all import matrix 

from sage.modular.arithgroup.all import is_Gamma0 

from sage.modular.cusps import Cusp 

from sage.arith.all import gcd 

class CuspidalSubgroup_generic(FiniteSubgroup): 

def _compute_lattice(self, rational_only=False, rational_subgroup=False): 

r""" 

Return a list of vectors that define elements of the rational 

homology that generate this finite subgroup. 

INPUT: 

- ``rational_only`` - bool (default: False); if 

``True``, only use rational cusps. 

OUTPUT: 

- ``list`` - list of vectors 

EXAMPLES:: 

sage: J = J0(37) 

sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J) 

sage: C._compute_lattice() 

Free module of degree 4 and rank 4 over Integer Ring 

Echelon basis matrix: 

[ 1 0 0 0] 

[ 0 1 0 0] 

[ 0 0 1 0] 

[ 0 0 0 1/3] 

sage: J = J0(43) 

sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J) 

sage: C._compute_lattice() 

Free module of degree 6 and rank 6 over Integer Ring 

Echelon basis matrix: 

[ 1 0 0 0 0 0] 

[ 0 1/7 0 6/7 0 5/7] 

[ 0 0 1 0 0 0] 

[ 0 0 0 1 0 0] 

[ 0 0 0 0 1 0] 

[ 0 0 0 0 0 1] 

sage: J = J0(22) 

sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J) 

sage: C._compute_lattice() 

Free module of degree 4 and rank 4 over Integer Ring 

Echelon basis matrix: 

[1/5 1/5 4/5 0] 

[ 0 1 0 0] 

[ 0 0 1 0] 

[ 0 0 0 1/5] 

sage: J = J1(13) 

sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J) 

sage: C._compute_lattice() 

Free module of degree 4 and rank 4 over Integer Ring 

Echelon basis matrix: 

[ 1/19 0 0 9/19] 

[ 0 1/19 1/19 18/19] 

[ 0 0 1 0] 

[ 0 0 0 1] 

We compute with and without the optional 

``rational_only`` option. 

:: 

sage: J = J0(27); G = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J) 

sage: G._compute_lattice() 

Free module of degree 2 and rank 2 over Integer Ring 

Echelon basis matrix: 

[1/3 0] 

[ 0 1/3] 

sage: G._compute_lattice(rational_only=True) 

Free module of degree 2 and rank 2 over Integer Ring 

Echelon basis matrix: 

[1/3 0] 

[ 0 1] 

""" 

A = self.abelian_variety() 

Cusp = A.modular_symbols() 

Amb = Cusp.ambient_module() 

Eis = Amb.eisenstein_submodule() 

C = Amb.cusps() 

N = Amb.level() 

if rational_subgroup: 

# QQ-rational subgroup of cuspidal subgroup 

assert A.is_ambient() 

Q = Cusp.abvarquo_rational_cuspidal_subgroup() 

return Q.V() 

if rational_only: 

# subgroup generated by differences of rational cusps 

if not is_Gamma0(A.group()): 

raise NotImplementedError('computation of rational cusps only implemented in Gamma0 case.') 

if not N.is_squarefree(): 

data = [n for n in range(2,N) if gcd(n,N) == 1] 

C = [c for c in C if is_rational_cusp_gamma0(c, N, data)] 

v = [Amb([infinity, alpha]).element() for alpha in C] 

cusp_matrix = matrix(QQ, len(v), Amb.dimension(), v) 

# TODO -- refactor something out here 

# Now we project onto the cuspidal part. 

B = Cusp.free_module().basis_matrix().stack(Eis.free_module().basis_matrix()) 

X = B.solve_left(cusp_matrix) 

X = X.matrix_from_columns(range(Cusp.dimension())) 

lattice = X.row_module(ZZ) + A.lattice() 

return lattice 

class CuspidalSubgroup(CuspidalSubgroup_generic): 

""" 

EXAMPLES:: 

sage: a = J0(65)[2] 

sage: t = a.cuspidal_subgroup() 

sage: t.order() 

6 

""" 

def _repr_(self): 

""" 

String representation of the cuspidal subgroup. 

EXAMPLES:: 

sage: G = J0(27).cuspidal_subgroup() 

sage: G._repr_() 

'Finite subgroup with invariants [3, 3] over QQ of Abelian variety J0(27) of dimension 1' 

""" 

return "Cuspidal subgroup %sover QQ of %s"%(self._invariants_repr(), self.abelian_variety()) 

def lattice(self): 

""" 

Returned cached tuple of vectors that define elements of the 

rational homology that generate this finite subgroup. 

OUTPUT: 

- ``tuple`` - cached 

EXAMPLES:: 

sage: J = J0(27) 

sage: G = J.cuspidal_subgroup() 

sage: G.lattice() 

Free module of degree 2 and rank 2 over Integer Ring 

Echelon basis matrix: 

[1/3 0] 

[ 0 1/3] 

Test that the result is cached:: 

sage: G.lattice() is G.lattice() 

True 

""" 

try: 

return self.__lattice 

except AttributeError: 

lattice = self._compute_lattice(rational_only = False) 

self.__lattice = lattice 

return lattice 

class RationalCuspSubgroup(CuspidalSubgroup_generic): 

""" 

EXAMPLES:: 

sage: a = J0(65)[2] 

sage: t = a.rational_cusp_subgroup() 

sage: t.order() 

6 

""" 

def _repr_(self): 

""" 

String representation of the cuspidal subgroup. 

EXAMPLES:: 

sage: G = J0(27).rational_cusp_subgroup() 

sage: G._repr_() 

'Finite subgroup with invariants [3] over QQ of Abelian variety J0(27) of dimension 1' 

""" 

return "Subgroup generated by differences of rational cusps %sover QQ of %s"%(self._invariants_repr(), self.abelian_variety()) 

def lattice(self): 

""" 

Return lattice that defines this group. 

OUTPUT: lattice 

EXAMPLES:: 

sage: G = J0(27).rational_cusp_subgroup() 

sage: G.lattice() 

Free module of degree 2 and rank 2 over Integer Ring 

Echelon basis matrix: 

[1/3 0] 

[ 0 1] 

Test that the result is cached. 

:: 

sage: G.lattice() is G.lattice() 

True 

""" 

try: 

return self.__lattice 

except AttributeError: 

lattice = self._compute_lattice(rational_only = True) 

self.__lattice = lattice 

return lattice 

class RationalCuspidalSubgroup(CuspidalSubgroup_generic): 

""" 

EXAMPLES:: 

sage: a = J0(65)[2] 

sage: t = a.rational_cuspidal_subgroup() 

sage: t.order() 

6 

""" 

def _repr_(self): 

""" 

String representation of the cuspidal subgroup. 

EXAMPLES:: 

sage: G = J0(27).rational_cuspidal_subgroup() 

sage: G._repr_() 

'Finite subgroup with invariants [3] over QQ of Abelian variety J0(27) of dimension 1' 

""" 

return "Rational cuspidal subgroup %sover QQ of %s"%(self._invariants_repr(), self.abelian_variety()) 

def lattice(self): 

""" 

Return lattice that defines this group. 

OUTPUT: lattice 

EXAMPLES:: 

sage: G = J0(27).rational_cuspidal_subgroup() 

sage: G.lattice() 

Free module of degree 2 and rank 2 over Integer Ring 

Echelon basis matrix: 

[1/3 0] 

[ 0 1] 

Test that the result is cached. 

:: 

sage: G.lattice() is G.lattice() 

True 

""" 

try: 

return self.__lattice 

except AttributeError: 

lattice = self._compute_lattice(rational_subgroup = True) 

self.__lattice = lattice 

return lattice 

def is_rational_cusp_gamma0(c, N, data): 

""" 

Return True if the rational number c is a rational cusp of level N. 

This uses remarks in Glenn Steven's Ph.D. thesis. 

INPUT: 

- ``c`` - a cusp 

- ``N`` - a positive integer 

- ``data`` - the list [n for n in range(2,N) if 

gcd(n,N) == 1], which is passed in as a parameter purely for 

efficiency reasons. 

EXAMPLES:: 

sage: from sage.modular.abvar.cuspidal_subgroup import is_rational_cusp_gamma0 

sage: N = 27 

sage: data = [n for n in range(2,N) if gcd(n,N) == 1] 

sage: is_rational_cusp_gamma0(Cusp(1/3), N, data) 

False 

sage: is_rational_cusp_gamma0(Cusp(1), N, data) 

True 

sage: is_rational_cusp_gamma0(Cusp(oo), N, data) 

True 

sage: is_rational_cusp_gamma0(Cusp(2/9), N, data) 

False 

""" 

num = c.numerator() 

den = c.denominator() 

for d in data: 

if not c.is_gamma0_equiv(Cusp(num,d*den), N): 

return False 

return True