Coverage for local/lib/python2.7/site-packages/sage/modular/abvar/finite_subgroup.py : 70%
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r""" Finite subgroups of modular abelian varieties
Sage can compute with fairly general finite subgroups of modular abelian varieties. Elements of finite order are represented by equivalence classes of elements in `H_1(A,\QQ)` modulo `H_1(A,\ZZ)`. A finite subgroup can be defined by giving generators and via various other constructions. Given a finite subgroup, one can compute generators, as well as the structure as an abstract group. Arithmetic on subgroups is also supported, including adding two subgroups together, checking inclusion, etc.
TODO: Intersection, action of Hecke operators.
AUTHORS:
- William Stein (2007-03)
EXAMPLES::
sage: J = J0(33) sage: C = J.cuspidal_subgroup() sage: C Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3 sage: C.order() 100 sage: C.gens() [[(1/10, 0, 1/10, 1/10, 1/10, 3/10)], [(0, 1/5, 1/10, 0, 1/10, 9/10)], [(0, 0, 1/2, 0, 1/2, 1/2)]] sage: C.0 + C.1 [(1/10, 1/5, 1/5, 1/10, 1/5, 6/5)] sage: 10*(C.0 + C.1) [(0, 0, 0, 0, 0, 0)] sage: G = C.subgroup([C.0 + C.1]); G Finite subgroup with invariants [10] over QQbar of Abelian variety J0(33) of dimension 3 sage: G.gens() [[(1/10, 1/5, 1/5, 1/10, 1/5, 1/5)]] sage: G.order() 10 sage: G <= C True sage: G >= C False
We make a table of the order of the cuspidal subgroup for the first few levels::
sage: for N in range(11,40): ....: print("{} {}".format(N, J0(N).cuspidal_subgroup().order())) 11 5 12 1 13 1 14 6 15 8 16 1 17 4 18 1 19 3 20 6 21 8 22 25 23 11 24 8 25 1 26 21 27 9 28 36 29 7 30 192 31 5 32 8 33 100 34 48 35 48 36 12 37 3 38 135 39 56
TESTS::
sage: G = J0(11).finite_subgroup([[1/3,0], [0,1/5]]); G Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1 sage: loads(dumps(G)) == G True sage: loads(dumps(G.0)) == G.0 True """
#***************************************************************************** # 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 __future__ import print_function from __future__ import absolute_import
from sage.modular.abvar.torsion_point import TorsionPoint from sage.modules.module import Module from sage.modules.free_module import is_FreeModule from sage.structure.element import ModuleElement from sage.structure.gens_py import abelian_iterator from sage.structure.sequence import Sequence from sage.structure.richcmp import richcmp_method, richcmp from sage.rings.all import QQ, ZZ, QQbar, Integer from sage.arith.all import gcd, lcm from sage.misc.all import prod from sage.structure.element import coercion_model
@richcmp_method class FiniteSubgroup(Module): r""" A finite subgroup of a modular abelian variety.
INPUT:
- ``abvar`` -- a modular abelian variety
- ``field_of_definition`` -- a field over which this group is defined
EXAMPLES:
This is an abstract base class, so there are no instances of this class itself::
sage: A = J0(37) sage: G = A.torsion_subgroup(3); G Finite subgroup with invariants [3, 3, 3, 3] over QQ of Abelian variety J0(37) of dimension 2 sage: type(G) <class 'sage.modular.abvar.finite_subgroup.FiniteSubgroup_lattice_with_category'> sage: from sage.modular.abvar.finite_subgroup import FiniteSubgroup sage: isinstance(G, FiniteSubgroup) True """
Element = TorsionPoint
def __init__(self, abvar, field_of_definition=QQ): """ Initialize ``self``.
TESTS::
sage: A = J0(11) sage: G = A.torsion_subgroup(2) sage: TestSuite(G).run() # long time """ raise TypeError("field_of_definition must be a field") raise TypeError("abvar must be a modular abelian variety")
################################################################ # DERIVED CLASS MUST OVERRIDE THE lattice METHOD ################################################################ def lattice(self): """ Return the lattice corresponding to this subgroup in the rational homology of the modular Jacobian product. The elements of the subgroup are represented by vectors in the ambient vector space (the rational homology), and this returns the lattice they span. EXAMPLES::
sage: J = J0(33); C = J[0].cuspidal_subgroup(); C Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: C.lattice() Free module of degree 6 and rank 2 over Integer Ring Echelon basis matrix: [ 1/5 13/5 -2 -4/5 2 -1/5] [ 0 3 -2 -1 2 0] """ raise NotImplementedError
def _relative_basis_matrix(self): """ Return matrix of this finite subgroup, but relative to the homology of the parent abelian variety.
EXAMPLES::
sage: A = J0(43)[1]; A Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43) sage: C = A.cuspidal_subgroup(); C Finite subgroup with invariants [7] over QQ of Simple abelian subvariety 43b(1,43) of dimension 2 of J0(43) sage: C._relative_basis_matrix() [ 1 0 0 0] [ 0 1/7 6/7 5/7] [ 0 0 1 0] [ 0 0 0 1] """
# General functionality def __richcmp__(self, other, op): """ Compare ``self`` to ``other``.
If ``other`` is not a :class:`FiniteSubgroup`, then ``NotImplemented`` is returned. If ``other`` is a :class:`FiniteSubgroup` and the ambient abelian varieties are not equal, then the ambient abelian varieties are compared. If ``other`` is a :class:`FiniteSubgroup` and the ambient abelian varieties are equal, then the subgroups are compared via their corresponding lattices.
EXAMPLES:
We first compare two subgroups of `J_0(37)`::
sage: A = J0(37) sage: G = A.torsion_subgroup(3); G.order() 81 sage: H = A.cuspidal_subgroup(); H.order() 3 sage: H < G True sage: H.is_subgroup(G) True
The ambient varieties are compared::
sage: A[0].cuspidal_subgroup() > J0(11).cuspidal_subgroup() True
Comparing subgroups sitting in different abelian varieties::
sage: A[0].cuspidal_subgroup() < A[1].cuspidal_subgroup() True """ # order gets reversed in passing to lattices.
def is_subgroup(self, other): """ Return True exactly if self is a subgroup of other, and both are defined as subgroups of the same ambient abelian variety.
EXAMPLES::
sage: C = J0(22).cuspidal_subgroup() sage: H = C.subgroup([C.0]) sage: K = C.subgroup([C.1]) sage: H.is_subgroup(K) False sage: K.is_subgroup(H) False sage: K.is_subgroup(C) True sage: H.is_subgroup(C) True """ # We use that self is contained in other, whether other is # either a finite group or an abelian variety, if and only # if self doesn't shrink when intersected with other. except TypeError: return False
def __add__(self, other): """ Return the sum of two subgroups.
EXAMPLES::
sage: C = J0(22).cuspidal_subgroup() sage: C.gens() [[(1/5, 1/5, 4/5, 0)], [(0, 0, 0, 1/5)]] sage: A = C.subgroup([C.0]); B = C.subgroup([C.1]) sage: A + B == C True """ raise TypeError("only addition of two finite subgroups is defined") raise ValueError("self and other must be in the same ambient Jacobian") lattice += C.lattice()
def exponent(self): """ Return the exponent of this finite abelian group.
OUTPUT: Integer
EXAMPLES::
sage: t = J0(33).hecke_operator(7) sage: G = t.kernel()[0]; G Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of Abelian variety J0(33) of dimension 3 sage: G.exponent() 4 """
def intersection(self, other): """ Return the intersection of the finite subgroups self and other.
INPUT:
- ``other`` - a finite group
OUTPUT: a finite group
EXAMPLES::
sage: E11a0, E11a1, B = J0(33) sage: G = E11a0.torsion_subgroup(6); H = E11a0.torsion_subgroup(9) sage: G.intersection(H) Finite subgroup with invariants [3, 3] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: W = E11a1.torsion_subgroup(15) sage: G.intersection(W) Finite subgroup with invariants [] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: E11a0.intersection(E11a1)[0] Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect subgroups of different abelian varieties.
::
sage: E11a0, E11a1, B = J0(33) sage: G = E11a0.torsion_subgroup(5); H = E11a1.torsion_subgroup(5) sage: G.intersection(H) Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: E11a0.intersection(E11a1)[0] Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
We intersect abelian varieties with subgroups::
sage: t = J0(33).hecke_operator(7) sage: G = t.kernel()[0]; G Finite subgroup with invariants [2, 2, 2, 2, 4, 4] over QQ of Abelian variety J0(33) of dimension 3 sage: A = J0(33).old_subvariety() sage: A.intersection(G) Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33) sage: A.hecke_operator(7).kernel()[0] Finite subgroup with invariants [2, 2, 2, 2] over QQ of Abelian subvariety of dimension 2 of J0(33) sage: B = J0(33).new_subvariety() sage: B.intersection(G) Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33) sage: B.hecke_operator(7).kernel()[0] Finite subgroup with invariants [4, 4] over QQ of Abelian subvariety of dimension 1 of J0(33) sage: A.intersection(B)[0] Finite subgroup with invariants [3, 3] over QQ of Abelian subvariety of dimension 2 of J0(33) """ else: raise TypeError("only intersection with a finite subgroup or " "modular abelian variety is defined") raise TypeError("finite subgroups must be in the same ambient product Jacobian")
# TODO: This might be way slower than what we could do if # we think more carefully.
def __mul__(self, right): """ Multiply this subgroup by the rational number right.
If right is an integer the result is a subgroup of self. If right is a rational number `n/m`, then this group is first divided by `m` then multiplied by `n`.
INPUT:
- ``right`` - a rational number
OUTPUT: a subgroup
EXAMPLES::
sage: J = J0(37) sage: H = J.cuspidal_subgroup(); H.order() 3 sage: G = H * 3; G.order() 1 sage: G = H * (1/2); G.order() 48 sage: J.torsion_subgroup(2) + H == G True sage: G = H*(3/2); G.order() 16 sage: J = J0(42) sage: G = J.cuspidal_subgroup(); factor(G.order()) 2^8 * 3^2 sage: (G * 3).order() 256 sage: (G * 0).order() 1 sage: (G * (1/5)).order() 22500000000 """ field_of_definition = self.field_of_definition())
def __rmul__(self, left): """ Multiply this finite subgroup on the left by an integer.
EXAMPLES::
sage: J = J0(42) sage: G = J.cuspidal_subgroup(); factor(G.order()) 2^8 * 3^2 sage: H = G.__rmul__(2) sage: H.order().factor() 2^4 * 3^2 sage: 2*G Finite subgroup with invariants [6, 24] over QQ of Abelian variety J0(42) of dimension 5 """
def abelian_variety(self): """ Return the abelian variety that this is a finite subgroup of.
EXAMPLES::
sage: J = J0(42) sage: G = J.rational_torsion_subgroup(); G Torsion subgroup of Abelian variety J0(42) of dimension 5 sage: G.abelian_variety() Abelian variety J0(42) of dimension 5 """
def field_of_definition(self): """ Return the field over which this finite modular abelian variety subgroup is defined. This is a field over which this subgroup is defined.
EXAMPLES::
sage: J = J0(42) sage: G = J.rational_torsion_subgroup(); G Torsion subgroup of Abelian variety J0(42) of dimension 5 sage: G.field_of_definition() Rational Field """
def _repr_(self): """ Return string representation of this finite subgroup.
EXAMPLES::
sage: J = J0(42) sage: G = J.torsion_subgroup(3); G._repr_() 'Finite subgroup with invariants [3, 3, 3, 3, 3, 3, 3, 3, 3, 3] over QQ of Abelian variety J0(42) of dimension 5' """ else: field = str(K)
def _invariants_repr(self): """ The string representation of the 'invariants' part of this group.
We make this a separate function so it is possible to create finite subgroups that don't print their invariants, since printing them could be expensive.
EXAMPLES::
sage: J0(42).cuspidal_subgroup()._invariants_repr() 'with invariants [2, 2, 12, 48] ' """
def order(self): """ Return the order (number of elements) of this finite subgroup.
EXAMPLES::
sage: J = J0(42) sage: C = J.cuspidal_subgroup() sage: C.order() 2304 """
def gens(self): """ Return generators for this finite subgroup.
EXAMPLES: We list generators for several cuspidal subgroups::
sage: J0(11).cuspidal_subgroup().gens() [[(0, 1/5)]] sage: J0(37).cuspidal_subgroup().gens() [[(0, 0, 0, 1/3)]] sage: J0(43).cuspidal_subgroup().gens() [[(0, 1/7, 0, 6/7, 0, 5/7)]] sage: J1(13).cuspidal_subgroup().gens() [[(1/19, 0, 0, 9/19)], [(0, 1/19, 1/19, 18/19)]] sage: J0(22).torsion_subgroup(6).gens() [[(1/6, 0, 0, 0)], [(0, 1/6, 0, 0)], [(0, 0, 1/6, 0)], [(0, 0, 0, 1/6)]] """
def gen(self, n): r""" Return `n^{th}` generator of self.
EXAMPLES::
sage: J = J0(23) sage: C = J.torsion_subgroup(3) sage: C.gens() [[(1/3, 0, 0, 0)], [(0, 1/3, 0, 0)], [(0, 0, 1/3, 0)], [(0, 0, 0, 1/3)]] sage: C.gen(0) [(1/3, 0, 0, 0)] sage: C.gen(3) [(0, 0, 0, 1/3)] sage: C.gen(4) Traceback (most recent call last): ... IndexError: list index out of range
Negative indices wrap around::
sage: C.gen(-1) [(0, 0, 0, 1/3)] """
def _element_constructor_(self, x, check=True): r""" Convert `x` into this finite subgroup.
This works when the abelian varieties that contain `x` and ``self`` are the same, or if `x` is convertible into the rational homology (viewed as an abstract `\QQ`-vector space).
EXAMPLES: We first construct the `11`-torsion subgroup of `J_0(23)`::
sage: J = J0(23) sage: G = J.torsion_subgroup(11) sage: G.invariants() [11, 11, 11, 11]
We also construct the cuspidal subgroup::
sage: C = J.cuspidal_subgroup() sage: C.invariants() [11]
sage: G(G.0) is G.0 True
We convert an element from the cuspidal subgroup into the `11`-torsion subgroup::
sage: z = G(C.0); z [(1/11, 10/11, 0, 8/11)] sage: z.parent() == G True
We convert a list, which defines an element of the underlying ``full_module`` into `G`, and verify an equality::
sage: x = G([1/11, 1/11, 0, -1/11]) sage: x == G([1/11, 1/11, 0, 10/11]) True
Finally we attempt to convert some elements that shouldn't work, since they are not in `G`::
sage: G(J.torsion_subgroup(3).0) Traceback (most recent call last): ... TypeError: element [1/3, 0, 0, 0] is not in free module
sage: G(J0(27).cuspidal_subgroup()(0)) Traceback (most recent call last): ... ValueError: ambient abelian varieties are different
"""
def __contains__(self, x): """ Return ``True`` if ``x`` is contained in this finite subgroup.
EXAMPLES:
We define two distinct finite subgroups of `J_0(27)`::
sage: G1 = J0(27).rational_cusp_subgroup(); G1 Finite subgroup with invariants [3] over QQ of Abelian variety J0(27) of dimension 1 sage: G1.0 [(1/3, 0)] sage: G2 = J0(27).cuspidal_subgroup(); G2 Finite subgroup with invariants [3, 3] over QQ of Abelian variety J0(27) of dimension 1 sage: G2.gens() [[(1/3, 0)], [(0, 1/3)]]
Now we check whether various elements are in `G_1` and `G_2`::
sage: G2.0 in G1 True sage: G2.1 in G1 False sage: G1.0 in G1 True sage: G1.0 in G2 True
The integer `0` is in `G_1`::
sage: 0 in G1 True
Elements that have a completely different ambient product Jacobian are never in `G`::
sage: J0(23).cuspidal_subgroup().0 in G1 False sage: J0(23).cuspidal_subgroup()(0) in G1 False """
def subgroup(self, gens): """ Return the subgroup of ``self`` spanned by the given generators, which must all be elements of ``self``.
EXAMPLES::
sage: J = J0(23) sage: G = J.torsion_subgroup(11); G Finite subgroup with invariants [11, 11, 11, 11] over QQ of Abelian variety J0(23) of dimension 2
We create the subgroup of the 11-torsion subgroup of `J_0(23)` generated by the first `11`-torsion point::
sage: H = G.subgroup([G.0]); H Finite subgroup with invariants [11] over QQbar of Abelian variety J0(23) of dimension 2 sage: H.invariants() [11]
We can also create a subgroup from a list of objects that can be converted into the ambient rational homology::
sage: H == G.subgroup([[1/11,0,0,0]]) True """ raise TypeError("gens must be a list or tuple")
def invariants(self): r""" Return elementary invariants of this abelian group, by which we mean a nondecreasing (immutable) sequence of integers `n_i`, `1 \leq i \leq k`, with `n_i` dividing `n_{i+1}`, and such that this group is abstractly isomorphic to `\ZZ/n_1\ZZ \times\cdots\times \ZZ/n_k\ZZ.`
EXAMPLES::
sage: J = J0(38) sage: C = J.cuspidal_subgroup(); C Finite subgroup with invariants [3, 45] over QQ of Abelian variety J0(38) of dimension 4 sage: v = C.invariants(); v [3, 45] sage: v[0] = 5 Traceback (most recent call last): ... ValueError: object is immutable; please change a copy instead. sage: type(v[0]) <type 'sage.rings.integer.Integer'>
::
sage: C * 3 Finite subgroup with invariants [15] over QQ of Abelian variety J0(38) of dimension 4
An example involving another cuspidal subgroup::
sage: C = J0(22).cuspidal_subgroup(); C Finite subgroup with invariants [5, 5] over QQ of Abelian variety J0(22) of dimension 2 sage: C.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: C.invariants() [5, 5] """
__iter__ = abelian_iterator
class FiniteSubgroup_lattice(FiniteSubgroup): def __init__(self, abvar, lattice, field_of_definition=QQbar, check=True): """ A finite subgroup of a modular abelian variety that is defined by a given lattice.
INPUT:
- ``abvar`` - a modular abelian variety
- ``lattice`` - a lattice that contains the lattice of abvar
- ``field_of_definition`` - the field of definition of this finite group scheme
- ``check`` - bool (default: True) whether or not to check that lattice contains the abvar lattice.
EXAMPLES::
sage: J = J0(11) sage: G = J.finite_subgroup([[1/3,0], [0,1/5]]); G Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1 """ raise TypeError("lattice must be a free module over ZZ") raise TypeError("abvar must be a modular abelian variety") raise ValueError("lattice must contain the lattice of abvar with finite index")
def lattice(self): r""" Return lattice that defines this finite subgroup.
EXAMPLES::
sage: J = J0(11) sage: G = J.finite_subgroup([[1/3,0], [0,1/5]]); G Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1 sage: G.lattice() Free module of degree 2 and rank 2 over Integer Ring Echelon basis matrix: [1/3 0] [ 0 1/5] """ |