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# -*- coding: utf-8 -*- """ Base class for modular abelian varieties
AUTHORS:
- William Stein (2007-03)
TESTS::
sage: A = J0(33) sage: D = A.decomposition(); D [ Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33), Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33), Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33) ] sage: loads(dumps(D)) == D True sage: loads(dumps(A)) == A 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 sage.categories.all import ModularAbelianVarieties from sage.structure.sequence import Sequence, Sequence_generic from sage.structure.richcmp import (richcmp_method, richcmp, richcmp_not_equal, rich_to_bool) from sage.structure.parent_base import ParentWithBase from .morphism import HeckeOperator, Morphism, DegeneracyMap from .torsion_subgroup import RationalTorsionSubgroup, QQbarTorsionSubgroup from .finite_subgroup import (FiniteSubgroup_lattice, FiniteSubgroup, TorsionPoint) from .cuspidal_subgroup import CuspidalSubgroup, RationalCuspidalSubgroup, RationalCuspSubgroup from sage.rings.all import ZZ, QQ, QQbar, Integer from sage.arith.all import LCM, divisors, prime_range, next_prime from sage.rings.ring import is_Ring from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing from sage.rings.polynomial.polynomial_element import Polynomial from sage.rings.infinity import infinity from sage.rings.fraction_field import FractionField from sage.modules.free_module import is_FreeModule from sage.modular.arithgroup.all import is_CongruenceSubgroup, is_Gamma0, is_Gamma1, is_GammaH from sage.modular.modsym.all import ModularSymbols from sage.modular.modsym.space import ModularSymbolsSpace from sage.modular.modform.constructor import Newform from sage.matrix.all import matrix, block_diagonal_matrix, identity_matrix from sage.modules.all import vector from sage.groups.all import AbelianGroup from sage.databases.cremona import cremona_letter_code from sage.misc.all import prod from sage.arith.misc import is_prime from sage.databases.cremona import CremonaDatabase from sage.schemes.elliptic_curves.constructor import EllipticCurve from sage.sets.primes import Primes from sage.functions.other import real
from copy import copy
from . import homspace from . import lseries
def is_ModularAbelianVariety(x): """ Return True if x is a modular abelian variety.
INPUT:
- ``x`` - object
EXAMPLES::
sage: from sage.modular.abvar.abvar import is_ModularAbelianVariety sage: is_ModularAbelianVariety(5) False sage: is_ModularAbelianVariety(J0(37)) True
Returning True is a statement about the data type not whether or not some abelian variety is modular::
sage: is_ModularAbelianVariety(EllipticCurve('37a')) False """
@richcmp_method class ModularAbelianVariety_abstract(ParentWithBase): def __init__(self, groups, base_field, is_simple=None, newform_level=None, isogeny_number=None, number=None, check=True): """ Abstract base class for modular abelian varieties.
INPUT:
- ``groups`` - a tuple of congruence subgroups
- ``base_field`` - a field
- ``is_simple`` - bool; whether or not self is simple
- ``newform_level`` - if self is isogenous to a newform abelian variety, returns the level of that abelian variety
- ``isogeny_number`` - which isogeny class the corresponding newform is in; this corresponds to the Cremona letter code
- ``number`` - the t number of the degeneracy map that this abelian variety is the image under
- ``check`` - whether to do some type checking on the defining data
EXAMPLES: One should not create an instance of this class, but we do so anyways here as an example::
sage: A = sage.modular.abvar.abvar.ModularAbelianVariety_abstract((Gamma0(37),), QQ) sage: type(A) <class 'sage.modular.abvar.abvar.ModularAbelianVariety_abstract_with_category'>
All hell breaks loose if you try to do anything with `A`::
sage: A <repr(<sage.modular.abvar.abvar.ModularAbelianVariety_abstract_with_category at 0x...>) failed: NotImplementedError: BUG -- lattice method must be defined in derived class>
All instances of this class are in the category of modular abelian varieties::
sage: A.category() Category of modular abelian varieties over Rational Field sage: J0(23).category() Category of modular abelian varieties over Rational Field """ raise TypeError("groups must be a tuple") raise TypeError("each element of groups must be a congruence subgroup") raise TypeError("base_field must be a field")
def groups(self): r""" Return an ordered tuple of the congruence subgroups that the ambient product Jacobian is attached to.
Every modular abelian variety is a finite quotient of an abelian subvariety of a product of modular Jacobians `J_\Gamma`. This function returns a tuple containing the groups `\Gamma`.
EXAMPLES::
sage: A = (J0(37) * J1(13))[0]; A Simple abelian subvariety 13aG1(1,13) of dimension 2 of J0(37) x J1(13) sage: A.groups() (Congruence Subgroup Gamma0(37), Congruence Subgroup Gamma1(13)) """
def is_J0(self): """ Return whether of not self is of the form J0(N).
OUTPUT: bool
EXAMPLES::
sage: J0(23).is_J0() True sage: J1(11).is_J0() False sage: (J0(23) * J1(11)).is_J0() False sage: J0(37)[0].is_J0() False sage: (J0(23) * J0(21)).is_J0() False """ and self.is_ambient()
def is_J1(self): """ Return whether of not self is of the form J1(N).
OUTPUT: bool
EXAMPLES::
sage: J1(23).is_J1() True sage: J0(23).is_J1() False sage: (J1(11) * J1(13)).is_J1() False sage: (J1(11) * J0(13)).is_J1() False sage: J1(23)[0].is_J1() False """ and self.is_ambient()
############################################################################# # lattice() *must* be defined by every derived class!!!! def lattice(self): """ Return lattice in ambient cuspidal modular symbols product that defines this modular abelian variety.
This must be defined in each derived class.
OUTPUT: a free module over `\ZZ`
EXAMPLES::
sage: A = sage.modular.abvar.abvar.ModularAbelianVariety_abstract((Gamma0(37),), QQ) sage: A <repr(<sage.modular.abvar.abvar.ModularAbelianVariety_abstract_with_category at 0x...>) failed: NotImplementedError: BUG -- lattice method must be defined in derived class> """ #############################################################################
def free_module(self): r""" Synonym for ``self.lattice()``.
OUTPUT: a free module over `\ZZ`
EXAMPLES::
sage: J0(37).free_module() Ambient free module of rank 4 over the principal ideal domain Integer Ring sage: J0(37)[0].free_module() Free module of degree 4 and rank 2 over Integer Ring Echelon basis matrix: [ 1 -1 1 0] [ 0 0 2 -1] """
def vector_space(self): r""" Return vector space corresponding to the modular abelian variety.
This is the lattice tensored with `\QQ`.
EXAMPLES::
sage: J0(37).vector_space() Vector space of dimension 4 over Rational Field sage: J0(37)[0].vector_space() Vector space of degree 4 and dimension 2 over Rational Field Basis matrix: [ 1 -1 0 1/2] [ 0 0 1 -1/2] """
def base_field(self): r""" Synonym for ``self.base_ring()``.
EXAMPLES::
sage: J0(11).base_field() Rational Field """
def base_extend(self, K): """ EXAMPLES::
sage: A = J0(37); A Abelian variety J0(37) of dimension 2 sage: A.base_extend(QQbar) Abelian variety J0(37) over Algebraic Field of dimension 2 sage: A.base_extend(GF(7)) Abelian variety J0(37) over Finite Field of size 7 of dimension 2 """
def __contains__(self, x): """ Determine whether or not self contains x.
EXAMPLES::
sage: J = J0(67); G = (J[0] + J[1]).intersection(J[1] + J[2]) sage: G[0] Finite subgroup with invariants [5, 10] over QQbar of Abelian subvariety of dimension 3 of J0(67) sage: a = G[0].0; a [(1/10, 1/10, 3/10, 1/2, 1, -2, -3, 33/10, 0, -1/2)] sage: a in J[0] False sage: a in (J[0]+J[1]) True sage: a in (J[1]+J[2]) True sage: C = G[1] # abelian variety in kernel sage: G[0].0 [(1/10, 1/10, 3/10, 1/2, 1, -2, -3, 33/10, 0, -1/2)] sage: 5*G[0].0 [(1/2, 1/2, 3/2, 5/2, 5, -10, -15, 33/2, 0, -5/2)] sage: 5*G[0].0 in C True """ return False return False
def __richcmp__(self, other, op): """ Compare two modular abelian varieties.
If other is not a modular abelian variety, compares the types of self and other. If other is a modular abelian variety, compares the groups, then if those are the same, compares the newform level and isogeny class number and degeneracy map numbers. If those are not defined or matched up, compare the underlying lattices.
EXAMPLES::
sage: J0(37)[0] < J0(37)[1] True sage: J0(37)[0] == J0(37)[1] False sage: J0(33)[0] < J0(33)[1] True sage: J0(33)[0] >= J0(33)[1] False """ return NotImplemented
# NOTE!! having the same newform level, isogeny class number, # and degen_t does not imply two abelian varieties are equal. # See the docstring for self.label.
def __radd__(self,other): """ Return other + self when other is 0. Otherwise raise a TypeError.
EXAMPLES::
sage: int(0) + J0(37) Abelian variety J0(37) of dimension 2 """ raise TypeError
def _repr_(self): """ Return string representation of this modular abelian variety.
This is just the generic base class, so it's unlikely to be called in practice.
EXAMPLES::
sage: A = J0(23) sage: import sage.modular.abvar.abvar as abvar sage: abvar.ModularAbelianVariety_abstract._repr_(A) 'Abelian variety J0(23) of dimension 2'
::
sage: (J0(11) * J0(33))._repr_() 'Abelian variety J0(11) x J0(33) of dimension 4' """ #if self.newform_level(none_if_not_known=True) is None: else:
else: simple, sub, label, self.dimension(), self._ambient_repr(), field)
def label(self): r""" Return the label associated to this modular abelian variety.
The format of the label is [level][isogeny class][group](t, ambient level)
If this abelian variety `B` has the above label, this implies only that `B` is isogenous to the newform abelian variety `A_f` associated to the newform with label [level][isogeny class][group]. The [group] is empty for `\Gamma_0(N)`, is G1 for `\Gamma_1(N)` and is GH[...] for `\Gamma_H(N)`.
.. warning::
The sum of `\delta_s(A_f)` for all `s\mid t` contains `A`, but no sum for a proper divisor of `t` contains `A`. It need *not* be the case that `B` is equal to `\delta_t(A_f)`!!!
OUTPUT: string
EXAMPLES::
sage: J0(11).label() '11a(1,11)' sage: J0(11)[0].label() '11a(1,11)' sage: J0(33)[2].label() '33a(1,33)' sage: J0(22).label() Traceback (most recent call last): ... ValueError: self must be simple
We illustrate that self need not equal `\delta_t(A_f)`::
sage: J = J0(11); phi = J.degeneracy_map(33, 1) + J.degeneracy_map(33,3) sage: B = phi.image(); B Abelian subvariety of dimension 1 of J0(33) sage: B.decomposition() [ Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33) ] sage: C = J.degeneracy_map(33,3).image(); C Abelian subvariety of dimension 1 of J0(33) sage: C == B False """
def newform(self, names=None): """ Return the newform `f` such that this abelian variety is isogenous to the newform abelian variety `A_f`. If this abelian variety is not simple, raise a ValueError.
INPUT:
- ``names`` -- (default: None) If the newform has coefficients in a number field, then a generator name must be specified.
OUTPUT: A newform `f` so that self is isogenous to `A_f`.
EXAMPLES::
sage: J0(11).newform() q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6)
sage: f = J0(23).newform(names='a') sage: AbelianVariety(f) == J0(23) True
sage: J = J0(33) sage: [s.newform('a') for s in J.decomposition()] [q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6), q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6), q + q^2 - q^3 - q^4 - 2*q^5 + O(q^6)]
The following fails since `J_0(33)` is not simple::
sage: J0(33).newform() Traceback (most recent call last): ... ValueError: self must be simple """
def newform_decomposition(self, names=None): """ Return the newforms of the simple subvarieties in the decomposition of self as a product of simple subvarieties, up to isogeny.
OUTPUT:
- an array of newforms
EXAMPLES::
sage: J = J1(11) * J0(23) sage: J.newform_decomposition('a') [q - 2*q^2 - q^3 + 2*q^4 + q^5 + O(q^6), q + a0*q^2 + (-2*a0 - 1)*q^3 + (-a0 - 1)*q^4 + 2*a0*q^5 + O(q^6)] """
def newform_label(self): """ Return the label [level][isogeny class][group] of the newform `f` such that this abelian variety is isogenous to the newform abelian variety `A_f`. If this abelian variety is not simple, raise a ValueError.
OUTPUT: string
EXAMPLES::
sage: J0(11).newform_label() '11a' sage: J0(33)[2].newform_label() '33a'
The following fails since `J_0(33)` is not simple::
sage: J0(33).newform_label() Traceback (most recent call last): ... ValueError: self must be simple """
def elliptic_curve(self): """ Return an elliptic curve isogenous to self. If self is not dimension 1 with rational base ring, raise a ValueError.
The elliptic curve is found by looking it up in the CremonaDatabase. The CremonaDatabase contains all curves up to some large conductor. If a curve is not found in the CremonaDatabase, a RuntimeError will be raised. In practice, only the most committed users will see this RuntimeError.
OUTPUT: an elliptic curve isogenous to self.
EXAMPLES::
sage: J = J0(11) sage: J.elliptic_curve() Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
sage: J = J0(49) sage: J.elliptic_curve() Elliptic Curve defined by y^2 + x*y = x^3 - x^2 - 2*x - 1 over Rational Field
sage: A = J0(37)[1] sage: E = A.elliptic_curve() sage: A.lseries()(1) 0.725681061936153 sage: E.lseries()(1) 0.725681061936153
Elliptic curves are of dimension 1. ::
sage: J = J0(23) sage: J.elliptic_curve() Traceback (most recent call last): ... ValueError: self must be of dimension 1
This is only implemented for curves over QQ. ::
sage: J = J0(11).change_ring(CC) sage: J.elliptic_curve() Traceback (most recent call last): ... ValueError: base ring must be QQ """
raise RuntimeError("Elliptic curve not found" + " in installed database")
def _isogeny_to_newform_abelian_variety(self): r""" Return an isogeny from self to an abelian variety `A_f` attached to a newform. If self is not simple (so that no such isogeny exists), raise a ValueError.
EXAMPLES::
sage: J0(22)[0]._isogeny_to_newform_abelian_variety() Abelian variety morphism: From: Simple abelian subvariety 11a(1,22) of dimension 1 of J0(22) To: Newform abelian subvariety 11a of dimension 1 of J0(11) sage: J = J0(11); phi = J.degeneracy_map(33, 1) + J.degeneracy_map(33,3) sage: A = phi.image() sage: A._isogeny_to_newform_abelian_variety().matrix() [-3 3] [ 0 -3] """
raise ValueError("self is not simple")
def _simple_isogeny(self, other): """ Given self and other, if both are simple, and correspond to the same newform with the same congruence subgroup, return an isogeny. Otherwise, raise a ValueError.
INPUT:
- ``self, other`` - modular abelian varieties
OUTPUT: an isogeny
EXAMPLES::
sage: J = J0(33); J Abelian variety J0(33) of dimension 3 sage: J[0]._simple_isogeny(J[1]) Abelian variety morphism: From: Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) To: Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33)
The following illustrates how simple isogeny is only implemented when the ambients are the same::
sage: J[0]._simple_isogeny(J1(11)) Traceback (most recent call last): ... NotImplementedError: _simple_isogeny only implemented when both abelian variety have the same ambient product Jacobian """ raise TypeError("other must be a modular abelian variety")
raise ValueError("self is not simple")
raise ValueError("other is not simple")
# The issue here is that the stuff below probably won't make any sense at all if we don't know # that the two newform abelian varieties $A_f$ are identical.
(self.isogeny_number() != other.isogeny_number()): raise ValueError("self and other do not correspond to the same newform")
self._isogeny_to_newform_abelian_variety()
def _Hom_(self, B, cat=None): """ INPUT:
- ``B`` - modular abelian varieties
- ``cat`` - category
EXAMPLES::
sage: J0(37)._Hom_(J1(37)) Space of homomorphisms from Abelian variety J0(37) of dimension 2 to Abelian variety J1(37) of dimension 40 sage: J0(37)._Hom_(J1(37)).homset_category() Category of modular abelian varieties over Rational Field """ elif K == QQbar or L == QQbar: F = QQbar else: # TODO -- improve this raise ValueError("please specify a category") else:
def in_same_ambient_variety(self, other): """ Return True if self and other are abelian subvarieties of the same ambient product Jacobian.
EXAMPLES::
sage: A,B,C = J0(33) sage: A.in_same_ambient_variety(B) True sage: A.in_same_ambient_variety(J0(11)) False """ return False return False
def modular_kernel(self): """ Return the modular kernel of this abelian variety, which is the kernel of the canonical polarization of self.
EXAMPLES::
sage: A = AbelianVariety('33a'); A Newform abelian subvariety 33a of dimension 1 of J0(33) sage: A.modular_kernel() Finite subgroup with invariants [3, 3] over QQ of Newform abelian subvariety 33a of dimension 1 of J0(33) """
def modular_degree(self): """ Return the modular degree of this abelian variety, which is the square root of the degree of the modular kernel.
EXAMPLES::
sage: A = AbelianVariety('37a') sage: A.modular_degree() 2 """
def intersection(self, other): """ Return the intersection of self and other inside a common ambient Jacobian product.
INPUT:
- ``other`` - a modular abelian variety or a finite group
OUTPUT: If other is a modular abelian variety:
- ``G`` - finite subgroup of self
- ``A`` - abelian variety (identity component of intersection) If other is a finite group:
- ``G`` - a finite group
EXAMPLES: We intersect some abelian varieties with finite intersection.
::
sage: J = J0(37) sage: J[0].intersection(J[1]) (Finite subgroup with invariants [2, 2] over QQ of Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37), Simple abelian subvariety of dimension 0 of J0(37))
::
sage: D = list(J0(65)); D [Simple abelian subvariety 65a(1,65) of dimension 1 of J0(65), Simple abelian subvariety 65b(1,65) of dimension 2 of J0(65), Simple abelian subvariety 65c(1,65) of dimension 2 of J0(65)] sage: D[0].intersection(D[1]) (Finite subgroup with invariants [2] over QQ of Simple abelian subvariety 65a(1,65) of dimension 1 of J0(65), Simple abelian subvariety of dimension 0 of J0(65)) sage: (D[0]+D[1]).intersection(D[1]+D[2]) (Finite subgroup with invariants [2] over QQbar of Abelian subvariety of dimension 3 of J0(65), Abelian subvariety of dimension 2 of J0(65))
::
sage: J = J0(33) sage: J[0].intersection(J[1]) (Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33), Simple abelian subvariety of dimension 0 of J0(33))
Next we intersect two abelian varieties with non-finite intersection::
sage: J = J0(67); D = J.decomposition(); D [ Simple abelian subvariety 67a(1,67) of dimension 1 of J0(67), Simple abelian subvariety 67b(1,67) of dimension 2 of J0(67), Simple abelian subvariety 67c(1,67) of dimension 2 of J0(67) ] sage: (D[0] + D[1]).intersection(D[1] + D[2]) (Finite subgroup with invariants [5, 10] over QQbar of Abelian subvariety of dimension 3 of J0(67), Abelian subvariety of dimension 2 of J0(67)) """ # First check whether we are intersecting an abelian variety # with a finite subgroup. If so, call the intersection method # for the finite group, which does know how to intersect with # an abelian variety.
# Now both self and other are abelian varieties. We require # at least that the ambient Jacobian product is the same for # them. raise TypeError("other must be an abelian variety in the same ambient space")
# 1. Compute the abelian variety (connected) part of the intersection # If there is a nonzero abelian variety, get the actual # lattice that defines it. We intersect (=saturate) in # the sum of the lattices, to ensure that the intersection # is an abelian subvariety of both self and other (even if # they aren't subvarieties of the ambient Jacobian). else:
# 2. Compute the finite intersection group when the # intersection is finite, or a group that maps surjectively # onto the component group in general.
# First we get basis matrices for the lattices that define # both abelian varieties.
# Then we stack matrices and find a subset that forms a # basis. # Finally we project onto the L factor. for v in V.coordinate_module(S).basis()]
else:
def __add__(self, other): r""" Return the sum of the *images* of self and other inside the ambient Jacobian product.
Here self and other must be abelian subvarieties of the ambient Jacobian product.
.. WARNING::
The sum of course only makes sense in some ambient variety, and by definition this function takes the sum of the images of both self and other in the ambient product Jacobian.
EXAMPLES:
We compute the sum of two abelian varieties of `J_0(33)`::
sage: J = J0(33) sage: J[0] + J[1] Abelian subvariety of dimension 2 of J0(33)
We sum all three and get the full `J_0(33)`::
sage: (J[0] + J[1]) + (J[1] + J[2]) Abelian variety J0(33) of dimension 3
Adding to zero works::
sage: J[0] + 0 Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)
Hence the sum command works::
sage: sum([J[0], J[2]]) Abelian subvariety of dimension 2 of J0(33)
We try to add something in `J_0(33)` to something in `J_0(11)`; this shouldn't and doesn't work.
::
sage: J[0] + J0(11) Traceback (most recent call last): ... TypeError: sum not defined since ambient spaces different
We compute the diagonal image of `J_0(11)` in `J_0(33)`, then add the result to the new elliptic curve of level `33`.
::
sage: A = J0(11) sage: B = (A.degeneracy_map(33,1) + A.degeneracy_map(33,3)).image() sage: B + J0(33)[2] Abelian subvariety of dimension 2 of J0(33)
TESTS:
This exposed a bug in HNF (see :trac:`4527`)::
sage: A = J0(206).new_subvariety().decomposition()[3] ; A # long time Simple abelian subvariety 206d(1,206) of dimension 4 of J0(206) sage: B = J0(206).old_subvariety(2) ; B # long time Abelian subvariety of dimension 16 of J0(206) sage: A+B # long time Abelian subvariety of dimension 20 of J0(206) """ if other == 0: return self raise TypeError("other must be a modular abelian variety") raise ValueError("incompatible ambient Jacobians")
def direct_product(self, other): """ Compute the direct product of self and other.
INPUT:
- ``self, other`` - modular abelian varieties
OUTPUT: abelian variety
EXAMPLES::
sage: J0(11).direct_product(J1(13)) Abelian variety J0(11) x J1(13) of dimension 3 sage: A = J0(33)[0].direct_product(J0(33)[1]); A Abelian subvariety of dimension 2 of J0(33) x J0(33) sage: A.lattice() Free module of degree 12 and rank 4 over Integer Ring Echelon basis matrix: [ 1 1 -2 0 2 -1 0 0 0 0 0 0] [ 0 3 -2 -1 2 0 0 0 0 0 0 0] [ 0 0 0 0 0 0 1 0 0 0 -1 2] [ 0 0 0 0 0 0 0 1 -1 1 0 -2] """
def __pow__(self, n): """ Return `n^{th}` power of self.
INPUT:
- ``n`` - a nonnegative integer
OUTPUT: an abelian variety
EXAMPLES::
sage: J = J0(37) sage: J^0 Simple abelian subvariety of dimension 0 of J0(37) sage: J^1 Abelian variety J0(37) of dimension 2 sage: J^1 is J True """ raise ValueError("n must be nonnegative") groups = self.groups() * n L = self.lattice().basis_matrix() lattice = block_diagonal_matrix([L]*n).row_module(ZZ) return ModularAbelianVariety(groups, lattice, self.base_field(), check=False)
def __mul__(self, other): """ Compute the direct product of self and other.
EXAMPLES: Some modular Jacobians::
sage: J0(11) * J0(33) Abelian variety J0(11) x J0(33) of dimension 4 sage: J0(11) * J0(33) * J0(11) Abelian variety J0(11) x J0(33) x J0(11) of dimension 5
We multiply some factors of `J_0(65)`::
sage: d = J0(65).decomposition() sage: d[0] * d[1] * J0(11) Abelian subvariety of dimension 4 of J0(65) x J0(65) x J0(11) """ raise TypeError("other must be a modular abelian variety") raise TypeError("self and other must have the same base ring")
def quotient(self, other): """ Compute the quotient of self and other, where other is either an abelian subvariety of self or a finite subgroup of self.
INPUT:
- ``other`` - a finite subgroup or subvariety
OUTPUT: a pair (A, phi) with phi the quotient map from self to A
EXAMPLES: We quotient `J_0(33)` out by an abelian subvariety::
sage: Q, f = J0(33).quotient(J0(33)[0]) sage: Q Abelian variety factor of dimension 2 of J0(33) sage: f Abelian variety morphism: From: Abelian variety J0(33) of dimension 3 To: Abelian variety factor of dimension 2 of J0(33)
We quotient `J_0(33)` by the cuspidal subgroup::
sage: C = J0(33).cuspidal_subgroup() sage: Q, f = J0(33).quotient(C) sage: Q Abelian variety factor of dimension 3 of J0(33) sage: f.kernel()[0] Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3 sage: C Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3 sage: J0(11).direct_product(J1(13)) Abelian variety J0(11) x J1(13) of dimension 3 """
def __truediv__(self, other): """ Compute the quotient of self and other, where other is either an abelian subvariety of self or a finite subgroup of self.
INPUT:
- ``other`` - a finite subgroup or subvariety
EXAMPLES: Quotient out by a finite group::
sage: J = J0(67); G = (J[0] + J[1]).intersection(J[1] + J[2]) sage: Q, _ = J/G[0]; Q Abelian variety factor of dimension 5 of J0(67) over Algebraic Field sage: Q.base_field() Algebraic Field sage: Q.lattice() Free module of degree 10 and rank 10 over Integer Ring Echelon basis matrix: [1/10 1/10 3/10 1/2 0 0 0 3/10 0 1/2] [ 0 1/5 4/5 4/5 0 0 0 0 0 3/5] ...
Quotient out by an abelian subvariety::
sage: A, B, C = J0(33) sage: Q, phi = J0(33)/A sage: Q Abelian variety factor of dimension 2 of J0(33) sage: phi.domain() Abelian variety J0(33) of dimension 3 sage: phi.codomain() Abelian variety factor of dimension 2 of J0(33) sage: phi.kernel() (Finite subgroup with invariants [2] over QQbar of Abelian variety J0(33) of dimension 3, Abelian subvariety of dimension 1 of J0(33)) sage: phi.kernel()[1] == A True
The abelian variety we quotient out by must be an abelian subvariety.
::
sage: Q = (A + B)/C; Q Traceback (most recent call last): ... TypeError: other must be a subgroup or abelian subvariety """ else:
def degeneracy_map(self, M_ls, t_ls): """ Return the degeneracy map with domain self and given level/parameter. If self.ambient_variety() is a product of Jacobians (as opposed to a single Jacobian), then one can provide a list of new levels and parameters, corresponding to the ambient Jacobians in order. (See the examples below.)
INPUT:
- ``M, t`` - integers level and `t`, or
- ``Mlist, tlist`` - if self is in a nontrivial product ambient Jacobian, input consists of a list of levels and corresponding list of `t`'s.
OUTPUT: a degeneracy map
EXAMPLES: We make several degeneracy maps related to `J_0(11)` and `J_0(33)` and compute their matrices.
::
sage: d1 = J0(11).degeneracy_map(33, 1); d1 Degeneracy map from Abelian variety J0(11) of dimension 1 to Abelian variety J0(33) of dimension 3 defined by [1] sage: d1.matrix() [ 0 -3 2 1 -2 0] [ 1 -2 0 1 0 -1] sage: d2 = J0(11).degeneracy_map(33, 3); d2 Degeneracy map from Abelian variety J0(11) of dimension 1 to Abelian variety J0(33) of dimension 3 defined by [3] sage: d2.matrix() [-1 0 0 0 1 -2] [-1 -1 1 -1 1 0] sage: d3 = J0(33).degeneracy_map(11, 1); d3 Degeneracy map from Abelian variety J0(33) of dimension 3 to Abelian variety J0(11) of dimension 1 defined by [1]
He we verify that first mapping from level `11` to level `33`, then back is multiplication by `4`::
sage: d1.matrix() * d3.matrix() [4 0] [0 4]
We compute a more complicated degeneracy map involving nontrivial product ambient Jacobians; note that this is just the block direct sum of the two matrices at the beginning of this example::
sage: d = (J0(11)*J0(11)).degeneracy_map([33,33], [1,3]); d Degeneracy map from Abelian variety J0(11) x J0(11) of dimension 2 to Abelian variety J0(33) x J0(33) of dimension 6 defined by [1, 3] sage: d.matrix() [ 0 -3 2 1 -2 0 0 0 0 0 0 0] [ 1 -2 0 1 0 -1 0 0 0 0 0 0] [ 0 0 0 0 0 0 -1 0 0 0 1 -2] [ 0 0 0 0 0 0 -1 -1 1 -1 1 0] """
raise ValueError("must have same number of Ms and ts") raise ValueError("must have same number of Ms and groups in ambient variety")
raise ValueError("one level must divide the other in %s-th component"%i) raise ValueError("each t must divide the quotient of the levels")
for i in range(length) ])
def _quotient_by_finite_subgroup(self, G): """ Return the quotient of self by the finite subgroup `G`. This is used internally by the quotient and __div__ commands.
INPUT:
- ``G`` - a finite subgroup of self
OUTPUT: abelian variety - the quotient `Q` of self by `G`
- ``morphism`` - from self to the quotient `Q`
EXAMPLES: We quotient the elliptic curve `J_0(11)` out by its cuspidal subgroup.
::
sage: A = J0(11) sage: G = A.cuspidal_subgroup(); G Finite subgroup with invariants [5] over QQ of Abelian variety J0(11) of dimension 1 sage: Q, f = A._quotient_by_finite_subgroup(G) sage: Q Abelian variety factor of dimension 1 of J0(11) sage: f Abelian variety morphism: From: Abelian variety J0(11) of dimension 1 To: Abelian variety factor of dimension 1 of J0(11)
We compute the finite kernel of `f` (hence the [0]) and note that it equals the subgroup `G` that we quotiented out by::
sage: f.kernel()[0] == G True """ return self
def _quotient_by_abelian_subvariety(self, B): """ Return the quotient of self by the abelian variety `B`. This is used internally by the quotient and __div__ commands.
INPUT:
- ``B`` - an abelian subvariety of self
OUTPUT:
- ``abelian variety`` - quotient `Q` of self by B
- ``morphism`` - from self to the quotient `Q`
EXAMPLES: We compute the new quotient of `J_0(33)`.
::
sage: A = J0(33); B = A.old_subvariety() sage: Q, f = A._quotient_by_abelian_subvariety(B)
Note that the quotient happens to also be an abelian subvariety::
sage: Q Abelian subvariety of dimension 1 of J0(33) sage: Q.lattice() Free module of degree 6 and rank 2 over Integer Ring Echelon basis matrix: [ 1 0 0 -1 0 0] [ 0 0 1 0 1 -1] sage: f Abelian variety morphism: From: Abelian variety J0(33) of dimension 3 To: Abelian subvariety of dimension 1 of J0(33)
We verify that `B` is equal to the kernel of the quotient map.
::
sage: f.kernel()[1] == B True
Next we quotient `J_0(33)` out by `Q` itself::
sage: C, g = A._quotient_by_abelian_subvariety(Q)
The result is not a subvariety::
sage: C Abelian variety factor of dimension 2 of J0(33) sage: C.lattice() Free module of degree 6 and rank 4 over Integer Ring Echelon basis matrix: [ 1/3 0 0 2/3 -1 0] [ 0 1 0 0 -1 1] [ 0 0 1/3 0 -2/3 2/3] [ 0 0 0 1 -1 -1] """
# We first compute the complement of B in self to get # an abelian variety C also in self such that self/B # is isogenous to C. This is the case because the # projection map pi:self --> C is surjective and has # kernel a finite extension of the abelian variety B.
# Now that we have C we need to find some abelian variety Q # isogenous to C and a map self --> Q whose kernel is exactly # B. We do this by computing the kernel of the map pi below, # which is an extension of the abelian variety B by a finite # group Phi of complements. Our strategy is to enlarge the # lattice that defines C so that the map pi below suddenly # has connected kernel.
def projection(self, A, check=True): """ Given an abelian subvariety A of self, return a projection morphism from self to A. Note that this morphism need not be unique.
INPUT:
- ``A`` - an abelian variety
OUTPUT: a morphism
EXAMPLES::
sage: a,b,c = J0(33) sage: pi = J0(33).projection(a); pi.matrix() [ 3 -2] [-5 5] [-4 1] [ 3 -2] [ 5 0] [ 1 1] sage: pi = (a+b).projection(a); pi.matrix() [ 0 0] [-3 2] [-4 1] [-1 -1] sage: pi = a.projection(a); pi.matrix() [1 0] [0 1]
We project onto a factor in a product of two Jacobians::
sage: A = J0(11)*J0(11); A Abelian variety J0(11) x J0(11) of dimension 2 sage: A[0] Simple abelian subvariety 11a(1,11) of dimension 1 of J0(11) x J0(11) sage: A.projection(A[0]) Abelian variety morphism: From: Abelian variety J0(11) x J0(11) of dimension 2 To: Simple abelian subvariety 11a(1,11) of dimension 1 of J0(11) x J0(11) sage: A.projection(A[0]).matrix() [0 0] [0 0] [1 0] [0 1] sage: A.projection(A[1]).matrix() [1 0] [0 1] [0 0] [0 0] """ raise ValueError("A must be an abelian subvariety of self")
# solve X * mat = self, i.e. write each row of self in terms of the # rows of mat.
# The projection map is got from the first 2*dim(A) columns of X.
def project_to_factor(self, n): """ If self is an ambient product of Jacobians, return a projection from self to the nth such Jacobian.
EXAMPLES::
sage: J = J0(33) sage: J.project_to_factor(0) Abelian variety endomorphism of Abelian variety J0(33) of dimension 3
::
sage: J = J0(33) * J0(37) * J0(11) sage: J.project_to_factor(2) Abelian variety morphism: From: Abelian variety J0(33) x J0(37) x J0(11) of dimension 6 To: Abelian variety J0(11) of dimension 1 sage: J.project_to_factor(2).matrix() [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [0 0] [1 0] [0 1] """ raise ValueError("self is not ambient") raise IndexError("index (=%s) too large (max = %s)"%(n, len(self.groups())))
for gp in self.groups()[0:n] ])
def is_subvariety_of_ambient_jacobian(self): """ Return True if self is (presented as) a subvariety of the ambient product Jacobian.
Every abelian variety in Sage is a quotient of a subvariety of an ambient Jacobian product by a finite subgroup.
EXAMPLES::
sage: J0(33).is_subvariety_of_ambient_jacobian() True sage: A = J0(33)[0]; A Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: A.is_subvariety_of_ambient_jacobian() True sage: B, phi = A / A.torsion_subgroup(2) sage: B Abelian variety factor of dimension 1 of J0(33) sage: phi.matrix() [2 0] [0 2] sage: B.is_subvariety_of_ambient_jacobian() False """
def ambient_variety(self): """ Return the ambient modular abelian variety that contains this abelian variety. The ambient variety is always a product of Jacobians of modular curves.
OUTPUT: abelian variety
EXAMPLES::
sage: A = J0(33)[0]; A Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: A.ambient_variety() Abelian variety J0(33) of dimension 3 """ self.base_field(), check=False)
def ambient_morphism(self): """ Return the morphism from self to the ambient variety. This is injective if self is natural a subvariety of the ambient product Jacobian.
OUTPUT: morphism
The output is cached.
EXAMPLES: We compute the ambient structure morphism for an abelian subvariety of `J_0(33)`::
sage: A,B,C = J0(33) sage: phi = A.ambient_morphism() sage: phi.domain() Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: phi.codomain() Abelian variety J0(33) of dimension 3 sage: phi.matrix() [ 1 1 -2 0 2 -1] [ 0 3 -2 -1 2 0]
phi is of course injective
::
sage: phi.kernel() (Finite subgroup with invariants [] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33), Abelian subvariety of dimension 0 of J0(33))
This is the same as the basis matrix for the lattice corresponding to self::
sage: A.lattice() Free module of degree 6 and rank 2 over Integer Ring Echelon basis matrix: [ 1 1 -2 0 2 -1] [ 0 3 -2 -1 2 0]
We compute a non-injective map to an ambient space::
sage: Q,pi = J0(33)/A sage: phi = Q.ambient_morphism() sage: phi.matrix() [ 1 4 1 9 -1 -1] [ 0 15 0 0 30 -75] [ 0 0 5 10 -5 15] [ 0 0 0 15 -15 30] sage: phi.kernel()[0] Finite subgroup with invariants [5, 15, 15] over QQ of Abelian variety factor of dimension 2 of J0(33) """
def is_ambient(self): """ Return True if self equals the ambient product Jacobian.
OUTPUT: bool
EXAMPLES::
sage: A,B,C = J0(33) sage: A.is_ambient() False sage: J0(33).is_ambient() True sage: (A+B).is_ambient() False sage: (A+B+C).is_ambient() True """
def dimension(self): """ Return the dimension of this abelian variety.
EXAMPLES::
sage: A = J0(23) sage: A.dimension() 2 """
def conductor(self): """ Return the conductor of this abelian variety.
EXAMPLES::
sage: A = J0(23) sage: A.conductor().factor() 23^2
sage: A = J1(25) sage: A.conductor().factor() 5^24
sage: A = J0(11^2); A.decomposition() [ Simple abelian subvariety 11a(1,121) of dimension 1 of J0(121), Simple abelian subvariety 11a(11,121) of dimension 1 of J0(121), Simple abelian subvariety 121a(1,121) of dimension 1 of J0(121), Simple abelian subvariety 121b(1,121) of dimension 1 of J0(121), Simple abelian subvariety 121c(1,121) of dimension 1 of J0(121), Simple abelian subvariety 121d(1,121) of dimension 1 of J0(121) ] sage: A.conductor().factor() 11^10
sage: A = J0(33)[0]; A Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: A.conductor() 11 sage: A.elliptic_curve().conductor() 11 """ raise ValueError("base ring must be QQ") for f in self.newform_decomposition('a'))
def rank(self): """ Return the rank of the underlying lattice of self.
EXAMPLES::
sage: J = J0(33) sage: J.rank() 6 sage: J[1] Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33) sage: (J[1] * J[1]).rank() 4 """
def degree(self): """ Return the degree of this abelian variety, which is the dimension of the ambient Jacobian product.
EXAMPLES::
sage: A = J0(23) sage: A.dimension() 2 """
def endomorphism_ring(self, category=None): """ Return the endomorphism ring of self.
OUTPUT: b = self.sturm_bound()
EXAMPLES: We compute a few endomorphism rings::
sage: J0(11).endomorphism_ring() Endomorphism ring of Abelian variety J0(11) of dimension 1 sage: J0(37).endomorphism_ring() Endomorphism ring of Abelian variety J0(37) of dimension 2 sage: J0(33)[2].endomorphism_ring() Endomorphism ring of Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
No real computation is done::
sage: J1(123456).endomorphism_ring() Endomorphism ring of Abelian variety J1(123456) of dimension 423185857 """
def sturm_bound(self): r""" Return a bound `B` such that all Hecke operators `T_n` for `n\leq B` generate the Hecke algebra.
OUTPUT: integer
EXAMPLES::
sage: J0(11).sturm_bound() 2 sage: J0(33).sturm_bound() 8 sage: J1(17).sturm_bound() 48 sage: J1(123456).sturm_bound() 1693483008 sage: JH(37,[2,3]).sturm_bound() 7 sage: J1(37).sturm_bound() 228 """
def is_hecke_stable(self): """ Return True if self is stable under the Hecke operators of its ambient Jacobian.
OUTPUT: bool
EXAMPLES::
sage: J0(11).is_hecke_stable() True sage: J0(33)[2].is_hecke_stable() True sage: J0(33)[0].is_hecke_stable() False sage: (J0(33)[0] + J0(33)[1]).is_hecke_stable() True """
#b = self.modular_symbols().sturm_bound()
def is_subvariety(self, other): """ Return True if self is a subvariety of other as they sit in a common ambient modular Jacobian. In particular, this function will only return True if self and other have exactly the same ambient Jacobians.
EXAMPLES::
sage: J = J0(37); J Abelian variety J0(37) of dimension 2 sage: A = J[0]; A Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37) sage: A.is_subvariety(A) True sage: A.is_subvariety(J) True """ return False return True return False # self is an abelian subvariety of other if and only if # 1. L is a subset of M (so the abelian subvarieties of # the ambient J are equal), and # 2. L is relatively saturated in M, i.e., M/L is # torsion free. return False # To determine if L is relatively saturated we compute the # intersection of M with (L tensor Q) and see if that equals # L.
def change_ring(self, R): """ Change the base ring of this modular abelian variety.
EXAMPLES::
sage: A = J0(23) sage: A.change_ring(QQ) Abelian variety J0(23) of dimension 2 """
def level(self): """ Return the level of this modular abelian variety, which is an integer N (usually minimal) such that this modular abelian variety is a quotient of `J_1(N)`. In the case that the ambient variety of self is a product of Jacobians, return the LCM of their levels.
EXAMPLES::
sage: J1(5077).level() 5077 sage: JH(389,[4]).level() 389 sage: (J0(11)*J0(17)).level() 187 """
def newform_level(self, none_if_not_known=False): """ Write self as a product (up to isogeny) of newform abelian varieties `A_f`. Then this function return the least common multiple of the levels of the newforms `f`, along with the corresponding group or list of groups (the groups do not appear with multiplicity).
INPUT:
- ``none_if_not_known`` - (default: False) if True, return None instead of attempting to compute the newform level, if it isn't already known. This None result is not cached.
OUTPUT: integer group or list of distinct groups
EXAMPLES::
sage: J0(33)[0].newform_level() (11, Congruence Subgroup Gamma0(33)) sage: J0(33)[0].newform_level(none_if_not_known=True) (11, Congruence Subgroup Gamma0(33))
Here there are multiple groups since there are in fact multiple newforms::
sage: (J0(11) * J1(13)).newform_level() (143, [Congruence Subgroup Gamma0(11), Congruence Subgroup Gamma1(13)]) """ return None self.newform_decomposition('a')]))
def zero_subvariety(self): """ Return the zero subvariety of self.
EXAMPLES::
sage: J = J0(37) sage: J.zero_subvariety() Simple abelian subvariety of dimension 0 of J0(37) sage: J.zero_subvariety().level() 37 sage: J.zero_subvariety().newform_level() (1, []) """ is_simple=True, check=False)
############################################################################### # Properties of the ambient product of Jacobians ############################################################################### def _ambient_repr(self): """ OUTPUT: string
EXAMPLES::
sage: (J0(33)*J1(11))._ambient_repr() 'J0(33) x J1(11)' """
def _ambient_latex_repr(self): """ Return Latex representation of the ambient product.
OUTPUT: string
EXAMPLES::
sage: (J0(11) * J0(33))._ambient_latex_repr() 'J_0(11) \\times J_0(33)' """
def _ambient_lattice(self): """ Return free lattice of rank twice the degree of self. This is the lattice corresponding to the ambient product Jacobian.
OUTPUT: lattice
EXAMPLES: We compute the ambient lattice of a product::
sage: (J0(33)*J1(11))._ambient_lattice() Ambient free module of rank 8 over the principal ideal domain Integer Ring
We compute the ambient lattice of an abelian subvariety `J_0(33)`, which is the same as the lattice for the `J_0(33)` itself::
sage: A = J0(33)[0]; A._ambient_lattice() Ambient free module of rank 6 over the principal ideal domain Integer Ring sage: J0(33)._ambient_lattice() Ambient free module of rank 6 over the principal ideal domain Integer Ring """
def _ambient_modular_symbols_spaces(self): """ Return a tuple of the ambient cuspidal modular symbols spaces that make up the Jacobian product that contains self.
OUTPUT: tuple of cuspidal modular symbols spaces
EXAMPLES::
sage: (J0(11) * J0(33))._ambient_modular_symbols_spaces() (Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field, 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: (J0(11) * J0(33)[0])._ambient_modular_symbols_spaces() (Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field, 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) """
def _ambient_modular_symbols_abvars(self): """ Return a tuple of the ambient modular symbols abelian varieties that make up the Jacobian product that contains self.
OUTPUT: tuple of modular symbols abelian varieties
EXAMPLES::
sage: (J0(11) * J0(33))._ambient_modular_symbols_abvars() (Abelian variety J0(11) of dimension 1, Abelian variety J0(33) of dimension 3) """ return self.ambient_variety()._ambient_modular_symbols_abvars()
def _ambient_dimension(self): """ Return the dimension of the ambient Jacobian product.
EXAMPLES::
sage: A = J0(37) * J1(13); A Abelian variety J0(37) x J1(13) of dimension 4 sage: A._ambient_dimension() 4 sage: B = A[0]; B Simple abelian subvariety 13aG1(1,13) of dimension 2 of J0(37) x J1(13) sage: B._ambient_dimension() 4
This example is fast because it implicitly calls _ambient_dimension.
::
sage: J0(902834082394) Abelian variety J0(902834082394) of dimension 113064825881 """
def _ambient_hecke_matrix_on_modular_symbols(self, n): r""" Return block direct sum of the matrix of the Hecke operator `T_n` acting on each of the ambient modular symbols spaces.
INPUT:
- ``n`` - an integer `\geq 1`.
OUTPUT: a matrix
EXAMPLES::
sage: (J0(11) * J1(13))._ambient_hecke_matrix_on_modular_symbols(2) [-2 0 0 0 0 0] [ 0 -2 0 0 0 0] [ 0 0 -2 0 -1 1] [ 0 0 1 -1 0 -1] [ 0 0 1 1 -2 0] [ 0 0 0 1 -1 -1] """ return self.ambient_variety()._ambient_hecke_matrix_on_modular_symbols(n) return matrix(QQ,0)
def rational_torsion_order(self, proof=True): """ Return the order of the rational torsion subgroup of this modular abelian variety.
This function is really an alias for :meth:`~sage.modular.abvar.torsion_subgroup.RationalTorsionSubgroup.order` See the docstring there for a more in-depth reference and more interesting examples.
INPUT:
- ``proof`` -- a boolean (default: True)
OUTPUT:
The order of the rational torsion subgroup of this modular abelian variety.
EXAMPLES::
sage: J0(11).rational_torsion_subgroup().order() 5 sage: J0(11).rational_torsion_order() 5 """
def number_of_rational_points(self): """ Return the number of rational points of this modular abelian variety.
It is not always possible to compute the order of the torsion subgroup. The BSD conjecture is assumed to compute the algebraic rank.
OUTPUT: a positive integer or infinity
EXAMPLES::
sage: J0(23).number_of_rational_points() 11 sage: J0(29).number_of_rational_points() 7 sage: J0(37).number_of_rational_points() +Infinity
sage: J0(12); J0(12).number_of_rational_points() Abelian variety J0(12) of dimension 0 1
sage: J1(17).number_of_rational_points() 584
sage: J1(16).number_of_rational_points() Traceback (most recent call last): ... RuntimeError: Unable to compute order of torsion subgroup (it is in [1, 2, 4, 5, 10, 20]) """ # Check easy dimension zero case
else:
def frobenius_polynomial(self, p, var='x'): """ Computes the frobenius polynomial at `p`.
INPUT:
- ``p`` -- prime number
OUTPUT:
a monic integral polynomial
EXAMPLES::
sage: f = Newform('39b','a') sage: A=AbelianVariety(f) sage: A.frobenius_polynomial(5) x^4 + 2*x^2 + 25
sage: J=J0(23) sage: J.frobenius_polynomial(997) x^4 + 20*x^3 + 1374*x^2 + 19940*x + 994009
sage: J = J0(33) sage: J.frobenius_polynomial(7) x^6 + 9*x^4 - 16*x^3 + 63*x^2 + 343
sage: J = J0(19) sage: J.frobenius_polynomial(3, var='y') y^2 + 2*y + 3
sage: J = J0(3); J Abelian variety J0(3) of dimension 0 sage: J.frobenius_polynomial(11) 1
sage: A = J1(27)[1]; A Simple abelian subvariety 27bG1(1,27) of dimension 12 of J1(27) sage: A.frobenius_polynomial(11) x^24 - 3*x^23 - 15*x^22 + 126*x^21 - 201*x^20 - 1488*x^19 + 7145*x^18 - 1530*x^17 - 61974*x^16 + 202716*x^15 - 19692*x^14 - 1304451*x^13 + 4526883*x^12 - 14348961*x^11 - 2382732*x^10 + 269814996*x^9 - 907361334*x^8 - 246408030*x^7 + 12657803345*x^6 - 28996910448*x^5 - 43086135081*x^4 + 297101409066*x^3 - 389061369015*x^2 - 855935011833*x + 3138428376721
sage: J = J1(33) sage: J.frobenius_polynomial(11) Traceback (most recent call last): ... ValueError: p must not divide the level of self sage: J.frobenius_polynomial(4) Traceback (most recent call last): ... ValueError: p must be prime """ self.newform_decomposition('a')] decomp))
# relativize number fields to compute charpoly of # left multiplication of ap on Kf as a Qe-vector # space.
# Re is just Qe with a different name
# take Qe norm
############################################################################### # Rational and Integral Homology ############################################################################### def _rational_homology_space(self): """ Return the rational homology of this modular abelian variety.
EXAMPLES::
sage: J = J0(11) sage: J._rational_homology_space() Vector space of dimension 2 over Rational Field
The result is cached::
sage: J._rational_homology_space() is J._rational_homology_space() True """
def homology(self, base_ring=ZZ): """ Return the homology of this modular abelian variety.
.. warning::
For efficiency reasons the basis of the integral homology need not be the same as the basis for the rational homology.
EXAMPLES::
sage: J0(389).homology(GF(7)) Homology with coefficients in Finite Field of size 7 of Abelian variety J0(389) of dimension 32 sage: J0(389).homology(QQ) Rational Homology of Abelian variety J0(389) of dimension 32 sage: J0(389).homology(ZZ) Integral Homology of Abelian variety J0(389) of dimension 32 """ else:
def integral_homology(self): """ Return the integral homology of this modular abelian variety.
EXAMPLES::
sage: H = J0(43).integral_homology(); H Integral Homology of Abelian variety J0(43) of dimension 3 sage: H.rank() 6 sage: H = J1(17).integral_homology(); H Integral Homology of Abelian variety J1(17) of dimension 5 sage: H.rank() 10
If you just ask for the rank of the homology, no serious calculations are done, so the following is fast::
sage: H = J0(50000).integral_homology(); H Integral Homology of Abelian variety J0(50000) of dimension 7351 sage: H.rank() 14702
A product::
sage: H = (J0(11) * J1(13)).integral_homology() sage: H.hecke_operator(2) Hecke operator T_2 on Integral Homology of Abelian variety J0(11) x J1(13) of dimension 3 sage: H.hecke_operator(2).matrix() [-2 0 0 0 0 0] [ 0 -2 0 0 0 0] [ 0 0 -2 0 -1 1] [ 0 0 1 -1 0 -1] [ 0 0 1 1 -2 0] [ 0 0 0 1 -1 -1] """
def rational_homology(self): """ Return the rational homology of this modular abelian variety.
EXAMPLES::
sage: H = J0(37).rational_homology(); H Rational Homology of Abelian variety J0(37) of dimension 2 sage: H.rank() 4 sage: H.base_ring() Rational Field sage: H = J1(17).rational_homology(); H Rational Homology of Abelian variety J1(17) of dimension 5 sage: H.rank() 10 sage: H.base_ring() Rational Field """
############################################################################### # L-series ############################################################################### def lseries(self): """ Return the complex `L`-series of this modular abelian variety.
EXAMPLES::
sage: A = J0(37) sage: A.lseries() Complex L-series attached to Abelian variety J0(37) of dimension 2 """
def padic_lseries(self, p): """ Return the `p`-adic `L`-series of this modular abelian variety.
EXAMPLES::
sage: A = J0(37) sage: A.padic_lseries(7) 7-adic L-series attached to Abelian variety J0(37) of dimension 2 """
############################################################################### # Hecke Operators ############################################################################### def hecke_operator(self, n): """ Return the `n^{th}` Hecke operator on the modular abelian variety, if this makes sense [[elaborate]]. Otherwise raise a ValueError.
EXAMPLES: We compute `T_2` on `J_0(37)`.
::
sage: t2 = J0(37).hecke_operator(2); t2 Hecke operator T_2 on Abelian variety J0(37) of dimension 2 sage: t2.charpoly().factor() x * (x + 2) sage: t2.index() 2
Note that there is no matrix associated to Hecke operators on modular abelian varieties. For a matrix, instead consider, e.g., the Hecke operator on integral or rational homology.
::
sage: t2.action_on_homology().matrix() [-1 1 1 -1] [ 1 -1 1 0] [ 0 0 -2 1] [ 0 0 0 0] """
def hecke_polynomial(self, n, var='x'): r""" Return the characteristic polynomial of the `n^{th}` Hecke operator `T_n` acting on self. Raises an ArithmeticError if self is not Hecke equivariant.
INPUT:
- ``n`` - integer `\geq 1`
- ``var`` - string (default: 'x'); valid variable name
EXAMPLES::
sage: J0(33).hecke_polynomial(2) x^3 + 3*x^2 - 4 sage: f = J0(33).hecke_polynomial(2, 'y'); f y^3 + 3*y^2 - 4 sage: f.parent() Univariate Polynomial Ring in y over Rational Field sage: J0(33)[2].hecke_polynomial(3) x + 1 sage: J0(33)[0].hecke_polynomial(5) x - 1 sage: J0(33)[0].hecke_polynomial(11) x - 1 sage: J0(33)[0].hecke_polynomial(3) Traceback (most recent call last): ... ArithmeticError: subspace is not invariant under matrix """ raise ValueError("n must be a positive integer")
def _compute_hecke_polynomial(self, n, var='x'): """ Return the Hecke polynomial of index `n` in terms of the given variable.
INPUT:
- ``n`` - positive integer
- ``var`` - string (default: 'x')
EXAMPLES::
sage: A = J0(33)*J0(11) sage: A._compute_hecke_polynomial(2) x^4 + 5*x^3 + 6*x^2 - 4*x - 8 """
def _integral_hecke_matrix(self, n): """ Return the matrix of the Hecke operator `T_n` acting on the integral homology of this modular abelian variety, if the modular abelian variety is stable under `T_n`. Otherwise, raise an ArithmeticError.
EXAMPLES::
sage: A = J0(23) sage: t = A._integral_hecke_matrix(2); t [ 0 1 -1 0] [ 0 1 -1 1] [-1 2 -2 1] [-1 1 0 -1] sage: t.parent() Full MatrixSpace of 4 by 4 dense matrices over Integer Ring """
def _rational_hecke_matrix(self, n): r""" Return the matrix of the Hecke operator `T_n` acting on the rational homology `H_1(A,\QQ)` of this modular abelian variety, if this action is defined. Otherwise, raise an ArithmeticError.
EXAMPLES::
sage: A = J0(23) sage: t = A._rational_hecke_matrix(2); t [ 0 1 -1 0] [ 0 1 -1 1] [-1 2 -2 1] [-1 1 0 -1] sage: t.parent() Full MatrixSpace of 4 by 4 dense matrices over Rational Field """
############################################################################### # Subgroups ############################################################################### def qbar_torsion_subgroup(self): r""" Return the group of all points of finite order in the algebraic closure of this abelian variety.
EXAMPLES::
sage: T = J0(33).qbar_torsion_subgroup(); T Group of all torsion points in QQbar on Abelian variety J0(33) of dimension 3
The field of definition is the same as the base field of the abelian variety.
::
sage: T.field_of_definition() Rational Field
On the other hand, T is a module over `\ZZ`.
::
sage: T.base_ring() Integer Ring """
def rational_torsion_subgroup(self): """ Return the maximal torsion subgroup of self defined over QQ.
EXAMPLES::
sage: J = J0(33) sage: A = J.new_subvariety() sage: A Abelian subvariety of dimension 1 of J0(33) sage: t = A.rational_torsion_subgroup(); t Torsion subgroup of Abelian subvariety of dimension 1 of J0(33) sage: t.multiple_of_order() 4 sage: t.divisor_of_order() 4 sage: t.order() 4 sage: t.gens() [[(1/2, 0, 0, -1/2, 0, 0)], [(0, 0, 1/2, 0, 1/2, -1/2)]] """
def cuspidal_subgroup(self): """ Return the cuspidal subgroup of this modular abelian variety. This is the subgroup generated by rational cusps.
EXAMPLES::
sage: J = J0(54) sage: C = J.cuspidal_subgroup() 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.invariants() [3, 3, 3, 3, 3, 9] sage: J1(13).cuspidal_subgroup() Finite subgroup with invariants [19, 19] over QQ of Abelian variety J1(13) of dimension 2 sage: A = J0(33)[0] sage: A.cuspidal_subgroup() Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) """ raise ValueError("self must be a subvariety of the ambient variety") else:
def _ambient_cuspidal_subgroup(self, rational_only=False, rational_subgroup=False): """ EXAMPLES::
sage: (J1(13)*J0(11))._ambient_cuspidal_subgroup() Finite subgroup with invariants [19, 95] over QQ of Abelian variety J1(13) x J0(11) of dimension 3 sage: (J0(33))._ambient_cuspidal_subgroup() Finite subgroup with invariants [10, 10] over QQ of Abelian variety J0(33) of dimension 3 sage: (J0(33)*J0(33))._ambient_cuspidal_subgroup() Finite subgroup with invariants [10, 10, 10, 10] over QQ of Abelian variety J0(33) x J0(33) of dimension 6 """ else:
def shimura_subgroup(self): r""" Return the Shimura subgroup of this modular abelian variety. This is the kernel of $J_0(N) \rightarrow J_1(N)$ under the natural map. Here we compute the Shimura subgroup as the kernel of $J_0(N) \rightarrow J_0(Np)$ where the map is the difference between the two degeneracy maps.
EXAMPLES::
sage: J=J0(11) sage: J.shimura_subgroup() Finite subgroup with invariants [5] over QQ of Abelian variety J0(11) of dimension 1
sage: J=J0(17) sage: G=J.cuspidal_subgroup(); G Finite subgroup with invariants [4] over QQ of Abelian variety J0(17) of dimension 1 sage: S=J.shimura_subgroup(); S Finite subgroup with invariants [4] over QQ of Abelian variety J0(17) of dimension 1 sage: G.intersection(S) Finite subgroup with invariants [2] over QQ of Abelian variety J0(17) of dimension 1
sage: J=J0(33) sage: A=J.decomposition()[0] sage: A.shimura_subgroup() Finite subgroup with invariants [5] over QQ of Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) sage: J.shimura_subgroup() Finite subgroup with invariants [10] over QQ of Abelian variety J0(33) of dimension 3 """
def rational_cusp_subgroup(self): r""" Return the subgroup of this modular abelian variety generated by rational cusps.
This is a subgroup of the group of rational points in the cuspidal subgroup.
.. warning::
This is only currently implemented for `\Gamma_0(N)`.
EXAMPLES::
sage: J = J0(54) sage: CQ = J.rational_cusp_subgroup(); CQ Finite subgroup with invariants [3, 3, 9] over QQ of Abelian variety J0(54) of dimension 4 sage: CQ.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: factor(CQ.order()) 3^4 sage: CQ.invariants() [3, 3, 9]
In this example the rational cuspidal subgroup and the cuspidal subgroup differ by a lot.
::
sage: J = J0(49) sage: J.cuspidal_subgroup() Finite subgroup with invariants [2, 14] over QQ of Abelian variety J0(49) of dimension 1 sage: J.rational_cusp_subgroup() Finite subgroup with invariants [2] over QQ of Abelian variety J0(49) of dimension 1
Note that computation of the rational cusp subgroup isn't implemented for `\Gamma_1`.
::
sage: J = J1(13) sage: J.cuspidal_subgroup() Finite subgroup with invariants [19, 19] over QQ of Abelian variety J1(13) of dimension 2 sage: J.rational_cusp_subgroup() Traceback (most recent call last): ... NotImplementedError: computation of rational cusps only implemented in Gamma0 case. """ raise ValueError("self must be a subvariety of the ambient variety") else:
def rational_cuspidal_subgroup(self): r""" Return the rational subgroup of the cuspidal subgroup of this modular abelian variety.
This is a subgroup of the group of rational points in the cuspidal subgroup.
.. warning::
This is only currently implemented for `\Gamma_0(N)`.
EXAMPLES::
sage: J = J0(54) sage: CQ = J.rational_cuspidal_subgroup(); CQ Finite subgroup with invariants [3, 3, 9] over QQ of Abelian variety J0(54) of dimension 4 sage: CQ.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: factor(CQ.order()) 3^4 sage: CQ.invariants() [3, 3, 9]
In this example the rational cuspidal subgroup and the cuspidal subgroup differ by a lot.
::
sage: J = J0(49) sage: J.cuspidal_subgroup() Finite subgroup with invariants [2, 14] over QQ of Abelian variety J0(49) of dimension 1 sage: J.rational_cuspidal_subgroup() Finite subgroup with invariants [2] over QQ of Abelian variety J0(49) of dimension 1
Note that computation of the rational cusp subgroup isn't implemented for `\Gamma_1`.
::
sage: J = J1(13) sage: J.cuspidal_subgroup() Finite subgroup with invariants [19, 19] over QQ of Abelian variety J1(13) of dimension 2 sage: J.rational_cuspidal_subgroup() Traceback (most recent call last): ... NotImplementedError: only implemented when group is Gamma0 """ raise ValueError("self must be a subvariety of the ambient variety") else:
def zero_subgroup(self): """ Return the zero subgroup of this modular abelian variety, as a finite group.
EXAMPLES::
sage: A =J0(54); G = A.zero_subgroup(); G Finite subgroup with invariants [] over QQ of Abelian variety J0(54) of dimension 4 sage: G.is_subgroup(A) True """
def finite_subgroup(self, X, field_of_definition=None, check=True): """ Return a finite subgroup of this modular abelian variety.
INPUT:
- ``X`` - list of elements of other finite subgroups of this modular abelian variety or elements that coerce into the rational homology (viewed as a rational vector space); also X could be a finite subgroup itself that is contained in this abelian variety.
- ``field_of_definition`` - (default: None) field over which this group is defined. If None try to figure out the best base field.
OUTPUT: a finite subgroup of a modular abelian variety
EXAMPLES::
sage: J = J0(11) sage: J.finite_subgroup([[1/5,0], [0,1/3]]) Finite subgroup with invariants [15] over QQbar of Abelian variety J0(11) of dimension 1
::
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: J.finite_subgroup([[0,0,0,0,0,1/6]]) Finite subgroup with invariants [6] over QQbar of Abelian variety J0(33) of dimension 3 sage: J.finite_subgroup(C) Finite subgroup with invariants [5] over QQ of Abelian variety J0(33) of dimension 3 """ raise ValueError("ambient product Jacobians must be equal") X = X.lattice() else: else: raise ValueError("X must be a subgroup of self.")
else:
def torsion_subgroup(self, n): """ If n is an integer, return the subgroup of points of order n. Return the `n`-torsion subgroup of elements of order dividing `n` of this modular abelian variety `A`, i.e., the group `A[n]`.
EXAMPLES::
sage: J1(13).torsion_subgroup(19) Finite subgroup with invariants [19, 19, 19, 19] over QQ of Abelian variety J1(13) of dimension 2
::
sage: A = J0(23) sage: G = A.torsion_subgroup(5); G Finite subgroup with invariants [5, 5, 5, 5] over QQ of Abelian variety J0(23) of dimension 2 sage: G.order() 625 sage: G.gens() [[(1/5, 0, 0, 0)], [(0, 1/5, 0, 0)], [(0, 0, 1/5, 0)], [(0, 0, 0, 1/5)]] sage: A = J0(23) sage: A.torsion_subgroup(2).order() 16 """
############################################################################### # Decomposition ############################################################################### def degen_t(self, none_if_not_known=False): """ If this abelian variety is obtained via decomposition then it gets labeled with the newform label along with some information about degeneracy maps. In particular, the label ends in a pair `(t,N)`, where `N` is the ambient level and `t` is an integer that divides the quotient of `N` by the newform level. This function returns the tuple `(t,N)`, or raises a ValueError if self isn't simple.
.. note::
It need not be the case that self is literally equal to the image of the newform abelian variety under the `t^{th}` degeneracy map. See the documentation for the label method for more details.
INPUT:
- ``none_if_not_known`` - (default: False) - if True, return None instead of attempting to compute the degen map's `t`, if it isn't known. This None result is not cached.
OUTPUT: a pair (integer, integer)
EXAMPLES::
sage: D = J0(33).decomposition(); D [ Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33), Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33), Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33) ] sage: D[0].degen_t() (1, 33) sage: D[1].degen_t() (3, 33) sage: D[2].degen_t() (1, 33) sage: J0(33).degen_t() Traceback (most recent call last): ... ValueError: self must be simple """
def isogeny_number(self, none_if_not_known=False): """ Return the number (starting at 0) of the isogeny class of new simple abelian varieties that self is in. If self is not simple, raises a ValueError exception.
INPUT:
- ``none_if_not_known`` - bool (default: False); if True then this function may return None instead of True of False if we don't already know the isogeny number of self.
EXAMPLES: We test the none_if_not_known flag first::
sage: J0(33).isogeny_number(none_if_not_known=True) is None True
Of course, `J_0(33)` is not simple, so this function raises a ValueError::
sage: J0(33).isogeny_number() Traceback (most recent call last): ... ValueError: self must be simple
Each simple factor has isogeny number 1, since that's the number at which the factor is new.
::
sage: J0(33)[1].isogeny_number() 0 sage: J0(33)[2].isogeny_number() 0
Next consider `J_0(37)` where there are two distinct newform factors::
sage: J0(37)[1].isogeny_number() 1 """ else:
def is_simple(self, none_if_not_known=False): """ Return whether or not this modular abelian variety is simple, i.e., has no proper nonzero abelian subvarieties.
INPUT:
- ``none_if_not_known`` - bool (default: False); if True then this function may return None instead of True of False if we don't already know whether or not self is simple.
EXAMPLES::
sage: J0(5).is_simple(none_if_not_known=True) is None # this may fail if J0(5) comes up elsewhere... True sage: J0(33).is_simple() False sage: J0(33).is_simple(none_if_not_known=True) False sage: J0(33)[1].is_simple() True sage: J1(17).is_simple() False """
def decomposition(self, simple=True, bound=None): """ Return a sequence of abelian subvarieties of self that are all simple, have finite intersection and sum to self.
INPUT: simple- bool (default: True) if True, all factors are simple. If False, each factor returned is isogenous to a power of a simple and the simples in each factor are distinct.
- ``bound`` - int (default: None) if given, only use Hecke operators up to this bound when decomposing. This can give wrong answers, so use with caution!
EXAMPLES::
sage: m = ModularSymbols(11).cuspidal_submodule() sage: d1 = m.degeneracy_map(33,1).matrix(); d3=m.degeneracy_map(33,3).matrix() sage: w = ModularSymbols(33).submodule((d1 + d3).image(), check=False) sage: A = w.abelian_variety(); A Abelian subvariety of dimension 1 of J0(33) sage: D = A.decomposition(); D [ Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33) ] sage: D[0] == A True sage: B = A + J0(33)[0]; B Abelian subvariety of dimension 2 of J0(33) sage: dd = B.decomposition(simple=False); dd [ Abelian subvariety of dimension 2 of J0(33) ] sage: dd[0] == B True sage: dd = B.decomposition(); dd [ Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33), Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33) ] sage: sum(dd) == B True
We decompose a product of two Jacobians::
sage: (J0(33) * J0(11)).decomposition() [ Simple abelian subvariety 11a(1,11) of dimension 1 of J0(33) x J0(11), Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33) x J0(11), Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33) x J0(11), Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33) x J0(11) ] """
# Decompose each piece, then lift D = [] else: # Decompose each ambient modular symbols factor. #X = [ModularAbelianVariety_modsym(ModularSymbols(G,sign=0).cuspidal_submodule()) for G in self.groups()] # Now lift each factor of the decomposition to self. else: isogeny_number=B.isogeny_number(none_if_not_known=True), number=B.degen_t(none_if_not_known=True))) # In this case decompose the ambient space into powers of # simple abelian varieties (i.e. with # \code{simple=False)}, and then intersect the lattice # corresponding to self with each of these factors.
else: # See the documentation for self._classify_ambient_factors # in order to understand what we're doing here. # Now self is isogenous to the sum of the factors in Z. # We use this isogeny to obtain a product decomposition of # self. # The easy case -- it is already such a decomposition else: # The hard case -- now we have to pull back the # factorization
# Suppose $B$ is an abelian variety and there is a # finite degree map $B\to J$, where $J$ is an ambient # Jacobian. Suppose further that we find abelian # subvarieties $E$ and $F$ of $J$ such that $E + F = # J$, $E$ and $F$ have finite intersection, the # composition $B \to J \to J/E$ is an isogeny, and we # know an explicit decomposition of $F$. Then we can # compute a decomposition of $B$ as follows. Let # $L_E$ and $L_F$ be the lattices corresponding to $E$ # and $F$ inside of $L_J$. Compute a matrix $\Phi$ # representing the composition $L_B \to L_J \to L_F # \otimes \QQ$, where the map $L_J$ to $L_F\otimes # \QQ$ is projection onto the second factor in the # decomposition of $L_J$ as $L_E + L_F$ (up to finite # index). Finally, for each factor $A_i$ of $F$ with # lattice $L_{A_i}$, compute the saturation $S_i$ of # $\Phi^{-1}(L_{A_i})$. Then the $S_i$ define a # decomposition of $B$. # Now we compute explicitly the ZZ-linear map (over # QQ) from L_B that is "projection onto L_F". This # means write each element of a basis for L_B in terms # of decomp_matrix, then take the bottom coordinates. # Now row of X gives each element of L_B as a linear # combination of the rows of decomp_matrix. We # project onto L_F by taking the right-most part of # this matrix. # Now proj is the matrix of projection that goes from # L_B to L_F, wrt the basis of those spaces.
# Now section maps L_F to L_B (tensor QQ). Now we # just take each factor of F, which corresponds to a # submodule of L_F, and map it over to L_B tensor QQ # and saturate. isogeny_number=A.isogeny_number(), number=A.degen_t())
################
else:
def _classify_ambient_factors(self, simple=True, bound=None): r""" This function implements the following algorithm, which produces data useful in finding a decomposition or complement of self.
#. Suppose `A_1 + \cdots + A_n` is a simple decomposition of the ambient space.
#. For each `i`, let `B_i = A_1 + \cdots + A_i`.
#. For each `i`, compute the intersection `C_i` of `B_i` and self.
#. For each `i`, if the dimension of `C_i` is bigger than `C_{i-1}` put `i` in the "in" list; otherwise put `i` in the "out" list.
Then one can show that self is isogenous to the sum of the `A_i` with `i` in the "in" list. Moreover, the sum of the `A_j` with `i` in the "out" list is a complement of self in the ambient space.
INPUT:
- ``simple`` - bool (default: True)
- ``bound`` - integer (default: None); if given, passed onto decomposition function
OUTPUT: IN list OUT list simple (or power of simple) factors
EXAMPLES::
sage: d1 = J0(11).degeneracy_map(33, 1); d1 Degeneracy map from Abelian variety J0(11) of dimension 1 to Abelian variety J0(33) of dimension 3 defined by [1] sage: d2 = J0(11).degeneracy_map(33, 3); d2 Degeneracy map from Abelian variety J0(11) of dimension 1 to Abelian variety J0(33) of dimension 3 defined by [3] sage: A = (d1 + d2).image(); A Abelian subvariety of dimension 1 of J0(33) sage: A._classify_ambient_factors() ([1], [0, 2], [ Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33), Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33), Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33) ]) """ # Decompose an arbitrary abelian variety else:
def _isogeny_to_product_of_simples(self): r""" Given an abelian variety `A`, return an isogeny `\phi: A \rightarrow B_1 \times \cdots \times B_n`, where each `B_i` is simple. Note that this isogeny is not unique.
EXAMPLES::
sage: J = J0(37) ; J.decomposition() [ Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37), Simple abelian subvariety 37b(1,37) of dimension 1 of J0(37) ] sage: phi = J._isogeny_to_product_of_simples() ; phi Abelian variety morphism: From: Abelian variety J0(37) of dimension 2 To: Abelian subvariety of dimension 2 of J0(37) x J0(37) sage: J[0].intersection(J[1]) == phi.kernel() True
::
sage: J = J0(22) * J0(37) sage: J._isogeny_to_product_of_simples() Abelian variety morphism: From: Abelian variety J0(22) x J0(37) of dimension 4 To: Abelian subvariety of dimension 4 of J0(11) x J0(11) x J0(37) x J0(37) """
def _isogeny_to_product_of_powers(self): r""" Given an abelian variety `A`, return an isogeny `\phi: A \rightarrow B_1 \times \cdots \times B_n`, where each `B_i` is a power of a simple abelian variety. These factors will be exactly those returned by self.decomposition(simple=False).Note that this isogeny is not unique.
EXAMPLES::
sage: J = J0(33) ; D = J.decomposition(simple=False) ; len(D) 2 sage: phi = J._isogeny_to_product_of_powers() ; phi Abelian variety morphism: From: Abelian variety J0(33) of dimension 3 To: Abelian subvariety of dimension 3 of J0(33) x J0(33)
::
sage: J = J0(22) * J0(37) sage: J._isogeny_to_product_of_powers() Abelian variety morphism: From: Abelian variety J0(22) x J0(37) of dimension 4 To: Abelian subvariety of dimension 4 of J0(22) x J0(37) x J0(22) x J0(37) x J0(22) x J0(37) """
def complement(self, A=None): """ Return a complement of this abelian variety.
INPUT:
- ``A`` - (default: None); if given, A must be an abelian variety that contains self, in which case the complement of self is taken inside A. Otherwise the complement is taken in the ambient product Jacobian.
OUTPUT: abelian variety
EXAMPLES::
sage: a,b,c = J0(33) sage: (a+b).complement() Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33) sage: (a+b).complement() == c True sage: a.complement(a+b) Abelian subvariety of dimension 1 of J0(33) """ if A is None: C = self.ambient_variety() else: C = A raise ValueError("self must be a subvariety of A") else: else:
def dual(self): r""" Return the dual of this abelian variety.
OUTPUT:
- dual abelian variety - morphism from self to dual - covering morphism from J to dual
.. warning::
This is currently only implemented when self is an abelian subvariety of the ambient Jacobian product, and the complement of self in the ambient product Jacobian share no common factors. A more general implementation will require implementing computation of the intersection pairing on integral homology and the resulting Weil pairing on torsion.
EXAMPLES: We compute the dual of the elliptic curve newform abelian variety of level `33`, and find the kernel of the modular map, which has structure `(\ZZ/3)^2`.
::
sage: A,B,C = J0(33) sage: C Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33) sage: Cd, f, pi = C.dual() sage: f.matrix() [3 0] [0 3] sage: f.kernel()[0] Finite subgroup with invariants [3, 3] over QQ of Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33)
By a theorem the modular degree must thus be `3`::
sage: E = EllipticCurve('33a') sage: E.modular_degree() 3
Next we compute the dual of a `2`-dimensional new simple abelian subvariety of `J_0(43)`.
::
sage: A = AbelianVariety('43b'); A Newform abelian subvariety 43b of dimension 2 of J0(43) sage: Ad, f, pi = A.dual()
The kernel shows that the modular degree is `2`::
sage: f.kernel()[0] Finite subgroup with invariants [2, 2] over QQ of Newform abelian subvariety 43b of dimension 2 of J0(43)
Unfortunately, the dual is not implemented in general::
sage: A = J0(22)[0]; A Simple abelian subvariety 11a(1,22) of dimension 1 of J0(22) sage: A.dual() Traceback (most recent call last): ... NotImplementedError: dual not implemented unless complement shares no simple factors with self. """ raise NotImplementedError("dual not implemented unless abelian variety is a subvariety of the ambient Jacobian product")
def _factors_with_same_label(self, other): """ Given two modular abelian varieties self and other, this function returns a list of simple abelian subvarieties appearing in the decomposition of self that have the same newform labels. Each simple factor with a given newform label appears at most one.
INPUT:
- ``other`` - abelian variety
OUTPUT: list of simple abelian varieties
EXAMPLES::
sage: D = J0(33).decomposition(); D [ Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33), Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33), Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33) ] sage: D[0]._factors_with_same_label(D[1]) [Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)] sage: D[0]._factors_with_same_label(D[2]) [] sage: (D[0]+D[1])._factors_with_same_label(D[1] + D[2]) [Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)]
This illustrates that the multiplicities in the returned list are 1::
sage: (D[0]+D[1])._factors_with_same_label(J0(33)) [Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)]
This illustrates that the ambient product Jacobians do not have to be the same::
sage: (D[0]+D[1])._factors_with_same_label(J0(22)) [Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33)]
This illustrates that the actual factor labels are relevant, not just the isogeny class.
::
sage: (D[0]+D[1])._factors_with_same_label(J1(11)) [] sage: J1(11)[0].newform_label() '11aG1' """ raise TypeError("other must be an abelian variety")
def _complement_shares_no_factors_with_same_label(self): """ Return True if no simple factor of self has the same newform_label as any factor in a Poincaré complement of self in the ambient product Jacobian.
EXAMPLES: `J_0(37)` is made up of two non-isogenous elliptic curves::
sage: J0(37)[0]._complement_shares_no_factors_with_same_label() True
`J_0(33)` decomposes as a product of two isogenous elliptic curves with a third nonisogenous curve::
sage: D = J0(33).decomposition(); D [ Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33), Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33), Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33) ] sage: D[0]._complement_shares_no_factors_with_same_label() False sage: (D[0]+D[1])._complement_shares_no_factors_with_same_label() True sage: D[2]._complement_shares_no_factors_with_same_label() True
This example illustrates the relevance of the ambient product Jacobian.
::
sage: D = (J0(11) * J0(11)).decomposition(); D [ Simple abelian subvariety 11a(1,11) of dimension 1 of J0(11) x J0(11), Simple abelian subvariety 11a(1,11) of dimension 1 of J0(11) x J0(11) ] sage: D[0]._complement_shares_no_factors_with_same_label() False
This example illustrates that it is the newform label, not the isogeny, class that matters::
sage: D = (J0(11)*J1(11)).decomposition(); D [ Simple abelian subvariety 11aG1(1,11) of dimension 1 of J0(11) x J1(11), Simple abelian subvariety 11a(1,11) of dimension 1 of J0(11) x J1(11) ] sage: D[0]._complement_shares_no_factors_with_same_label() True sage: D[0].newform_label() '11aG1' sage: D[1].newform_label() '11a' """
def __getitem__(self, i): """ Return the `i^{th}` decomposition factor of self or return the slice `i` of decompositions of self.
EXAMPLES::
sage: J = J0(389) sage: J.decomposition() [ Simple abelian subvariety 389a(1,389) of dimension 1 of J0(389), Simple abelian subvariety 389b(1,389) of dimension 2 of J0(389), Simple abelian subvariety 389c(1,389) of dimension 3 of J0(389), Simple abelian subvariety 389d(1,389) of dimension 6 of J0(389), Simple abelian subvariety 389e(1,389) of dimension 20 of J0(389) ] sage: J[2] Simple abelian subvariety 389c(1,389) of dimension 3 of J0(389) sage: J[-1] Simple abelian subvariety 389e(1,389) of dimension 20 of J0(389) sage: J = J0(125); J.decomposition() [ Simple abelian subvariety 125a(1,125) of dimension 2 of J0(125), Simple abelian subvariety 125b(1,125) of dimension 2 of J0(125), Simple abelian subvariety 125c(1,125) of dimension 4 of J0(125) ] sage: J[:2] [ Simple abelian subvariety 125a(1,125) of dimension 2 of J0(125), Simple abelian subvariety 125b(1,125) of dimension 2 of J0(125) ] """
class ModularAbelianVariety(ModularAbelianVariety_abstract): def __init__(self, groups, lattice=None, base_field=QQ, is_simple=None, newform_level=None, isogeny_number=None, number=None, check=True): r""" Create a modular abelian variety with given level and base field.
INPUT:
- ``groups`` - a tuple of congruence subgroups
- ``lattice`` - (default: `\ZZ^n`) a full lattice in `\ZZ^n`, where `n` is the sum of the dimensions of the spaces of cuspidal modular symbols corresponding to each `\Gamma \in` groups
- ``base_field`` - a field (default: `\QQ`)
EXAMPLES::
sage: J0(23) Abelian variety J0(23) of dimension 2 """ isogeny_number=isogeny_number, number=number, check=check) raise TypeError("lattice must be a free module") raise TypeError("lattice must be over ZZ") raise ValueError("lattice must have degree 2*n (=%s)"%(2*n)) raise ValueError("lattice must be full")
def lattice(self): """ Return the lattice that defines this abelian variety.
OUTPUT:
- ``lattice`` - a lattice embedded in the rational homology of the ambient product Jacobian
EXAMPLES::
sage: A = (J0(11) * J0(37))[1]; A Simple abelian subvariety 37a(1,37) of dimension 1 of J0(11) x J0(37) sage: type(A) <class 'sage.modular.abvar.abvar.ModularAbelianVariety_with_category'> sage: A.lattice() Free module of degree 6 and rank 2 over Integer Ring Echelon basis matrix: [ 0 0 1 -1 1 0] [ 0 0 0 0 2 -1] """
class ModularAbelianVariety_modsym_abstract(ModularAbelianVariety_abstract): # Anything that derives from this class must define the # modular_symbols method, which returns a cuspidal modular symbols # space over QQ. It can have any sign. def _modular_symbols(self): """ Return the space of modular symbols corresponding to this modular symbols abelian variety.
EXAMPLES: This function is in the abstract base class, so it raises a NotImplementedError::
sage: M = ModularSymbols(37).cuspidal_submodule() sage: A = M.abelian_variety(); A Abelian variety J0(37) of dimension 2 sage: sage.modular.abvar.abvar.ModularAbelianVariety_modsym_abstract._modular_symbols(A) Traceback (most recent call last): ... NotImplementedError: bug -- must define this
Of course this function isn't called in practice, so this works::
sage: A._modular_symbols() Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field """
def __add__(self, other): """ Add two modular abelian variety factors.
EXAMPLES::
sage: A = J0(42); D = A.decomposition(); D [ Simple abelian subvariety 14a(1,42) of dimension 1 of J0(42), Simple abelian subvariety 14a(3,42) of dimension 1 of J0(42), Simple abelian subvariety 21a(1,42) of dimension 1 of J0(42), Simple abelian subvariety 21a(2,42) of dimension 1 of J0(42), Simple abelian subvariety 42a(1,42) of dimension 1 of J0(42) ] sage: D[0] + D[1] Abelian subvariety of dimension 2 of J0(42) sage: D[1].is_subvariety(D[0] + D[1]) True sage: D[0] + D[1] + D[2] Abelian subvariety of dimension 3 of J0(42) sage: D[0] + D[0] Abelian subvariety of dimension 1 of J0(42) sage: D[0] + D[0] == D[0] True sage: sum(D, D[0]) == A True """ raise TypeError("sum not defined")
def groups(self): """ Return the tuple of groups associated to the modular symbols abelian variety. This is always a 1-tuple.
OUTPUT: tuple
EXAMPLES::
sage: A = ModularSymbols(33).cuspidal_submodule().abelian_variety(); A Abelian variety J0(33) of dimension 3 sage: A.groups() (Congruence Subgroup Gamma0(33),) sage: type(A) <class 'sage.modular.abvar.abvar.ModularAbelianVariety_modsym_with_category'> """
def lattice(self): r""" Return the lattice defining this modular abelian variety.
OUTPUT:
A free `\ZZ`-module embedded in an ambient `\QQ`-vector space.
EXAMPLES::
sage: A = ModularSymbols(33).cuspidal_submodule()[0].abelian_variety(); A Abelian subvariety of dimension 1 of J0(33) sage: A.lattice() Free module of degree 6 and rank 2 over Integer Ring User basis matrix: [ 1 0 0 -1 0 0] [ 0 0 1 0 1 -1] sage: type(A) <class 'sage.modular.abvar.abvar.ModularAbelianVariety_modsym_with_category'> """ else:
def _set_lattice(self, lattice): """ Set the lattice of this modular symbols abelian variety.
.. warning::
This is only for internal use. Do not use this unless you really really know what you're doing. That's why there is an underscore in this method name.
INPUT:
- ``lattice`` - a lattice
EXAMPLES: We do something evil - there's no type checking since this function is for internal use only::
sage: A = ModularSymbols(33).cuspidal_submodule().abelian_variety() sage: A._set_lattice(5) sage: A.lattice() 5 """
def modular_symbols(self, sign=0): """ Return space of modular symbols (with given sign) associated to this modular abelian variety, if it can be found by cutting down using Hecke operators. Otherwise raise a RuntimeError exception.
EXAMPLES::
sage: A = J0(37) sage: A.modular_symbols() Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field sage: A.modular_symbols(1) Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(37) of weight 2 with sign 1 over Rational Field
More examples::
sage: J0(11).modular_symbols() Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field sage: J0(11).modular_symbols(sign=1) Modular Symbols subspace of dimension 1 of Modular Symbols space of dimension 2 for Gamma_0(11) of weight 2 with sign 1 over Rational Field sage: J0(11).modular_symbols(sign=0) Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field sage: J0(11).modular_symbols(sign=-1) Modular Symbols space of dimension 1 for Gamma_0(11) of weight 2 with sign -1 over Rational Field
Even more examples::
sage: A = J0(33)[1]; A Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33) sage: A.modular_symbols() 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
It is not always possible to determine the sign subspaces::
sage: A.modular_symbols(1) Traceback (most recent call last): ... RuntimeError: unable to determine sign (=1) space of modular symbols
::
sage: A.modular_symbols(-1) Traceback (most recent call last): ... RuntimeError: unable to determine sign (=-1) space of modular symbols """
def _compute_hecke_polynomial(self, n, var='x'): """ Return the characteristic polynomial of the `n^{th}` Hecke operator on self.
.. note::
If self has dimension d, then this is a polynomial of degree d. It is not of degree 2\*d, so it is the square root of the characteristic polynomial of the Hecke operator on integral or rational homology (which has degree 2\*d).
EXAMPLES::
sage: J0(11).hecke_polynomial(2) x + 2 sage: J0(23)._compute_hecke_polynomial(2) x^2 + x - 1 sage: J1(13).hecke_polynomial(2) x^2 + 3*x + 3 sage: factor(J0(43).hecke_polynomial(2)) (x + 2) * (x^2 - 2)
The Hecke polynomial is the square root of the characteristic polynomial::
sage: factor(J0(43).hecke_operator(2).charpoly()) (x + 2) * (x^2 - 2) """
def _integral_hecke_matrix(self, n, sign=0): """ Return the action of the Hecke operator `T_n` on the integral homology of self.
INPUT:
- ``n`` - a positive integer
- ``sign`` - 0, +1, or -1; if 1 or -1 act on the +1 or -1 quotient of the integral homology.
EXAMPLES::
sage: J1(13)._integral_hecke_matrix(2) # slightly random choice of basis [-2 0 -1 1] [ 1 -1 0 -1] [ 1 1 -2 0] [ 0 1 -1 -1] sage: J1(13)._integral_hecke_matrix(2,sign=1) # slightly random choice of basis [-1 1] [-1 -2] sage: J1(13)._integral_hecke_matrix(2,sign=-1) # slightly random choice of basis [-2 -1] [ 1 -1] """
def _rational_hecke_matrix(self, n, sign=0): """ Return the action of the Hecke operator `T_n` on the rational homology of self.
INPUT:
- ``n`` - a positive integer
- ``sign`` - 0, +1, or -1; if 1 or -1 act on the +1 or -1 quotient of the rational homology.
EXAMPLES::
sage: J1(13)._rational_hecke_matrix(2) # slightly random choice of basis [-2 0 -1 1] [ 1 -1 0 -1] [ 1 1 -2 0] [ 0 1 -1 -1] sage: J0(43)._rational_hecke_matrix(2,sign=1) # slightly random choice of basis [-2 0 1] [-1 -2 2] [-2 0 2] """
def group(self): """ Return the congruence subgroup associated that this modular abelian variety is associated to.
EXAMPLES::
sage: J0(13).group() Congruence Subgroup Gamma0(13) sage: J1(997).group() Congruence Subgroup Gamma1(997) sage: JH(37,[3]).group() Congruence Subgroup Gamma_H(37) with H generated by [3] sage: J0(37)[1].groups() (Congruence Subgroup Gamma0(37),) """
def is_subvariety(self, other): """ Return True if self is a subvariety of other.
EXAMPLES::
sage: J = J0(37); J Abelian variety J0(37) of dimension 2 sage: A = J[0]; A Simple abelian subvariety 37a(1,37) of dimension 1 of J0(37) sage: A.is_subvariety(J) True sage: A.is_subvariety(J0(11)) False
There may be a way to map `A` into `J_0(74)`, but `A` is not equipped with any special structure of an embedding.
::
sage: A.is_subvariety(J0(74)) False
Some ambient examples::
sage: J = J0(37) sage: J.is_subvariety(J) True sage: J.is_subvariety(25) False
More examples::
sage: A = J0(42); D = A.decomposition(); D [ Simple abelian subvariety 14a(1,42) of dimension 1 of J0(42), Simple abelian subvariety 14a(3,42) of dimension 1 of J0(42), Simple abelian subvariety 21a(1,42) of dimension 1 of J0(42), Simple abelian subvariety 21a(2,42) of dimension 1 of J0(42), Simple abelian subvariety 42a(1,42) of dimension 1 of J0(42) ] sage: D[0].is_subvariety(A) True sage: D[1].is_subvariety(D[0] + D[1]) True sage: D[2].is_subvariety(D[0] + D[1]) False """
def is_ambient(self): """ Return True if this abelian variety attached to a modular symbols space is attached to the cuspidal subspace of the ambient modular symbols space.
OUTPUT: bool
EXAMPLES::
sage: A = ModularSymbols(43).cuspidal_subspace().abelian_variety(); A Abelian variety J0(43) of dimension 3 sage: A.is_ambient() True sage: type(A) <class 'sage.modular.abvar.abvar.ModularAbelianVariety_modsym_with_category'> sage: A = ModularSymbols(43).cuspidal_subspace()[1].abelian_variety(); A Abelian subvariety of dimension 2 of J0(43) sage: A.is_ambient() False """
def dimension(self): """ Return the dimension of this modular abelian variety.
EXAMPLES::
sage: J0(37)[0].dimension() 1 sage: J0(43)[1].dimension() 2 sage: J1(17)[1].dimension() 4 """ else: d = M.dimension()
def new_subvariety(self, p=None): """ Return the new or `p`-new subvariety of self.
INPUT:
- ``self`` - a modular abelian variety
- ``p`` - prime number or None (default); if p is a prime, return the p-new subvariety. Otherwise return the full new subvariety.
EXAMPLES::
sage: J0(34).new_subvariety() Abelian subvariety of dimension 1 of J0(34) sage: J0(100).new_subvariety() Abelian subvariety of dimension 1 of J0(100) sage: J1(13).new_subvariety() Abelian variety J1(13) of dimension 2 """ except KeyError: pass
def old_subvariety(self, p=None): """ Return the old or `p`-old abelian variety of self.
INPUT:
- ``self`` - a modular abelian variety
- ``p`` - prime number or None (default); if p is a prime, return the p-old subvariety. Otherwise return the full old subvariety.
EXAMPLES::
sage: J0(33).old_subvariety() Abelian subvariety of dimension 2 of J0(33) sage: J0(100).old_subvariety() Abelian subvariety of dimension 6 of J0(100) sage: J1(13).old_subvariety() Abelian subvariety of dimension 0 of J1(13) """ except KeyError: pass
def decomposition(self, simple=True, bound=None): r""" Decompose this modular abelian variety as a product of abelian subvarieties, up to isogeny.
INPUT: simple- bool (default: True) if True, all factors are simple. If False, each factor returned is isogenous to a power of a simple and the simples in each factor are distinct.
- ``bound`` - int (default: None) if given, only use Hecke operators up to this bound when decomposing. This can give wrong answers, so use with caution!
EXAMPLES::
sage: J = J0(33) sage: J.decomposition() [ Simple abelian subvariety 11a(1,33) of dimension 1 of J0(33), Simple abelian subvariety 11a(3,33) of dimension 1 of J0(33), Simple abelian subvariety 33a(1,33) of dimension 1 of J0(33) ] sage: J1(17).decomposition() [ Simple abelian subvariety 17aG1(1,17) of dimension 1 of J1(17), Simple abelian subvariety 17bG1(1,17) of dimension 4 of J1(17) ] """ else: else: X = A.decomposition(bound = bound) is_simple=True, newform_level=(N, G), isogeny_number=isogeny_number, number=(t,M))) else: D = ModularAbelianVariety_abstract.decomposition(self, simple=simple, bound=bound)
class ModularAbelianVariety_modsym(ModularAbelianVariety_modsym_abstract):
def __init__(self, modsym, lattice=None, newform_level=None, is_simple=None, isogeny_number=None, number=None, check=True): """ Modular abelian variety that corresponds to a Hecke stable space of cuspidal modular symbols.
EXAMPLES: We create a modular abelian variety attached to a space of modular symbols.
::
sage: M = ModularSymbols(23).cuspidal_submodule() sage: A = M.abelian_variety(); A Abelian variety J0(23) of dimension 2 """ raise TypeError("modsym must be a modular symbols space") raise TypeError("modular symbols space must have sign 0") raise ValueError("modsym must be cuspidal")
newform_level=newform_level, is_simple=is_simple, isogeny_number=isogeny_number, number=number, check=check)
def _modular_symbols(self): """ Return the modular symbols space that defines this modular abelian variety.
OUTPUT: space of modular symbols
EXAMPLES::
sage: M = ModularSymbols(37).cuspidal_submodule() sage: A = M.abelian_variety(); A Abelian variety J0(37) of dimension 2 sage: A._modular_symbols() Modular Symbols subspace of dimension 4 of Modular Symbols space of dimension 5 for Gamma_0(37) of weight 2 with sign 0 over Rational Field """
def component_group_order(self, p): """ Return the order of the component group of the special fiber at p of the Neron model of self.
NOTE: For bad primes, this is only implemented when the group if Gamma0 and p exactly divides the level.
NOTE: the input abelian variety must be simple
ALGORITHM: See "Component Groups of Quotients of J0(N)" by Kohel and Stein. That paper is about optimal quotients; however, section 4.1 of Conrad-Stein "Component Groups of Purely Toric Quotients", one sees that the component group of an optimal quotient is the same as the component group of its dual (which is the subvariety).
INPUT:
- p -- a prime number
OUTPUT:
- Integer
EXAMPLES::
sage: A = J0(37)[1] sage: A.component_group_order(37) 3 sage: A = J0(43)[1] sage: A.component_group_order(37) 1 sage: A.component_group_order(43) 7 sage: A = J0(23)[0] sage: A.component_group_order(23) 11 """ raise ValueError("self must be simple") raise ValueError("p must be a prime integer") # Easy special case -- a prime of good reduction # Cases that we don't know how to handle yet. raise NotImplementedError("computation of component group not implemented when group isn't Gamma0") raise NotImplementedError("computation of component group not implemented when p^2 divides the level")
# Now we're on Gamma0(p*M) with gcd(p,M) = 1. # 1. Compute factor of Brandt module space, and put integral structure on it.
# TODO -- in case self.level() is prime, should use # supersingular module instead for massive speedup... Of # course, then one can just use Emertons theorem that the # component group order equals the torsion order, and avoid # all of this!
# X_ZZ is the submodule of degree 0 divisors
# 2. Compute the map alpha: X --> Hom(X[I],Z) over ZZ # todo -- this could be done more quickly with a clever matrix multiply
# 3. Compute invariants: # * Phi_X = #coker(alpha) # * m_X = #(alpha(X)/alpha(X[I]))
# 4. Compute the modular degree
# 5. Obtain the component group order using Theorem 1 of [Kohel-Stein]
# 6. Record the answer
def _invariants_of_image_of_component_group_of_J0(self, p): """ Return the elementary invariants of the image of the component group of J0(N). The API of this function is subject to change, which is why it starts with an underscore.
INPUT:
- p -- integer
OUTPUT:
- list -- of elementary invariants
EXAMPLES::
sage: A = J0(62).new_subvariety()[1]; A Simple abelian subvariety 62b(1,62) of dimension 2 of J0(62) sage: A._invariants_of_image_of_component_group_of_J0(2) [1, 6] sage: A.component_group_order(2) 66 """
def tamagawa_number(self, p): """ Return the Tamagawa number of this abelian variety at p.
NOTE: For bad primes, this is only implemented when the group if Gamma0 and p exactly divides the level and Atkin-Lehner acts diagonally on this abelian variety (e.g., if this variety is new and simple). See the self.component_group command for more information.
NOTE: the input abelian variety must be simple
In cases where this function doesn't work, consider using the self.tamagawa_number_bounds functions.
INPUT:
- p -- a prime number
OUTPUT:
- Integer
EXAMPLES::
sage: A = J0(37)[1] sage: A.tamagawa_number(37) 3 sage: A = J0(43)[1] sage: A.tamagawa_number(37) 1 sage: A.tamagawa_number(43) 7 sage: A = J0(23)[0] sage: A.tamagawa_number(23) 11 """ raise ValueError("self must be simple") except NotImplementedError: raise NotImplementedError("Tamagawa number can't be determined using known algorithms, so consider using the tamagawa_number_bounds function instead") else: raise NotImplementedError("the Tamagawa number at %s is a power of 2, but the exact power can't be determined using known algorithms. Consider using the tamagawa_number_bounds function instead."%p)
def tamagawa_number_bounds(self, p): """ Return a divisor and multiple of the Tamagawa number of self at `p`.
NOTE: the input abelian variety must be simple.
INPUT:
- `p` -- a prime number
OUTPUT:
- div -- integer; divisor of Tamagawa number at `p` - mul -- integer; multiple of Tamagawa number at `p` - mul_primes -- tuple; in case mul==0, a list of all primes that can possibly divide the Tamagawa number at `p`
EXAMPLES::
sage: A = J0(63).new_subvariety()[1]; A Simple abelian subvariety 63b(1,63) of dimension 2 of J0(63) sage: A.tamagawa_number_bounds(7) (3, 3, ()) sage: A.tamagawa_number_bounds(3) (1, 0, (2, 3, 5)) """ raise ValueError("self must be simple") # Frob acts trivially elif W == 1: # Frob acts by -1 n = g.valuation(2) if n <= 1: div = 2**n else: phi_X_invs = self._invariants_of_image_of_component_group_of_J0(p) m = max(1, len([z for z in phi_X_invs if z%2==0])) div = 2**m mul = 2**n else: raise NotImplementedError("Atkin-Lehner at p must act as a scalar") else:
def brandt_module(self, p): """ Return the Brandt module at p that corresponds to self. This is the factor of the vector space on the ideal class set in an order of level N in the quaternion algebra ramified at p and infinity.
INPUT:
- p -- prime that exactly divides the level
OUTPUT:
- Brandt module space that corresponds to self.
EXAMPLES::
sage: J0(43)[1].brandt_module(43) Subspace of dimension 2 of Brandt module of dimension 4 of level 43 of weight 2 over Rational Field sage: J0(43)[1].brandt_module(43).basis() ((1, 0, -1/2, -1/2), (0, 1, -1/2, -1/2)) sage: J0(43)[0].brandt_module(43).basis() ((0, 0, 1, -1),) sage: J0(35)[0].brandt_module(5).basis() ((1, 0, -1, 0),) sage: J0(35)[0].brandt_module(7).basis() ((1, -1, 1, -1),) """ raise NotImplementedError("Brandt module only defined on Gamma0") raise ValueError("p must exactly divide the level") # now cut out version of self in B raise RuntimeError("unable to cut out Brandt module (got dimension %s instead of %s)"%(V.dimension(), self.dimension())) else: D = V.decomposition() D = [A for A in D if A.dimension() == self.dimension()] # now figure out which element of D is isomorphic to self. q = 2 while len(D) > 1 and q <= B: f = S.hecke_polynomial(q) D = [A for A in D if A.hecke_polynomial(q) == f] q = next_prime(q) if len(D) != 1: raise RuntimeError("unable to locate Brandt module (got %s candidates instead of 1)"%(len(D))) V = D[0]
def sqrt_poly(f): """ Return the square root of the polynomial `f`.
.. note::
At some point something like this should be a member of the polynomial class. For now this is just used internally by some charpoly functions above.
EXAMPLES::
sage: R.<x> = QQ[] sage: f = (x-1)*(x+2)*(x^2 + 1/3*x + 5) sage: f x^4 + 4/3*x^3 + 10/3*x^2 + 13/3*x - 10 sage: sage.modular.abvar.abvar.sqrt_poly(f^2) x^4 + 4/3*x^3 + 10/3*x^2 + 13/3*x - 10 sage: sage.modular.abvar.abvar.sqrt_poly(f) Traceback (most recent call last): ... ValueError: f must be a perfect square sage: sage.modular.abvar.abvar.sqrt_poly(2*f^2) Traceback (most recent call last): ... ValueError: f must be monic """
#################################################################################################### # Useful for decomposing exactly the sort of modular symbols spaces that come up here. from random import randrange
def random_hecke_operator(M, t=None, p=2): """ Return a random Hecke operator acting on `M`, got by adding to `t` a random multiple of `T_p`
INPUT:
- ``M`` - modular symbols space
- ``t`` - None or a Hecke operator
- ``p`` - a prime
OUTPUT: Hecke operator prime
EXAMPLES::
sage: M = ModularSymbols(11).cuspidal_subspace() sage: t, p = sage.modular.abvar.abvar.random_hecke_operator(M) sage: p 3 sage: t, p = sage.modular.abvar.abvar.random_hecke_operator(M, t, p) sage: p 5 """
def factor_new_space(M): """ Given a new space `M` of modular symbols, return the decomposition into simple of `M` under the Hecke operators.
INPUT:
- ``M`` - modular symbols space
OUTPUT: list of factors
EXAMPLES::
sage: M = ModularSymbols(37).cuspidal_subspace() sage: sage.modular.abvar.abvar.factor_new_space(M) [ 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, 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 ] """ raise RuntimeError("unable to factor new space -- this should not happen") # should never happen
def factor_modsym_space_new_factors(M): """ Return the factorizations of all the new subspaces of `M`.
INPUT:
- ``M`` -- ambient modular symbols space
OUTPUT: list of decompositions corresponding to each new space.
EXAMPLES::
sage: M = ModularSymbols(33) sage: sage.modular.abvar.abvar.factor_modsym_space_new_factors(M) [[ Modular Symbols subspace of dimension 2 of Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field ], [ 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 ]] """ for d in M.level().divisors() if d%K == 0 and (d == 11 or d >= 13)]
def simple_factorization_of_modsym_space(M, simple=True): """ Return the canonical factorization of `M` into (simple) subspaces.
INPUT:
- ``M`` -- ambient modular symbols space
- ``simple`` -- boolean (default: ``True``); if set to ``False``, isogenous simple factors are grouped together
OUTPUT: sequence
EXAMPLES::
sage: M = ModularSymbols(33) sage: sage.modular.abvar.abvar.simple_factorization_of_modsym_space(M) [ (11, 0, 1, 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), (11, 0, 3, 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), (33, 0, 1, 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) ] sage: sage.modular.abvar.abvar.simple_factorization_of_modsym_space(M, simple=False) [ (11, 0, None, 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), (33, 0, None, 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) ]
TESTS:
Check that :trac:`21799` is fixed::
sage: JH(28, [15]).decomposition() [ Simple abelian subvariety 14aGH[15](1,28) of dimension 1 of JH(28,[15]), Simple abelian subvariety 14aGH[15](2,28) of dimension 1 of JH(28,[15]), Simple abelian subvariety 28aGH[15](1,28) of dimension 2 of JH(28,[15]) ] """ # Compute the matrices of the degeneracy maps up. # Construct a matrix with rows the basis for all the factors # stacked on top of each other. We just multiply this by each # degeneracy matrix to get the basis for the images of the # factors at higher level. By doing matrix multiplies, we # save time over taking images of individual factors.
# Compute the actual images
# Construct the corresponding subspaces at higher level. else:
def modsym_lattices(M, factors): """ Append lattice information to the output of simple_factorization_of_modsym_space.
INPUT:
- ``M`` - modular symbols spaces
- ``factors`` - Sequence (simple_factorization_of_modsym_space)
OUTPUT: sequence with more information for each factor (the lattice)
EXAMPLES::
sage: M = ModularSymbols(33) sage: factors = sage.modular.abvar.abvar.simple_factorization_of_modsym_space(M, simple=False) sage: sage.modular.abvar.abvar.modsym_lattices(M, factors) [ (11, 0, None, 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, Free module of degree 6 and rank 4 over Integer Ring Echelon basis matrix: [ 1 0 0 0 -1 2] [ 0 1 0 0 -1 1] [ 0 0 1 0 -2 2] [ 0 0 0 1 -1 -1]), (33, 0, None, 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, Free module of degree 6 and rank 2 over Integer Ring Echelon basis matrix: [ 1 0 0 -1 0 0] [ 0 0 1 0 1 -1]) ] """ # 1. Change basis of everything to the ambient integral modular symbols space # 2. Clear denominator. # 3. Echelonize/saturate each factor return factors
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