Torsion subgroups of modular abelian varieties

Sage can compute information about the structure of the torsion subgroup of a modular abelian variety. Sage computes a multiple of the order by computing the greatest common divisor of the orders of the torsion subgroup of the reduction of the abelian variety modulo p for various primes p. Sage computes a divisor of the order by computing the rational cuspidal subgroup. When these two bounds agree (which is often the case), we determine the exact structure of the torsion subgroup.

AUTHORS:

• William Stein (2007-03)

EXAMPLES: First we consider \(J_0(50)\) where everything works out nicely:

Sage

sage: J = J0(50)
sage: T = J.rational_torsion_subgroup(); T
Torsion subgroup of Abelian variety J0(50) of dimension 2
sage: T.multiple_of_order()
15
sage: T.divisor_of_order()
15
sage: T.gens()
[[(1/15, 3/5, 2/5, 14/15)]]
sage: T.invariants()
[15]
sage: d = J.decomposition(); d
[
Simple abelian subvariety 50a(1,50) of dimension 1 of J0(50),
Simple abelian subvariety 50b(1,50) of dimension 1 of J0(50)
]
sage: d[0].rational_torsion_subgroup().order()
3
sage: d[1].rational_torsion_subgroup().order()
5

Python

>>> from sage.all import *
>>> J = J0(Integer(50))
>>> T = J.rational_torsion_subgroup(); T
Torsion subgroup of Abelian variety J0(50) of dimension 2
>>> T.multiple_of_order()
15
>>> T.divisor_of_order()
15
>>> T.gens()
[[(1/15, 3/5, 2/5, 14/15)]]
>>> T.invariants()
[15]
>>> d = J.decomposition(); d
[
Simple abelian subvariety 50a(1,50) of dimension 1 of J0(50),
Simple abelian subvariety 50b(1,50) of dimension 1 of J0(50)
]
>>> d[Integer(0)].rational_torsion_subgroup().order()
3
>>> d[Integer(1)].rational_torsion_subgroup().order()
5

Next we make a table of the upper and lower bounds for each new factor.

Sage

sage: for N in range(1,38):
....:    for A in J0(N).new_subvariety().decomposition():
....:        T = A.rational_torsion_subgroup()
....:        print('%-5s%-5s%-5s%-5s'%(N, A.dimension(), T.divisor_of_order(), T.multiple_of_order()))
11   1    5    5
14   1    6    6
15   1    8    8
17   1    4    4
19   1    3    3
20   1    6    6
21   1    8    8
23   2    11   11
24   1    8    8
26   1    3    3
26   1    7    7
27   1    3    3
29   2    7    7
30   1    6    6
31   2    5    5
32   1    4    4
33   1    4    4
34   1    6    6
35   1    3    3
35   2    16   16
36   1    6    6
37   1    1    1
37   1    3    3

Python

>>> from sage.all import *
>>> for N in range(Integer(1),Integer(38)):
...    for A in J0(N).new_subvariety().decomposition():
...        T = A.rational_torsion_subgroup()
...        print('%-5s%-5s%-5s%-5s'%(N, A.dimension(), T.divisor_of_order(), T.multiple_of_order()))
11   1    5    5
14   1    6    6
15   1    8    8
17   1    4    4
19   1    3    3
20   1    6    6
21   1    8    8
23   2    11   11
24   1    8    8
26   1    3    3
26   1    7    7
27   1    3    3
29   2    7    7
30   1    6    6
31   2    5    5
32   1    4    4
33   1    4    4
34   1    6    6
35   1    3    3
35   2    16   16
36   1    6    6
37   1    1    1
37   1    3    3

class sage.modular.abvar.torsion_subgroup.QQbarTorsionSubgroup(abvar)

Bases: Module

Group of all torsion points over the algebraic closure on an abelian variety.

INPUT:

• abvar – an abelian variety

EXAMPLES:

Sage

sage: A = J0(23)
sage: A.qbar_torsion_subgroup()                                             # needs sage.rings.number_field
Group of all torsion points in QQbar on Abelian variety J0(23) of dimension 2

Python

>>> from sage.all import *
>>> A = J0(Integer(23))
>>> A.qbar_torsion_subgroup()                                             # needs sage.rings.number_field
Group of all torsion points in QQbar on Abelian variety J0(23) of dimension 2

Element

alias of TorsionPoint

abelian_variety()

Return the abelian variety that this is the set of all torsion points on.

OUTPUT: abelian variety

EXAMPLES:

Sage

sage: J0(23).qbar_torsion_subgroup().abelian_variety()                      # needs sage.rings.number_field
Abelian variety J0(23) of dimension 2

Python

>>> from sage.all import *
>>> J0(Integer(23)).qbar_torsion_subgroup().abelian_variety()                      # needs sage.rings.number_field
Abelian variety J0(23) of dimension 2

field_of_definition()

Return the field of definition of this subgroup. Since this is the group of all torsion it is defined over the base field of this abelian variety.

OUTPUT: a field

EXAMPLES:

Sage

sage: J0(23).qbar_torsion_subgroup().field_of_definition()                  # needs sage.rings.number_field
Rational Field

Python

>>> from sage.all import *
>>> J0(Integer(23)).qbar_torsion_subgroup().field_of_definition()                  # needs sage.rings.number_field
Rational Field

class sage.modular.abvar.torsion_subgroup.RationalTorsionSubgroup(abvar)

Bases: FiniteSubgroup

The torsion subgroup of a modular abelian variety.

divisor_of_order()

Return a divisor of the order of this torsion subgroup of a modular abelian variety.

OUTPUT: a divisor of this torsion subgroup

EXAMPLES:

Sage

sage: t = J0(37)[1].rational_torsion_subgroup()
sage: t.divisor_of_order()
3

sage: J = J1(19)
sage: J.rational_torsion_subgroup().divisor_of_order()
4383

sage: J = J0(45)
sage: J.rational_cusp_subgroup().order()
32
sage: J.rational_cuspidal_subgroup().order()
64
sage: J.rational_torsion_subgroup().divisor_of_order()
64

Python

>>> from sage.all import *
>>> t = J0(Integer(37))[Integer(1)].rational_torsion_subgroup()
>>> t.divisor_of_order()
3

>>> J = J1(Integer(19))
>>> J.rational_torsion_subgroup().divisor_of_order()
4383

>>> J = J0(Integer(45))
>>> J.rational_cusp_subgroup().order()
32
>>> J.rational_cuspidal_subgroup().order()
64
>>> J.rational_torsion_subgroup().divisor_of_order()
64

lattice()

Return lattice that defines this torsion subgroup, if possible.

Warning

There is no known algorithm in general to compute the rational torsion subgroup. Use rational_cusp_group to obtain a subgroup of the rational torsion subgroup in general.

EXAMPLES:

Sage

sage: J0(11).rational_torsion_subgroup().lattice()
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[  1   0]
[  0 1/5]

Python

>>> from sage.all import *
>>> J0(Integer(11)).rational_torsion_subgroup().lattice()
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[  1   0]
[  0 1/5]

The following fails because in fact I know of no (reasonable) algorithm to provably compute the torsion subgroup in general.

Sage

sage: T = J0(33).rational_torsion_subgroup()
sage: T.lattice()
Traceback (most recent call last):
...
NotImplementedError: unable to compute the rational torsion subgroup
in this case (there is no known general algorithm yet)

Python

>>> from sage.all import *
>>> T = J0(Integer(33)).rational_torsion_subgroup()
>>> T.lattice()
Traceback (most recent call last):
...
NotImplementedError: unable to compute the rational torsion subgroup
in this case (there is no known general algorithm yet)

The problem is that the multiple of the order obtained by counting points over finite fields is twice the divisor of the order got from the rational cuspidal subgroup.

Sage

sage: T.multiple_of_order(30)
200
sage: J0(33).rational_cusp_subgroup().order()
100

Python

>>> from sage.all import *
>>> T.multiple_of_order(Integer(30))
200
>>> J0(Integer(33)).rational_cusp_subgroup().order()
100

multiple_of_order(maxp=None, proof=True)

Return a multiple of the order.

INPUT:

• proof – boolean (default: True)

The computation of the rational torsion order of J1(p) is conjectural and will only be used if proof=False. See Section 6.2.3 of [CES2003].

EXAMPLES:

Sage

sage: J = J1(11); J
Abelian variety J1(11) of dimension 1
sage: J.rational_torsion_subgroup().multiple_of_order()
5

sage: J = J0(17)
sage: J.rational_torsion_subgroup().order()
4

Python

>>> from sage.all import *
>>> J = J1(Integer(11)); J
Abelian variety J1(11) of dimension 1
>>> J.rational_torsion_subgroup().multiple_of_order()
5

>>> J = J0(Integer(17))
>>> J.rational_torsion_subgroup().order()
4

This is an example where proof=False leads to a better bound and better performance.

Sage

sage: J = J1(23)
sage: J.rational_torsion_subgroup().multiple_of_order()  # long time (2s)
9406793
sage: J.rational_torsion_subgroup().multiple_of_order(proof=False)
408991

Python

>>> from sage.all import *
>>> J = J1(Integer(23))
>>> J.rational_torsion_subgroup().multiple_of_order()  # long time (2s)
9406793
>>> J.rational_torsion_subgroup().multiple_of_order(proof=False)
408991

multiple_of_order_using_frobp(maxp=None)

Return a multiple of the order of this torsion group.

In the \(Gamma_0\) case, the multiple is computed using characteristic polynomials of Hecke operators of odd index not dividing the level. In the \(Gamma_1\) case, the multiple is computed by expressing the frobenius polynomial in terms of the characteristic polynomial of left multiplication by \(a_p\) for odd primes p not dividing the level.

INPUT:

• maxp – (default: None) if maxp is None, return gcd of best bound computed so far with bound obtained by computing GCD’s of orders modulo \(p\) until this gcd stabilizes for 3 successive primes. If maxp is given, just use all primes up to and including maxp.

EXAMPLES:

Sage

sage: J = J0(11)
sage: G = J.rational_torsion_subgroup()
sage: G.multiple_of_order_using_frobp(11)
5

Python

>>> from sage.all import *
>>> J = J0(Integer(11))
>>> G = J.rational_torsion_subgroup()
>>> G.multiple_of_order_using_frobp(Integer(11))
5

Increasing maxp may yield a tighter bound. If maxp=None, then Sage will use more primes until the multiple stabilizes for 3 successive primes.

Sage

sage: J = J0(389)
sage: G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(389) of dimension 32
sage: G.multiple_of_order_using_frobp()
97
sage: [G.multiple_of_order_using_frobp(p) for p in prime_range(3,11)]
[92645296242160800, 7275, 291]
sage: [G.multiple_of_order_using_frobp(p) for p in prime_range(3,13)]
[92645296242160800, 7275, 291, 97]
sage: [G.multiple_of_order_using_frobp(p) for p in prime_range(3,19)]
[92645296242160800, 7275, 291, 97, 97, 97]

Python

>>> from sage.all import *
>>> J = J0(Integer(389))
>>> G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(389) of dimension 32
>>> G.multiple_of_order_using_frobp()
97
>>> [G.multiple_of_order_using_frobp(p) for p in prime_range(Integer(3),Integer(11))]
[92645296242160800, 7275, 291]
>>> [G.multiple_of_order_using_frobp(p) for p in prime_range(Integer(3),Integer(13))]
[92645296242160800, 7275, 291, 97]
>>> [G.multiple_of_order_using_frobp(p) for p in prime_range(Integer(3),Integer(19))]
[92645296242160800, 7275, 291, 97, 97, 97]

We can compute the multiple of order of the torsion subgroup for Gamma0 and Gamma1 varieties, and their products.

Sage

sage: A = J0(11) * J0(33)
sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp()
1000

sage: A = J1(23)
sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp()
9406793
sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp(maxp=50)
408991

sage: A = J1(19) * J0(21)
sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp()
35064

Python

>>> from sage.all import *
>>> A = J0(Integer(11)) * J0(Integer(33))
>>> A.rational_torsion_subgroup().multiple_of_order_using_frobp()
1000

>>> A = J1(Integer(23))
>>> A.rational_torsion_subgroup().multiple_of_order_using_frobp()
9406793
>>> A.rational_torsion_subgroup().multiple_of_order_using_frobp(maxp=Integer(50))
408991

>>> A = J1(Integer(19)) * J0(Integer(21))
>>> A.rational_torsion_subgroup().multiple_of_order_using_frobp()
35064

The next example illustrates calling this function with a larger input and how the result may be cached when maxp is None:

Sage

sage: T = J0(43)[1].rational_torsion_subgroup()
sage: T.multiple_of_order_using_frobp()
14
sage: T.multiple_of_order_using_frobp(50)
7
sage: T.multiple_of_order_using_frobp()
7

Python

>>> from sage.all import *
>>> T = J0(Integer(43))[Integer(1)].rational_torsion_subgroup()
>>> T.multiple_of_order_using_frobp()
14
>>> T.multiple_of_order_using_frobp(Integer(50))
7
>>> T.multiple_of_order_using_frobp()
7

This function is not implemented for general congruence subgroups unless the dimension is zero.

Sage

sage: A = JH(13,[2]); A
Abelian variety J0(13) of dimension 0
sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp()
1

sage: A = JH(15, [2]); A
Abelian variety JH(15,[2]) of dimension 1
sage: A.rational_torsion_subgroup().multiple_of_order_using_frobp()
Traceback (most recent call last):
...
NotImplementedError: torsion multiple only implemented for Gamma0 and Gamma1

Python

>>> from sage.all import *
>>> A = JH(Integer(13),[Integer(2)]); A
Abelian variety J0(13) of dimension 0
>>> A.rational_torsion_subgroup().multiple_of_order_using_frobp()
1

>>> A = JH(Integer(15), [Integer(2)]); A
Abelian variety JH(15,[2]) of dimension 1
>>> A.rational_torsion_subgroup().multiple_of_order_using_frobp()
Traceback (most recent call last):
...
NotImplementedError: torsion multiple only implemented for Gamma0 and Gamma1

order(proof=True)

Return the order of the torsion subgroup of this modular abelian variety.

This function may fail if the multiple obtained by counting points modulo \(p\) exceeds the divisor obtained from the rational cuspidal subgroup.

The computation of the rational torsion order of J1(p) is conjectural and will only be used if proof=False. See Section 6.2.3 of [CES2003].

INPUT:

• proof – boolean (default: True)

OUTPUT: the order of this torsion subgroup

EXAMPLES:

Sage

sage: A = J0(11)
sage: A.rational_torsion_subgroup().order()
5
sage: A = J0(23)
sage: A.rational_torsion_subgroup().order()
11
sage: T = J0(37)[1].rational_torsion_subgroup()
sage: T.order()
3

sage: J = J1(13)
sage: J.rational_torsion_subgroup().order()
19

Python

>>> from sage.all import *
>>> A = J0(Integer(11))
>>> A.rational_torsion_subgroup().order()
5
>>> A = J0(Integer(23))
>>> A.rational_torsion_subgroup().order()
11
>>> T = J0(Integer(37))[Integer(1)].rational_torsion_subgroup()
>>> T.order()
3

>>> J = J1(Integer(13))
>>> J.rational_torsion_subgroup().order()
19

Sometimes the order can only be computed with proof=False.

Sage

sage: J = J1(23)
sage: J.rational_torsion_subgroup().order()
Traceback (most recent call last):
...
RuntimeError: Unable to compute order of torsion subgroup
(it is in [408991, 9406793])

sage: J.rational_torsion_subgroup().order(proof=False)
408991

Python

>>> from sage.all import *
>>> J = J1(Integer(23))
>>> J.rational_torsion_subgroup().order()
Traceback (most recent call last):
...
RuntimeError: Unable to compute order of torsion subgroup
(it is in [408991, 9406793])

>>> J.rational_torsion_subgroup().order(proof=False)
408991

possible_orders(proof=True)

Return the possible orders of this torsion subgroup. Outside of special cases, this is done by computing a divisor and multiple of the order.

INPUT:

• proof – boolean (default: True)

OUTPUT: an array of positive integers

The computation of the rational torsion order of J1(p) is conjectural and will only be used if proof=False. See Section 6.2.3 of [CES2003].

EXAMPLES:

Sage

sage: J0(11).rational_torsion_subgroup().possible_orders()
[5]
sage: J0(33).rational_torsion_subgroup().possible_orders()
[100, 200]

sage: J1(13).rational_torsion_subgroup().possible_orders()
[19]
sage: J1(16).rational_torsion_subgroup().possible_orders()
[1, 2, 4, 5, 10, 20]

Python

>>> from sage.all import *
>>> J0(Integer(11)).rational_torsion_subgroup().possible_orders()
[5]
>>> J0(Integer(33)).rational_torsion_subgroup().possible_orders()
[100, 200]

>>> J1(Integer(13)).rational_torsion_subgroup().possible_orders()
[19]
>>> J1(Integer(16)).rational_torsion_subgroup().possible_orders()
[1, 2, 4, 5, 10, 20]