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: 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


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

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    12
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


TESTS:

sage: T = J0(54).rational_torsion_subgroup()
sage: loads(dumps(T)) == T
True

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

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

INPUT:

• abvar - an abelian variety

EXAMPLES:

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

abelian_variety()

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

OUTPUT: abelian variety

EXAMPLES:

sage: J0(23).qbar_torsion_subgroup().abelian_variety()
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: J0(23).qbar_torsion_subgroup().field_of_definition()
Rational Field

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

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.

EXAMPLES:

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

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: 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]


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

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)


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

multiple_of_order(maxp=None)

Return a multiple of the order of this torsion group.

The multiple is computed using characteristic polynomials of Hecke operators of odd index not dividing the level.

INPUT:

• maxp - (default: None) If maxp is None (the default), 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: J = J0(11)
sage: G = J.rational_torsion_subgroup()
sage: G.multiple_of_order(11)
5
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()
97
sage: [G.multiple_of_order(p) for p in prime_range(3,11)]
[92645296242160800, 7275, 291]
sage: [G.multiple_of_order(p) for p in prime_range(3,13)]
[92645296242160800, 7275, 291, 97]
sage: [G.multiple_of_order(p) for p in prime_range(3,19)]
[92645296242160800, 7275, 291, 97, 97, 97]

sage: J = J0(33) * J0(11) ; J.rational_torsion_subgroup().order()
Traceback (most recent call last):
...
NotImplementedError: torsion multiple only implemented for Gamma0


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

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

order()

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

This may fail if the multiple obtained by counting points modulo $$p$$ exceeds the divisor obtained from the rational cuspidal subgroup.

EXAMPLES:

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

possible_orders()

Return the possible orders of this torsion subgroup, computed from a known divisor and multiple of the order.

EXAMPLES:

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


Note that this function has not been implemented for $$J_1(N)$$, though it should be reasonably easy to do so soon (see Conrad, Edixhoven, Stein):

sage: J1(13).rational_torsion_subgroup().possible_orders()
Traceback (most recent call last):
...
NotImplementedError: torsion multiple only implemented for Gamma0


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