def upper_bound_index_cusps_in_JG_torsion(G, d, bound=60):
"""
INPUT:
- G - a congruence subgroup
- d - integer, the size of the rational cuspidal subgroup
- bound (optional, default = 60) - the bound for the primes p up to which to use
the hecke matrix `T_p - <p> - p` for bounding the torsion subgroup
OUTPUT:
- an integer `i` such that `(\#J_G(\QQ)_{tors})/d` is a divisor of `i`.
EXAMPLES::
sage: from mdsage import *
sage: d = rational_cuspidal_classgroup(Gamma1(23)).cardinality()
sage: upper_bound_index_cusps_in_JG_torsion(Gamma1(23),d)
1
"""
N = G.level()
M = ModularSymbols(G)
Sint = cuspidal_integral_structure(M)
kill_mat = (M.star_involution().matrix().restrict(Sint) - 1)
kill = kill_mat.transpose().change_ring(ZZ).row_module()
for p in prime_range(3, bound):
if not N % p == 0:
kill += kill_torsion_coprime_to_q(
p, M).restrict(Sint).change_ring(ZZ).transpose().row_module()
if kill.matrix().is_square() and kill.matrix().determinant() == d:
#print p
break
kill_mat = kill.matrix().transpose()
#print N,"index of torsion in stuff killed",kill.matrix().determinant()/d
if kill.matrix().determinant() == d:
return 1
pm = integral_period_mapping(M)
period_images1 = [
sum([M.coordinate_vector(M([c, infinity])) for c in cusps]) * pm
for cusps in galois_orbits(G)
]
m = (Matrix(period_images1) * kill_mat).stack(kill_mat)
diag = m.change_ring(ZZ).echelon_form().diagonal()
#print diag,prod(diag)
assert prod(diag) == kill.matrix().determinant() / d
period_images2 = [
M.coordinate_vector(M([c, infinity])) * pm for c in G.cusps()
if c != Cusp(oo)
]
m = (Matrix(period_images2) * kill_mat).stack(kill_mat)
m, denom = m._clear_denom()
diag = (m.change_ring(ZZ).echelon_form() / denom).diagonal()
#print diag
#print prod(i.numerator() for i in diag),"if this is 1 then :)"
return prod(i.numerator() for i in diag)
def upper_bound_index_cusps_in_JG_torsion(G,d, bound = 60):
"""
INPUT:
- G - a congruence subgroup
- d - integer, the size of the rational cuspidal subgroup
- bound (optional, default = 60) - the bound for the primes p up to which to use
the hecke matrix `T_p - <p> - p` for bounding the torsion subgroup
OUTPUT:
- an integer `i` such that `(\#J_G(\QQ)_{tors})/d` is a divisor of `i`.
EXAMPLES::
sage: from mdsage import *
sage: d = rational_cuspidal_classgroup(Gamma1(23)).cardinality()
sage: upper_bound_index_cusps_in_JG_torsion(Gamma1(23),d)
1
"""
N = G.level()
M=ModularSymbols(G);
Sint=cuspidal_integral_structure(M)
kill_mat=(M.star_involution().matrix().restrict(Sint)-1)
kill=kill_mat.transpose().change_ring(ZZ).row_module()
for p in prime_range(3,bound):
if not N % p ==0:
kill+=kill_torsion_coprime_to_q(p,M).restrict(Sint).change_ring(ZZ).transpose().row_module()
if kill.matrix().is_square() and kill.matrix().determinant()==d:
#print p
break
kill_mat=kill.matrix().transpose()
#print N,"index of torsion in stuff killed",kill.matrix().determinant()/d
if kill.matrix().determinant()==d:
return 1
pm=integral_period_mapping(M)
period_images1=[sum([M.coordinate_vector(M([c,infinity])) for c in cusps])*pm for cusps in galois_orbits(G)]
m=(Matrix(period_images1)*kill_mat).stack(kill_mat)
diag=m.change_ring(ZZ).echelon_form().diagonal()
#print diag,prod(diag)
assert prod(diag)==kill.matrix().determinant()/d
period_images2=[M.coordinate_vector(M([c,infinity]))*pm for c in G.cusps() if c != Cusp(oo)]
m=(Matrix(period_images2)*kill_mat).stack(kill_mat)
m,denom=m._clear_denom()
diag=(m.change_ring(ZZ).echelon_form()/denom).diagonal()
#print diag
#print prod(i.numerator() for i in diag),"if this is 1 then :)"
return prod(i.numerator() for i in diag)
def JG_torsion_upperbound(G, bound=60):
"""
INPUT:
- G - a congruence subgroup
- bound (optional, default = 60) - the bound for the primes p up to which to use
the hecke matrix `T_p - <p> - p` for bounding the torsion subgroup
OUTPUT:
- A subgroup of `(S_2(G) \otimes \QQ) / S_2(G)` that is guaranteed to contain
the rational torison subgroup, together with a subgroup generated by the
rational cusps.
The subgroup is given as a subgroup of `S_2(G)/NS_2(G)` for a suitable integer N
EXAMPLES::
sage: from mdsage import *
sage: d = rational_cuspidal_classgroup(Gamma1(23)).cardinality()
sage: upper_bound_index_cusps_in_JG_torsion(Gamma1(23),d)
1
"""
N = G.level()
M = ModularSymbols(G)
Sint = cuspidal_integral_structure(M)
kill_mat = (M.star_involution().matrix().restrict(Sint) - 1)
kill = kill_mat.transpose().change_ring(ZZ).row_module()
for p in prime_range(3, bound):
if not N % p == 0:
kill += kill_torsion_coprime_to_q(
p, M).restrict(Sint).change_ring(ZZ).transpose().row_module()
#if kill.matrix().is_square() and kill.matrix().determinant()==d:
# #print p
# break
kill_mat = kill.matrix().transpose()
#print N,"index of torsion in stuff killed",kill.matrix().determinant()/d
#if kill.matrix().determinant()==d:
# return 1
d = prod(kill_mat.smith_form()[0].diagonal())
pm = integral_period_mapping(M)
#period_images1=[sum([M.coordinate_vector(M([c,infinity])) for c in cusps])*pm for cusps in galois_orbits(G)]
period_images2 = [
M.coordinate_vector(M([c, infinity])) * pm for c in G.cusps()
if c != Cusp(oo)
]
m = (Matrix(period_images2) * kill_mat).stack(kill_mat)
m, d2 = m._clear_denom()
d = gcd(d, d2)
def JG_torsion_upperbound(G, bound = 60):
"""
INPUT:
- G - a congruence subgroup
- bound (optional, default = 60) - the bound for the primes p up to which to use
the hecke matrix `T_p - <p> - p` for bounding the torsion subgroup
OUTPUT:
- A subgroup of `(S_2(G) \otimes \QQ) / S_2(G)` that is guaranteed to contain
the rational torison subgroup, together with a subgroup generated by the
rational cusps.
The subgroup is given as a subgroup of `S_2(G)/NS_2(G)` for a suitable integer N
EXAMPLES::
sage: from mdsage import *
sage: d = rational_cuspidal_classgroup(Gamma1(23)).cardinality()
sage: upper_bound_index_cusps_in_JG_torsion(Gamma1(23),d)
1
"""
N = G.level()
M=ModularSymbols(G);
Sint=cuspidal_integral_structure(M)
kill_mat=(M.star_involution().matrix().restrict(Sint)-1)
kill=kill_mat.transpose().change_ring(ZZ).row_module()
for p in prime_range(3,bound):
if not N % p ==0:
kill+=kill_torsion_coprime_to_q(p,M).restrict(Sint).change_ring(ZZ).transpose().row_module()
#if kill.matrix().is_square() and kill.matrix().determinant()==d:
# #print p
# break
kill_mat=kill.matrix().transpose()
#print N,"index of torsion in stuff killed",kill.matrix().determinant()/d
#if kill.matrix().determinant()==d:
# return 1
d = prod(kill_mat.smith_form()[0].diagonal())
pm=integral_period_mapping(M)
#period_images1=[sum([M.coordinate_vector(M([c,infinity])) for c in cusps])*pm for cusps in galois_orbits(G)]
period_images2=[M.coordinate_vector(M([c,infinity]))*pm for c in G.cusps() if c != Cusp(oo)]
m=(Matrix(period_images2)*kill_mat).stack(kill_mat)
m,d2=m._clear_denom()
d=gcd(d,d2)
