def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
return_BSD=False):
r"""
Attempts to prove the Birch and Swinnerton-Dyer conjectural
formula for `E`, returning a list of primes `p` for which this
function fails to prove BSD(E,p). Here, BSD(E,p) is the
statement: "the Birch and Swinnerton-Dyer formula holds up to a
rational number coprime to `p`."
INPUT:
- ``E`` - an elliptic curve
- ``verbosity`` - int, how much information about the proof to print.
- 0 - print nothing
- 1 - print sketch of proof
- 2 - print information about remaining primes
- ``two_desc`` - string (default ``'mwrank'``), what to use for the
two-descent. Options are ``'mwrank', 'simon', 'sage'``
- ``proof`` - bool or None (default: None, see
proof.elliptic_curve or sage.structure.proof). If False, this
function just immediately returns the empty list.
- ``secs_hi`` - maximum number of seconds to try to compute the
Heegner index before switching over to trying to compute the
Heegner index bound. (Rank 0 only!)
- ``return_BSD`` - bool (default: False) whether to return an object
which contains information to reconstruct a proof
NOTE:
When printing verbose output, phrases such as "by Mazur" are referring
to the following list of papers:
REFERENCES:
.. [Cha] B. Cha. Vanishing of some cohomology goups and bounds for the
Shafarevich-Tate groups of elliptic curves. J. Number Theory, 111:154-
178, 2005.
.. [Jetchev] D. Jetchev. Global divisibility of Heegner points and
Tamagawa numbers. Compos. Math. 144 (2008), no. 4, 811--826.
.. [Kato] K. Kato. p-adic Hodge theory and values of zeta functions of
modular forms. Astérisque, (295):ix, 117-290, 2004.
.. [Kolyvagin] V. A. Kolyvagin. On the structure of Shafarevich-Tate
groups. Algebraic geometry, 94--121, Lecture Notes in Math., 1479,
Springer, Berlin, 1991.
.. [LumStein] A. Lum, W. Stein. Verification of the Birch and
Swinnerton-Dyer Conjecture for Elliptic Curves with Complex
Multiplication (unpublished)
.. [Mazur] B. Mazur. Modular curves and the Eisenstein ideal. Inst.
Hautes Études Sci. Publ. Math. No. 47 (1977), 33--186 (1978).
.. [Rubin] K. Rubin. The "main conjectures" of Iwasawa theory for
imaginary quadratic fields. Invent. Math. 103 (1991), no. 1, 25--68.
.. [SteinWuthrich] W. Stein and C. Wuthrich, Algorithms
for the Arithmetic of Elliptic Curves using Iwasawa Theory
Mathematics of Computation 82 (2013), 1757-1792.
.. [SteinEtAl] G. Grigorov, A. Jorza, S. Patrikis, W. Stein,
C. Tarniţǎ. Computational verification of the Birch and
Swinnerton-Dyer conjecture for individual elliptic curves.
Math. Comp. 78 (2009), no. 268, 2397--2425.
EXAMPLES::
sage: EllipticCurve('11a').prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 5} by Kolyvagin.
Kato further implies that #Sha[5] is trivial.
[]
sage: EllipticCurve('14a').prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
Kato further implies that #Sha[3] is trivial.
[]
sage: E = EllipticCurve("50b1")
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3, 5} by Kolyvagin.
Kolyvagin's bound for p = 3 applies by Stein et al.
True for p = 3 by Kolyvagin bound
Remaining primes:
p = 5: reducible, not surjective, additive, divides a Tamagawa number
(no bounds found)
ord_p(#Sha_an) = 0
[5]
sage: E.prove_BSD(two_desc='simon')
[5]
A rank two curve::
sage: E = EllipticCurve('389a')
We know nothing with proof=True::
sage: E.prove_BSD()
Set of all prime numbers: 2, 3, 5, 7, ...
We (think we) know everything with proof=False::
sage: E.prove_BSD(proof=False)
[]
A curve of rank 0 and prime conductor::
sage: E = EllipticCurve('19a')
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
Kato further implies that #Sha[3] is trivial.
[]
sage: E = EllipticCurve('37a')
sage: E.rank()
1
sage: E._EllipticCurve_rational_field__rank
{True: 1}
sage: E.analytic_rank = lambda : 0
sage: E.prove_BSD()
Traceback (most recent call last):
...
RuntimeError: It seems that the rank conjecture does not hold for this curve (Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field)! This may be a counterexample to BSD, but is more likely a bug.
We test the consistency check for the 2-part of Sha::
sage: E = EllipticCurve('37a')
sage: S = E.sha(); S
Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: def foo(use_database):
... return 4
sage: S.an = foo
sage: E.prove_BSD()
Traceback (most recent call last):
...
RuntimeError: Apparent contradiction: 0 <= rank(sha[2]) <= 0, but ord_2(sha_an) = 2
An example with a Tamagawa number at 5::
sage: E = EllipticCurve('123a1')
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 5} by Kolyvagin.
Remaining primes:
p = 5: reducible, not surjective, good ordinary, divides a Tamagawa number
(no bounds found)
ord_p(#Sha_an) = 0
[5]
A curve for which 3 divides the order of the Tate-Shafarevich group::
sage: E = EllipticCurve('681b')
sage: E.prove_BSD(verbosity=2) # long time
p = 2: True by 2-descent...
True for p not in {2, 3} by Kolyvagin....
Remaining primes:
p = 3: irreducible, surjective, non-split multiplicative
(0 <= ord_p <= 2)
ord_p(#Sha_an) = 2
[3]
A curve for which we need to use ``heegner_index_bound``::
sage: E = EllipticCurve('198b')
sage: E.prove_BSD(verbosity=1, secs_hi=1)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
[3]
The ``return_BSD`` option gives an object with detailed information
about the proof::
sage: E = EllipticCurve('26b')
sage: B = E.prove_BSD(return_BSD=True)
sage: B.two_tor_rk
0
sage: B.N
26
sage: B.gens
[]
sage: B.primes
[]
sage: B.heegner_indexes
{-23: 2}
TESTS:
This was fixed by trac #8184 and #7575::
sage: EllipticCurve('438e1').prove_BSD(verbosity=1)
p = 2: True by 2-descent...
True for p not in {2} by Kolyvagin.
[]
::
sage: E = EllipticCurve('960d1')
sage: E.prove_BSD(verbosity=1) # long time (4s on sage.math, 2011)
p = 2: True by 2-descent
True for p not in {2} by Kolyvagin.
[]
"""
if proof is None:
from sage.structure.proof.proof import get_flag
proof = get_flag(proof, "elliptic_curve")
else:
proof = bool(proof)
if not proof:
return []
from copy import copy
BSD = BSD_data()
# We replace this curve by the optimal curve, which we can do since
# truth of BSD(E,p) is invariant under isogeny.
BSD.curve = E.optimal_curve()
if BSD.curve.has_cm():
# ensure that CM is by a maximal order
non_max_j_invs = [-12288000, 54000, 287496, 16581375]
if BSD.curve.j_invariant() in non_max_j_invs: # is this possible for optimal curves?
if verbosity > 0:
print 'CM by non maximal order: switching curves'
for E in BSD.curve.isogeny_class():
if E.j_invariant() not in non_max_j_invs:
BSD.curve = E
break
BSD.update()
galrep = BSD.curve.galois_representation()
if two_desc=='mwrank':
M = mwrank_two_descent_work(BSD.curve, BSD.two_tor_rk)
elif two_desc=='simon':
M = simon_two_descent_work(BSD.curve, BSD.two_tor_rk)
elif two_desc=='sage':
M = native_two_isogeny_descent_work(BSD.curve, BSD.two_tor_rk)
else:
raise NotImplementedError()
rank_lower_bd, rank_upper_bd, sha2_lower_bd, sha2_upper_bd, gens = M
assert sha2_lower_bd <= sha2_upper_bd
if gens is not None: gens = BSD.curve.saturation(gens)[0]
if rank_lower_bd > rank_upper_bd:
raise RuntimeError("Apparent contradiction: %d <= rank <= %d."%(rank_lower_bd, rank_upper_bd))
BSD.two_selmer_rank = rank_upper_bd + sha2_lower_bd + BSD.two_tor_rk
if sha2_upper_bd == sha2_lower_bd:
BSD.rank = rank_lower_bd
BSD.bounds[2] = (sha2_lower_bd, sha2_upper_bd)
else:
BSD.rank = BSD.curve.rank(use_database=True)
sha2_upper_bd -= (BSD.rank - rank_lower_bd)
BSD.bounds[2] = (sha2_lower_bd, sha2_upper_bd)
if verbosity > 0:
print "Unable to compute the rank exactly -- used database."
if rank_lower_bd > 1:
# We do not know BSD(E,p) for even a single p, since it's
# an open problem to show that L^r(E,1)/(Reg*Omega) is
# rational for any curve with r >= 2.
from sage.sets.all import Primes
BSD.primes = Primes()
if return_BSD:
BSD.rank = rank_lower_bd
return BSD
return BSD.primes
if (BSD.sha_an.ord(2) == 0) != (BSD.bounds[2][1] == 0):
raise RuntimeError("Apparent contradiction: %d <= rank(sha[2]) <= %d, but ord_2(sha_an) = %d"%(sha2_lower_bd, sha2_upper_bd, BSD.sha_an.ord(2)))
if BSD.bounds[2][0] == BSD.sha_an.ord(2) and BSD.sha_an.ord(2) == BSD.bounds[2][1]:
if verbosity > 0:
print 'p = 2: True by 2-descent'
BSD.primes = []
BSD.bounds.pop(2)
BSD.proof[2] = ['2-descent']
else:
BSD.primes = [2]
BSD.proof[2] = [('2-descent',)+BSD.bounds[2]]
if len(gens) > rank_lower_bd or \
rank_lower_bd > rank_upper_bd:
raise RuntimeError("Something went wrong with 2-descent.")
if BSD.rank != len(gens):
if BSD.rank != len(BSD.curve._EllipticCurve_rational_field__gens[True]):
raise RuntimeError("Could not get generators")
gens = BSD.curve._EllipticCurve_rational_field__gens[True]
BSD.gens = [BSD.curve.point(x, check=True) for x in gens]
if BSD.rank != BSD.curve.analytic_rank():
raise RuntimeError("It seems that the rank conjecture does not hold for this curve (%s)! This may be a counterexample to BSD, but is more likely a bug."%(BSD.curve))
# reduce set of remaining primes to a finite set
import signal
kolyvagin_primes = []
heegner_index = None
if BSD.rank == 0:
for D in BSD.curve.heegner_discriminants_list(10):
max_height = max(13,BSD.curve.quadratic_twist(D).CPS_height_bound())
heegner_primes = -1
while heegner_primes == -1:
if max_height > 21: break
heegner_primes, _, exact = BSD.curve.heegner_index_bound(D, max_height=max_height)
max_height += 1
if isinstance(heegner_primes, list):
break
if not isinstance(heegner_primes, list):
raise RuntimeError("Tried 10 Heegner discriminants, and heegner_index_bound failed each time.")
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
else:
BSD.heegner_index_upper_bound[D] = max(heegner_primes+[1])
if 2 in heegner_primes:
heegner_primes.remove(2)
else: # rank 1
for D in BSD.curve.heegner_discriminants_list(10):
I = BSD.curve.heegner_index(D)
J = I.is_int()
if J[0] and J[1]>0:
I = J[1]
else:
J = (2*I).is_int()
if J[0] and J[1]>0:
I = J[1]
else:
continue
heegner_index = I
BSD.heegner_indexes[D] = I
break
heegner_primes = [p for p in arith.prime_divisors(heegner_index) if p!=2]
assert BSD.sha_an in ZZ and BSD.sha_an > 0
if BSD.curve.has_cm():
if BSD.curve.analytic_rank() == 0:
if verbosity > 0:
print ' p >= 5: true by Rubin'
BSD.primes.append(3)
else:
K = rings.QuadraticField(BSD.curve.cm_discriminant(), 'a')
D_K = K.disc()
D_E = BSD.curve.discriminant()
if len(K.factor(3)) == 1: # 3 does not split in K
BSD.primes.append(3)
for p in arith.prime_divisors(D_K):
if p >= 5:
BSD.primes.append(p)
for p in arith.prime_divisors(D_E):
if p >= 5 and D_K%p and len(K.factor(p)) == 1: # p is inert in K
BSD.primes.append(p)
for p in heegner_primes:
if p >= 5 and D_E%p != 0 and D_K%p != 0 and len(K.factor(p)) == 1: # p is good for E and inert in K
kolyvagin_primes.append(p)
for p in arith.prime_divisors(BSD.sha_an):
if p >= 5 and D_K%p != 0 and len(K.factor(p)) == 1:
if BSD.curve.is_good(p):
if verbosity > 2 and p in heegner_primes and heegner_index is None:
print 'ALERT: Prime p (%d) >= 5 dividing sha_an, good for E, inert in K, in heegner_primes, should not divide the actual Heegner index'
# Note that the following check is not entirely
# exhaustive, in case there is a p not dividing
# the Heegner index in heegner_primes,
# for which only an outer bound was computed
if p not in heegner_primes:
raise RuntimeError("p = %d divides sha_an, is of good reduction for E, inert in K, and does not divide the Heegner index. This may be a counterexample to BSD, but is more likely a bug. %s"%(p,BSD.curve))
if verbosity > 0:
print 'True for p not in {%s} by Kolyvagin (via Stein & Lum -- unpublished) and Rubin.'%str(list(set(BSD.primes).union(set(kolyvagin_primes))))[1:-1]
BSD.proof['finite'] = copy(BSD.primes)
else: # no CM
# do some tricks to get to a finite set without calling bound_kolyvagin
BSD.primes += [p for p in galrep.non_surjective() if p != 2]
for p in heegner_primes:
if p not in BSD.primes:
BSD.primes.append(p)
for p in arith.prime_divisors(BSD.sha_an):
if p not in BSD.primes and p != 2:
BSD.primes.append(p)
if verbosity > 0:
s = str(BSD.primes)[1:-1]
if 2 not in BSD.primes:
if len(s) == 0: s = '2'
else: s = '2, '+s
print 'True for p not in {' + s + '} by Kolyvagin.'
BSD.proof['finite'] = copy(BSD.primes)
primes_to_remove = []
for p in BSD.primes:
if p == 2: continue
if galrep.is_surjective(p) and not BSD.curve.has_additive_reduction(p):
if BSD.curve.has_nonsplit_multiplicative_reduction(p):
if BSD.rank > 0:
continue
if p==3:
if (not (BSD.curve.is_ordinary(p) and BSD.curve.is_good(p))) and (not BSD.curve.has_split_multiplicative_reduction(p)):
continue
if BSD.rank > 0:
continue
if verbosity > 1:
print ' p = %d: Trying p_primary_bound'%p
p_bound = BSD.Sha.p_primary_bound(p)
if p in BSD.proof:
BSD.proof[p].append(('Stein-Wuthrich', p_bound))
else:
BSD.proof[p] = [('Stein-Wuthrich', p_bound)]
if BSD.sha_an.ord(p) == 0 and p_bound == 0:
if verbosity > 0:
print 'True for p=%d by Stein-Wuthrich.'%p
primes_to_remove.append(p)
else:
if p in BSD.bounds:
BSD.bounds[p][1] = min(BSD.bounds[p][1], p_bound)
else:
BSD.bounds[p] = (0, p_bound)
print 'Analytic %d-rank is '%p + str(BSD.sha_an.ord(p)) + ', actual %d-rank is at most %d.'%(p, p_bound)
print ' by Stein-Wuthrich.\n'
for p in primes_to_remove:
BSD.primes.remove(p)
kolyvagin_primes = []
for p in BSD.primes:
if p == 2: continue
if galrep.is_surjective(p):
kolyvagin_primes.append(p)
for p in kolyvagin_primes:
BSD.primes.remove(p)
# apply other hypotheses which imply Kolyvagin's bound holds
bounded_primes = []
D_K = rings.QuadraticField(D, 'a').disc()
# Cha's hypothesis
for p in BSD.primes:
if p == 2: continue
if D_K%p != 0 and BSD.N%(p**2) != 0 and galrep.is_irreducible(p):
if verbosity > 0:
print 'Kolyvagin\'s bound for p = %d applies by Cha.'%p
if p in BSD.proof:
BSD.proof[p].append('Cha')
else:
BSD.proof[p] = ['Cha']
kolyvagin_primes.append(p)
# Stein et al.
if not BSD.curve.has_cm():
L = arith.lcm([F.torsion_order() for F in BSD.curve.isogeny_class()])
for p in BSD.primes:
if p in kolyvagin_primes or p == 2: continue
if L%p != 0:
if len(arith.prime_divisors(D_K)) == 1:
if D_K%p == 0: continue
if verbosity > 0:
print 'Kolyvagin\'s bound for p = %d applies by Stein et al.'%p
kolyvagin_primes.append(p)
if p in BSD.proof:
BSD.proof[p].append('Stein et al.')
else:
BSD.proof[p] = ['Stein et al.']
for p in kolyvagin_primes:
if p in BSD.primes:
BSD.primes.remove(p)
# apply Kolyvagin's bound
primes_to_remove = []
for p in kolyvagin_primes:
if p == 2: continue
if p not in heegner_primes:
ord_p_bound = 0
elif heegner_index is not None: # p must divide heegner_index
ord_p_bound = 2*heegner_index.ord(p)
# Here Jetchev's results apply.
m_max = max([BSD.curve.tamagawa_number(q).ord(p) for q in BSD.N.prime_divisors()])
if m_max > 0:
if verbosity > 0:
print 'Jetchev\'s results apply (at p = %d) with m_max ='%p, m_max
if p in BSD.proof:
BSD.proof[p].append(('Jetchev',m_max))
else:
BSD.proof[p] = [('Jetchev',m_max)]
ord_p_bound -= 2*m_max
else: # Heegner index is None
for D in BSD.heegner_index_upper_bound:
M = BSD.heegner_index_upper_bound[D]
ord_p_bound = 0
while p**(ord_p_bound+1) <= M**2:
ord_p_bound += 1
# now ord_p_bound is one on I_K!!!
ord_p_bound *= 2 # by Kolyvagin, now ord_p_bound is one on #Sha
break
if p in BSD.proof:
BSD.proof[p].append(('Kolyvagin',ord_p_bound))
else:
BSD.proof[p] = [('Kolyvagin',ord_p_bound)]
if BSD.sha_an.ord(p) == 0 and ord_p_bound == 0:
if verbosity > 0:
print 'True for p = %d by Kolyvagin bound'%p
primes_to_remove.append(p)
elif BSD.sha_an.ord(p) > ord_p_bound:
raise RuntimeError("p = %d: ord_p_bound == %d, but sha_an.ord(p) == %d. This appears to be a counterexample to BSD, but is more likely a bug."%(p,ord_p_bound,BSD.sha_an.ord(p)))
else: # BSD.sha_an.ord(p) <= ord_p_bound != 0:
if p in BSD.bounds:
low = BSD.bounds[p][0]
BSD.bounds[p] = (low, min(BSD.bounds[p][1], ord_p_bound))
else:
BSD.bounds[p] = (0, ord_p_bound)
for p in primes_to_remove:
kolyvagin_primes.remove(p)
BSD.primes = list( set(BSD.primes).union(set(kolyvagin_primes)) )
# Kato's bound
if BSD.rank == 0 and not BSD.curve.has_cm():
L_over_Omega = BSD.curve.lseries().L_ratio()
kato_primes = BSD.Sha.bound_kato()
primes_to_remove = []
for p in BSD.primes:
if p == 2: continue
if p not in kato_primes:
if verbosity > 0:
print 'Kato further implies that #Sha[%d] is trivial.'%p
primes_to_remove.append(p)
if p in BSD.proof:
BSD.proof[p].append(('Kato',0))
else:
BSD.proof[p] = [('Kato',0)]
if p not in [2,3] and BSD.N%p != 0:
if galrep.is_surjective(p):
bd = L_over_Omega.valuation(p)
if verbosity > 1:
print 'Kato implies that ord_p(#Sha[%d]) <= %d '%(p,bd)
if p in BSD.proof:
BSD.proof[p].append(('Kato',bd))
else:
BSD.proof[p] = [('Kato',bd)]
if p in BSD.bounds:
low = BSD.bounds[p][0]
BSD.bounds[p][1] = (low, min(BSD.bounds[p][1], bd))
else:
BSD.bounds[p] = (0, bd)
for p in primes_to_remove:
BSD.primes.remove(p)
# Mazur
primes_to_remove = []
if BSD.N.is_prime():
for p in BSD.primes:
if p == 2: continue
if galrep.is_reducible(p):
primes_to_remove.append(p)
if verbosity > 0:
print 'True for p=%s by Mazur'%p
for p in primes_to_remove:
BSD.primes.remove(p)
if p in BSD.proof:
BSD.proof[p].append('Mazur')
else:
BSD.proof[p] = ['Mazur']
BSD.primes.sort()
# Try harder to compute the Heegner index, where it matters
if heegner_index is None:
if max_height < 18:
max_height = 18
for D in BSD.heegner_index_upper_bound:
M = BSD.heegner_index_upper_bound[D]
for p in kolyvagin_primes:
if p not in BSD.primes or p == 3: continue
if verbosity > 0:
print ' p = %d: Trying harder for Heegner index'%p
obt = 0
while p**(BSD.sha_an.ord(p)/2+1) <= M and max_height < 22:
if verbosity > 2:
print ' trying max_height =', max_height
old_bound = M
M, _, exact = BSD.curve.heegner_index_bound(D, max_height=max_height, secs_dc=secs_dc)
if M == -1:
max_height += 1
continue
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
M = exact
if verbosity > 2:
print ' heegner index =', M
else:
M = max(M+[1])
if verbosity > 2:
print ' bound =', M
if old_bound == M:
obt += 1
if obt == 2:
break
max_height += 1
BSD.heegner_index_upper_bound[D] = min(M,BSD.heegner_index_upper_bound[D])
low, upp = BSD.bounds[p]
expn = 0
while p**(expn+1) <= M:
expn += 1
if 2*expn < upp:
upp = 2*expn
BSD.bounds[p] = (low,upp)
if verbosity > 0:
print ' got better bound on ord_p =', upp
if low == upp:
if upp != BSD.sha_an.ord(p):
raise RuntimeError
else:
if verbosity > 0:
print ' proven!'
BSD.primes.remove(p)
break
for p in kolyvagin_primes:
if p not in BSD.primes or p == 3: continue
for D in BSD.curve.heegner_discriminants_list(4):
if D in BSD.heegner_index_upper_bound: continue
print ' discriminant', D
if verbosity > 0:
print 'p = %d: Trying discriminant = %d for Heegner index'%(p,D)
max_height = max(10, BSD.curve.quadratic_twist(D).CPS_height_bound())
obt = 0
while True:
if verbosity > 2:
print ' trying max_height =', max_height
old_bound = M
if p**(BSD.sha_an.ord(p)/2+1) > M or max_height >= 22:
break
M, _, exact = BSD.curve.heegner_index_bound(D, max_height=max_height, secs_dc=secs_dc)
if M == -1:
max_height += 1
continue
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
M = exact
if verbosity > 2:
print ' heegner index =', M
else:
M = max(M+[1])
if verbosity > 2:
print ' bound =', M
if old_bound == M:
obt += 1
if obt == 2:
break
max_height += 1
BSD.heegner_index_upper_bound[D] = M
low, upp = BSD.bounds[p]
expn = 0
while p**(expn+1) <= M:
expn += 1
if 2*expn < upp:
upp = 2*expn
BSD.bounds[p] = (low,upp)
if verbosity > 0:
print ' got better bound =', upp
if low == upp:
if upp != BSD.sha_an.ord(p):
raise RuntimeError
else:
if verbosity > 0:
print ' proven!'
BSD.primes.remove(p)
break
# print some extra information
if verbosity > 1:
if len(BSD.primes) > 0:
print 'Remaining primes:'
for p in BSD.primes:
s = 'p = ' + str(p) + ': '
if galrep.is_irreducible(p):
s += 'ir'
s += 'reducible, '
if not galrep.is_surjective(p):
s += 'not '
s += 'surjective, '
a_p = BSD.curve.an(p)
if BSD.curve.is_good(p):
if a_p%p != 0:
s += 'good ordinary'
else:
s += 'good, non-ordinary'
else:
assert BSD.curve.is_minimal()
if a_p == 0:
s += 'additive'
elif a_p == 1:
s += 'split multiplicative'
elif a_p == -1:
s += 'non-split multiplicative'
if BSD.curve.tamagawa_product()%p==0:
s += ', divides a Tamagawa number'
if p in BSD.bounds:
s += '\n (%d <= ord_p <= %d)'%BSD.bounds[p]
else:
s += '\n (no bounds found)'
s += '\n ord_p(#Sha_an) = %d'%BSD.sha_an.ord(p)
if heegner_index is None:
may_divide = True
for D in BSD.heegner_index_upper_bound:
if p > BSD.heegner_index_upper_bound[D] or p not in kolyvagin_primes:
may_divide = False
if may_divide:
s += '\n may divide the Heegner index, for which only a bound was computed'
print s
if BSD.curve.has_cm():
if BSD.rank == 1:
BSD.proof['reason_finite'] = 'Rubin&Kolyvagin'
else:
BSD.proof['reason_finite'] = 'Rubin'
else:
BSD.proof['reason_finite'] = 'Kolyvagin'
# reduce memory footprint of BSD object:
BSD.curve = BSD.curve.label()
BSD.Sha = None
return BSD if return_BSD else BSD.primes
def prove_BSD(E,
verbosity=0,
two_desc='mwrank',
proof=None,
secs_hi=5,
return_BSD=False):
r"""
Attempt to prove the Birch and Swinnerton-Dyer conjectural
formula for `E`, returning a list of primes `p` for which this
function fails to prove BSD(E,p).
Here, BSD(E,p) is the
statement: "the Birch and Swinnerton-Dyer formula holds up to a
rational number coprime to `p`."
INPUT:
- ``E`` - an elliptic curve
- ``verbosity`` - int, how much information about the proof to print.
- 0 - print nothing
- 1 - print sketch of proof
- 2 - print information about remaining primes
- ``two_desc`` - string (default ``'mwrank'``), what to use for the
two-descent. Options are ``'mwrank', 'simon', 'sage'``
- ``proof`` - bool or None (default: None, see
proof.elliptic_curve or sage.structure.proof). If False, this
function just immediately returns the empty list.
- ``secs_hi`` - maximum number of seconds to try to compute the
Heegner index before switching over to trying to compute the
Heegner index bound. (Rank 0 only!)
- ``return_BSD`` - bool (default: False) whether to return an object
which contains information to reconstruct a proof
.. NOTE::
When printing verbose output, phrases such as "by Mazur" are referring
to the following list of papers:
REFERENCES:
- [Cha2005]_
- [Jet2008]_
- [Kat2004]_
- [Kol1991]_
- [LW2015]_
- [LS]
- [Maz1978]_
- [Rub1991]_
- [SW2013]_
- [GJPST2009]_
EXAMPLES::
sage: EllipticCurve('11a').prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 5} by Kolyvagin.
Kolyvagin's bound for p = 5 applies by Lawson-Wuthrich
True for p = 5 by Kolyvagin bound
[]
sage: EllipticCurve('14a').prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
Kolyvagin's bound for p = 3 applies by Lawson-Wuthrich
True for p = 3 by Kolyvagin bound
[]
sage: E = EllipticCurve("20a1")
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
Kato further implies that #Sha[3] is trivial.
[]
sage: E = EllipticCurve("50b1")
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3, 5} by Kolyvagin.
Kolyvagin's bound for p = 3 applies by Lawson-Wuthrich
Kolyvagin's bound for p = 5 applies by Lawson-Wuthrich
True for p = 3 by Kolyvagin bound
True for p = 5 by Kolyvagin bound
[]
sage: E.prove_BSD(two_desc='simon')
[]
A rank two curve::
sage: E = EllipticCurve('389a')
We know nothing with proof=True::
sage: E.prove_BSD()
Set of all prime numbers: 2, 3, 5, 7, ...
We (think we) know everything with proof=False::
sage: E.prove_BSD(proof=False)
[]
A curve of rank 0 and prime conductor::
sage: E = EllipticCurve('19a')
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
Kolyvagin's bound for p = 3 applies by Lawson-Wuthrich
True for p = 3 by Kolyvagin bound
[]
sage: E = EllipticCurve('37a')
sage: E.rank()
1
sage: E._EllipticCurve_rational_field__rank
(1, True)
sage: E.analytic_rank = lambda : 0
sage: E.prove_BSD()
Traceback (most recent call last):
...
RuntimeError: It seems that the rank conjecture does not hold for this curve (Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field)! This may be a counterexample to BSD, but is more likely a bug.
We test the consistency check for the 2-part of Sha::
sage: E = EllipticCurve('37a')
sage: S = E.sha(); S
Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: def foo(use_database):
....: return 4
sage: S.an = foo
sage: E.prove_BSD()
Traceback (most recent call last):
...
RuntimeError: Apparent contradiction: 0 <= rank(sha[2]) <= 0, but ord_2(sha_an) = 2
An example with a Tamagawa number at 5::
sage: E = EllipticCurve('123a1')
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 5} by Kolyvagin.
Kolyvagin's bound for p = 5 applies by Lawson-Wuthrich
True for p = 5 by Kolyvagin bound
[]
A curve for which 3 divides the order of the Tate-Shafarevich group::
sage: E = EllipticCurve('681b')
sage: E.prove_BSD(verbosity=2) # long time
p = 2: True by 2-descent...
True for p not in {2, 3} by Kolyvagin....
Remaining primes:
p = 3: irreducible, surjective, non-split multiplicative
(0 <= ord_p <= 2)
ord_p(#Sha_an) = 2
[3]
A curve for which we need to use ``heegner_index_bound``::
sage: E = EllipticCurve('198b')
sage: E.prove_BSD(verbosity=1, secs_hi=1)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
[3]
The ``return_BSD`` option gives an object with detailed information
about the proof::
sage: E = EllipticCurve('26b')
sage: B = E.prove_BSD(return_BSD=True)
sage: B.two_tor_rk
0
sage: B.N
26
sage: B.gens
[]
sage: B.primes
[]
sage: B.heegner_indexes
{-23: 2}
TESTS:
This was fixed by :trac:`8184` and :trac:`7575`::
sage: EllipticCurve('438e1').prove_BSD(verbosity=1)
p = 2: True by 2-descent...
True for p not in {2} by Kolyvagin.
[]
::
sage: E = EllipticCurve('960d1')
sage: E.prove_BSD(verbosity=1) # long time (4s on sage.math, 2011)
p = 2: True by 2-descent
True for p not in {2} by Kolyvagin.
[]
"""
if proof is None:
from sage.structure.proof.proof import get_flag
proof = get_flag(proof, "elliptic_curve")
else:
proof = bool(proof)
if not proof:
return []
from copy import copy
BSD = BSD_data()
# We replace this curve by the optimal curve, which we can do since
# truth of BSD(E,p) is invariant under isogeny.
BSD.curve = E.optimal_curve()
if BSD.curve.has_cm():
# ensure that CM is by a maximal order
non_max_j_invs = [-12288000, 54000, 287496, 16581375]
if BSD.curve.j_invariant(
) in non_max_j_invs: # is this possible for optimal curves?
if verbosity > 0:
print('CM by non maximal order: switching curves')
for E in BSD.curve.isogeny_class():
if E.j_invariant() not in non_max_j_invs:
BSD.curve = E
break
BSD.update()
galrep = BSD.curve.galois_representation()
if two_desc == 'mwrank':
M = mwrank_two_descent_work(BSD.curve, BSD.two_tor_rk)
elif two_desc == 'simon':
M = simon_two_descent_work(BSD.curve, BSD.two_tor_rk)
elif two_desc == 'sage':
M = native_two_isogeny_descent_work(BSD.curve, BSD.two_tor_rk)
else:
raise NotImplementedError()
rank_lower_bd, rank_upper_bd, sha2_lower_bd, sha2_upper_bd, gens = M
assert sha2_lower_bd <= sha2_upper_bd
if gens is not None:
gens = BSD.curve.saturation(gens)[0]
if rank_lower_bd > rank_upper_bd:
raise RuntimeError("Apparent contradiction: %d <= rank <= %d." %
(rank_lower_bd, rank_upper_bd))
BSD.two_selmer_rank = rank_upper_bd + sha2_lower_bd + BSD.two_tor_rk
if sha2_upper_bd == sha2_lower_bd:
BSD.rank = rank_lower_bd
BSD.bounds[2] = (sha2_lower_bd, sha2_upper_bd)
else:
BSD.rank = BSD.curve.rank(use_database=True)
sha2_upper_bd -= (BSD.rank - rank_lower_bd)
BSD.bounds[2] = (sha2_lower_bd, sha2_upper_bd)
if verbosity > 0:
print("Unable to compute the rank exactly -- used database.")
if rank_lower_bd > 1:
# We do not know BSD(E,p) for even a single p, since it's
# an open problem to show that L^r(E,1)/(Reg*Omega) is
# rational for any curve with r >= 2.
from sage.sets.all import Primes
BSD.primes = Primes()
if return_BSD:
BSD.rank = rank_lower_bd
return BSD
return BSD.primes
if (BSD.sha_an.ord(2) == 0) != (BSD.bounds[2][1] == 0):
raise RuntimeError(
"Apparent contradiction: %d <= rank(sha[2]) <= %d, but ord_2(sha_an) = %d"
% (sha2_lower_bd, sha2_upper_bd, BSD.sha_an.ord(2)))
if BSD.bounds[2][0] == BSD.sha_an.ord(2) and BSD.sha_an.ord(
2) == BSD.bounds[2][1]:
if verbosity > 0:
print('p = 2: True by 2-descent')
BSD.primes = []
BSD.bounds.pop(2)
BSD.proof[2] = ['2-descent']
else:
BSD.primes = [2]
BSD.proof[2] = [('2-descent', ) + BSD.bounds[2]]
if len(gens) > rank_lower_bd or rank_lower_bd > rank_upper_bd:
raise RuntimeError("Something went wrong with 2-descent.")
if BSD.rank != len(gens):
gens = BSD.curve.gens(proof=True)
if BSD.rank != len(gens):
raise RuntimeError("Could not get generators")
BSD.gens = [BSD.curve.point(x, check=True) for x in gens]
if BSD.rank != BSD.curve.analytic_rank():
raise RuntimeError(
"It seems that the rank conjecture does not hold for this curve (%s)! This may be a counterexample to BSD, but is more likely a bug."
% BSD.curve)
# reduce set of remaining primes to a finite set
kolyvagin_primes = []
heegner_index = None
if BSD.rank == 0:
for D in BSD.curve.heegner_discriminants_list(10):
max_height = max(13,
BSD.curve.quadratic_twist(D).CPS_height_bound())
heegner_primes = -1
while heegner_primes == -1:
if max_height > 21:
break
heegner_primes, _, exact = BSD.curve.heegner_index_bound(
D, max_height=max_height)
max_height += 1
if isinstance(heegner_primes, list):
break
if not isinstance(heegner_primes, list):
raise RuntimeError(
"Tried 10 Heegner discriminants, and heegner_index_bound failed each time."
)
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
else:
BSD.heegner_index_upper_bound[D] = max(heegner_primes + [1])
if 2 in heegner_primes:
heegner_primes.remove(2)
else: # rank 1
for D in BSD.curve.heegner_discriminants_list(10):
I = BSD.curve.heegner_index(D)
J = I.is_int()
if J[0] and J[1] > 0:
I = J[1]
else:
J = (2 * I).is_int()
if J[0] and J[1] > 0:
I = J[1]
else:
continue
heegner_index = I
BSD.heegner_indexes[D] = I
break
heegner_primes = [p for p in prime_divisors(heegner_index) if p != 2]
assert BSD.sha_an in ZZ and BSD.sha_an > 0
if BSD.curve.has_cm():
if BSD.curve.analytic_rank() == 0:
if verbosity > 0:
print(' p >= 5: true by Rubin')
BSD.primes.append(3)
else:
K = QuadraticField(BSD.curve.cm_discriminant(), 'a')
D_K = K.disc()
D_E = BSD.curve.discriminant()
if len(K.factor(3)) == 1: # 3 does not split in K
BSD.primes.append(3)
for p in prime_divisors(D_K):
if p >= 5:
BSD.primes.append(p)
for p in prime_divisors(D_E):
if p >= 5 and D_K % p and len(K.factor(p)) == 1:
# p is inert in K
BSD.primes.append(p)
for p in heegner_primes:
if p >= 5 and D_E % p and D_K % p and len(K.factor(p)) == 1:
# p is good for E and inert in K
kolyvagin_primes.append(p)
for p in prime_divisors(BSD.sha_an):
if p >= 5 and D_K % p and len(K.factor(p)) == 1:
if BSD.curve.is_good(p):
if verbosity > 2 and p in heegner_primes and heegner_index is None:
print(
'ALERT: Prime p (%d) >= 5 dividing sha_an, good for E, inert in K, in heegner_primes, should not divide the actual Heegner index'
)
# Note that the following check is not entirely
# exhaustive, in case there is a p not dividing
# the Heegner index in heegner_primes,
# for which only an outer bound was computed
if p not in heegner_primes:
raise RuntimeError(
"p = %d divides sha_an, is of good reduction for E, inert in K, and does not divide the Heegner index. This may be a counterexample to BSD, but is more likely a bug. %s"
% (p, BSD.curve))
if verbosity > 0:
print(
'True for p not in {%s} by Kolyvagin (via Stein & Lum -- unpublished) and Rubin.'
% str(list(set(BSD.primes).union(
set(kolyvagin_primes))))[1:-1])
BSD.proof['finite'] = copy(BSD.primes)
else: # no CM
# do some tricks to get to a finite set without calling bound_kolyvagin
BSD.primes += [p for p in galrep.non_surjective() if p != 2]
for p in heegner_primes:
if p not in BSD.primes:
BSD.primes.append(p)
for p in prime_divisors(BSD.sha_an):
if p not in BSD.primes and p != 2:
BSD.primes.append(p)
if verbosity > 0:
s = str(BSD.primes)[1:-1]
if 2 not in BSD.primes:
if not s:
s = '2'
else:
s = '2, ' + s
print('True for p not in {' + s + '} by Kolyvagin.')
BSD.proof['finite'] = copy(BSD.primes)
primes_to_remove = []
for p in BSD.primes:
if p == 2:
continue
if galrep.is_surjective(
p) and not BSD.curve.has_additive_reduction(p):
if BSD.curve.has_nonsplit_multiplicative_reduction(p):
if BSD.rank > 0:
continue
if p == 3:
if (not (BSD.curve.is_ordinary(p) and BSD.curve.is_good(p))
) and (not BSD.curve.
has_split_multiplicative_reduction(p)):
continue
if BSD.rank > 0:
continue
if verbosity > 1:
print(' p = %d: Trying p_primary_bound' % p)
p_bound = BSD.Sha.p_primary_bound(p)
if p in BSD.proof:
BSD.proof[p].append(('Stein-Wuthrich', p_bound))
else:
BSD.proof[p] = [('Stein-Wuthrich', p_bound)]
if BSD.sha_an.ord(p) == 0 and p_bound == 0:
if verbosity > 0:
print('True for p=%d by Stein-Wuthrich.' % p)
primes_to_remove.append(p)
else:
if p in BSD.bounds:
BSD.bounds[p][1] = min(BSD.bounds[p][1], p_bound)
else:
BSD.bounds[p] = (0, p_bound)
print('Analytic %d-rank is ' % p + str(BSD.sha_an.ord(p)) +
', actual %d-rank is at most %d.' % (p, p_bound))
print(' by Stein-Wuthrich.\n')
for p in primes_to_remove:
BSD.primes.remove(p)
kolyvagin_primes = []
for p in BSD.primes:
if p == 2:
continue
if galrep.is_surjective(p):
kolyvagin_primes.append(p)
for p in kolyvagin_primes:
BSD.primes.remove(p)
# apply other hypotheses which imply Kolyvagin's bound holds
D_K = QuadraticField(D, 'a').disc()
# Cha's hypothesis
for p in BSD.primes:
if p == 2:
continue
if D_K % p != 0 and BSD.N % (p**2) != 0 and galrep.is_irreducible(p):
if verbosity > 0:
print('Kolyvagin\'s bound for p = %d applies by Cha.' % p)
if p in BSD.proof:
BSD.proof[p].append('Cha')
else:
BSD.proof[p] = ['Cha']
kolyvagin_primes.append(p)
# Stein et al replaced
for p in BSD.primes:
# the lemma about the vanishing of H^1 is false in Stein et al for p=5 and 11
# here is the correction from Lawson-Wuthrich. Especially Theorem 14 in
# [LW2015] above.
if p in kolyvagin_primes or p == 2 or D_K % p == 0:
continue
crit_lw = False
if p > 11 or p == 7:
crit_lw = True
elif p == 11:
if BSD.N != 121 or BSD.curve.label() != "121c2":
crit_lw = True
elif galrep.is_irreducible(p):
crit_lw = True
else:
phis = BSD.curve.isogenies_prime_degree(p)
if len(phis) != 1:
crit_lw = True
else:
C = phis[0].codomain()
if p == 3:
if BSD.curve.torsion_order() % p != 0 and C.torsion_order(
) % p != 0:
crit_lw = True
else: # p == 5
Et = BSD.curve.quadratic_twist(5)
if Et.torsion_order() % p != 0 and C.torsion_order(
) % p != 0:
crit_lw = True
if crit_lw:
if verbosity > 0:
print(
'Kolyvagin\'s bound for p = %d applies by Lawson-Wuthrich'
% p)
kolyvagin_primes.append(p)
if p in BSD.proof:
BSD.proof[p].append('Lawson-Wuthrich')
else:
BSD.proof[p] = ['Lawson-Wuthrich']
for p in kolyvagin_primes:
if p in BSD.primes:
BSD.primes.remove(p)
# apply Kolyvagin's bound
primes_to_remove = []
for p in kolyvagin_primes:
if p == 2:
continue
if p not in heegner_primes:
ord_p_bound = 0
elif heegner_index is not None: # p must divide heegner_index
ord_p_bound = 2 * heegner_index.ord(p)
# Here Jetchev's results apply.
m_max = max([
BSD.curve.tamagawa_number(q).ord(p)
for q in BSD.N.prime_divisors()
])
if m_max > 0:
if verbosity > 0:
print(
'Jetchev\'s results apply (at p = %d) with m_max =' %
p, m_max)
if p in BSD.proof:
BSD.proof[p].append(('Jetchev', m_max))
else:
BSD.proof[p] = [('Jetchev', m_max)]
ord_p_bound -= 2 * m_max
else: # Heegner index is None
for D in BSD.heegner_index_upper_bound:
M = BSD.heegner_index_upper_bound[D]
ord_p_bound = 0
while p**(ord_p_bound + 1) <= M**2:
ord_p_bound += 1
# now ord_p_bound is one on I_K!!!
ord_p_bound *= 2 # by Kolyvagin, now ord_p_bound is one on #Sha
break
if p in BSD.proof:
BSD.proof[p].append(('Kolyvagin', ord_p_bound))
else:
BSD.proof[p] = [('Kolyvagin', ord_p_bound)]
if BSD.sha_an.ord(p) == 0 and ord_p_bound == 0:
if verbosity > 0:
print('True for p = %d by Kolyvagin bound' % p)
primes_to_remove.append(p)
elif BSD.sha_an.ord(p) > ord_p_bound:
raise RuntimeError(
"p = %d: ord_p_bound == %d, but sha_an.ord(p) == %d. This appears to be a counterexample to BSD, but is more likely a bug."
% (p, ord_p_bound, BSD.sha_an.ord(p)))
else: # BSD.sha_an.ord(p) <= ord_p_bound != 0:
if p in BSD.bounds:
low = BSD.bounds[p][0]
BSD.bounds[p] = (low, min(BSD.bounds[p][1], ord_p_bound))
else:
BSD.bounds[p] = (0, ord_p_bound)
for p in primes_to_remove:
kolyvagin_primes.remove(p)
BSD.primes = list(set(BSD.primes).union(set(kolyvagin_primes)))
# Kato's bound
if BSD.rank == 0 and not BSD.curve.has_cm():
L_over_Omega = BSD.curve.lseries().L_ratio()
kato_primes = BSD.Sha.bound_kato()
primes_to_remove = []
for p in BSD.primes:
if p == 2:
continue
if p not in kato_primes:
if verbosity > 0:
print('Kato further implies that #Sha[%d] is trivial.' % p)
primes_to_remove.append(p)
if p in BSD.proof:
BSD.proof[p].append(('Kato', 0))
else:
BSD.proof[p] = [('Kato', 0)]
if p not in [2, 3] and BSD.N % p != 0:
if galrep.is_surjective(p):
bd = L_over_Omega.valuation(p)
if verbosity > 1:
print('Kato implies that ord_p(#Sha[%d]) <= %d ' %
(p, bd))
if p in BSD.proof:
BSD.proof[p].append(('Kato', bd))
else:
BSD.proof[p] = [('Kato', bd)]
if p in BSD.bounds:
low = BSD.bounds[p][0]
BSD.bounds[p][1] = (low, min(BSD.bounds[p][1], bd))
else:
BSD.bounds[p] = (0, bd)
for p in primes_to_remove:
BSD.primes.remove(p)
# Mazur
primes_to_remove = []
if BSD.N.is_prime():
for p in BSD.primes:
if p == 2:
continue
if galrep.is_reducible(p):
primes_to_remove.append(p)
if verbosity > 0:
print('True for p=%s by Mazur' % p)
for p in primes_to_remove:
BSD.primes.remove(p)
if p in BSD.proof:
BSD.proof[p].append('Mazur')
else:
BSD.proof[p] = ['Mazur']
BSD.primes.sort()
# Try harder to compute the Heegner index, where it matters
if heegner_index is None:
if max_height < 18:
max_height = 18
for D in BSD.heegner_index_upper_bound:
M = BSD.heegner_index_upper_bound[D]
for p in kolyvagin_primes:
if p not in BSD.primes or p == 3:
continue
if verbosity > 0:
print(' p = %d: Trying harder for Heegner index' % p)
obt = 0
while p**(BSD.sha_an.ord(p) / 2 + 1) <= M and max_height < 22:
if verbosity > 2:
print(' trying max_height =', max_height)
old_bound = M
M, _, exact = BSD.curve.heegner_index_bound(
D, max_height=max_height, secs_dc=secs_hi)
if M == -1:
max_height += 1
continue
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
M = exact
if verbosity > 2:
print(' heegner index =', M)
else:
M = max(M + [1])
if verbosity > 2:
print(' bound =', M)
if old_bound == M:
obt += 1
if obt == 2:
break
max_height += 1
BSD.heegner_index_upper_bound[D] = min(
M, BSD.heegner_index_upper_bound[D])
low, upp = BSD.bounds[p]
expn = 0
while p**(expn + 1) <= M:
expn += 1
if 2 * expn < upp:
upp = 2 * expn
BSD.bounds[p] = (low, upp)
if verbosity > 0:
print(' got better bound on ord_p =', upp)
if low == upp:
if upp != BSD.sha_an.ord(p):
raise RuntimeError
else:
if verbosity > 0:
print(' proven!')
BSD.primes.remove(p)
break
for p in kolyvagin_primes:
if p not in BSD.primes or p == 3:
continue
for D in BSD.curve.heegner_discriminants_list(4):
if D in BSD.heegner_index_upper_bound:
continue
print(' discriminant', D)
if verbosity > 0:
print(
'p = %d: Trying discriminant = %d for Heegner index' %
(p, D))
max_height = max(
10,
BSD.curve.quadratic_twist(D).CPS_height_bound())
obt = 0
while True:
if verbosity > 2:
print(' trying max_height =', max_height)
old_bound = M
if p**(BSD.sha_an.ord(p) / 2 + 1) > M or max_height >= 22:
break
M, _, exact = BSD.curve.heegner_index_bound(
D, max_height=max_height, secs_dc=secs_hi)
if M == -1:
max_height += 1
continue
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
M = exact
if verbosity > 2:
print(' heegner index =', M)
else:
M = max(M + [1])
if verbosity > 2:
print(' bound =', M)
if old_bound == M:
obt += 1
if obt == 2:
break
max_height += 1
BSD.heegner_index_upper_bound[D] = M
low, upp = BSD.bounds[p]
expn = 0
while p**(expn + 1) <= M:
expn += 1
if 2 * expn < upp:
upp = 2 * expn
BSD.bounds[p] = (low, upp)
if verbosity > 0:
print(' got better bound =', upp)
if low == upp:
if upp != BSD.sha_an.ord(p):
raise RuntimeError
else:
if verbosity > 0:
print(' proven!')
BSD.primes.remove(p)
break
# print some extra information
if verbosity > 1:
if BSD.primes:
print('Remaining primes:')
for p in BSD.primes:
s = 'p = ' + str(p) + ': '
if galrep.is_irreducible(p):
s += 'ir'
s += 'reducible, '
if not galrep.is_surjective(p):
s += 'not '
s += 'surjective, '
a_p = BSD.curve.an(p)
if BSD.curve.is_good(p):
if a_p % p != 0:
s += 'good ordinary'
else:
s += 'good, non-ordinary'
else:
assert BSD.curve.is_minimal()
if a_p == 0:
s += 'additive'
elif a_p == 1:
s += 'split multiplicative'
elif a_p == -1:
s += 'non-split multiplicative'
if BSD.curve.tamagawa_product() % p == 0:
s += ', divides a Tamagawa number'
if p in BSD.bounds:
s += '\n (%d <= ord_p <= %d)' % BSD.bounds[p]
else:
s += '\n (no bounds found)'
s += '\n ord_p(#Sha_an) = %d' % BSD.sha_an.ord(p)
if heegner_index is None:
may_divide = True
for D in BSD.heegner_index_upper_bound:
if p > BSD.heegner_index_upper_bound[
D] or p not in kolyvagin_primes:
may_divide = False
if may_divide:
s += '\n may divide the Heegner index, for which only a bound was computed'
print(s)
if BSD.curve.has_cm():
if BSD.rank == 1:
BSD.proof['reason_finite'] = 'Rubin&Kolyvagin'
else:
BSD.proof['reason_finite'] = 'Rubin'
else:
BSD.proof['reason_finite'] = 'Kolyvagin'
# reduce memory footprint of BSD object:
BSD.curve = BSD.curve.label()
BSD.Sha = None
return BSD if return_BSD else BSD.primes
