def volcano_schoof(E, volcano_2=True):
q = E.base_field().order()
residues = []
moduli = []
l = 2
while prod(moduli) <= 4 * sqrt(q):
if q % l == 0:
l = next_prime(l)
res = finding_t_modln(E, l, volcano_2=volcano_2)
residues.append(res[0])
moduli.append(res[1])
l = next_prime(l)
t = CRT_list(residues, moduli)
if t > 2 * sqrt(q):
t = t - prod(moduli)
return t
def schoof(E, info=False):
q = E.base_field().order()
residues = [trace_of_frobenius_mod_2(E)]
moduli = [2]
l = 3
while prod(moduli) <= 4 * sqrt(q):
if q % l == 0:
l = next_prime(l)
residues.append(trace_of_frobenius_mod(E, l))
moduli.append(l)
l = next_prime(l)
t = CRT_list(residues, moduli)
if t > 2 * sqrt(q):
t = t - prod(moduli)
if info:
return t, moduli
return t
def project_onto_eigenspace(gamma,
ord_basis,
hord,
weight=2,
level=1,
epstwist=None,
derivative_order=1):
ell = 1
level = ZZ(level)
R = hord.parent().base_ring()
while True:
ell = next_prime(ell)
if level % ell == 0:
continue
verbose('Using ell = %s' % ell)
T = hecke_matrix_on_ord(ell, ord_basis, weight, level, epstwist)
aell = gamma[ell] / gamma[1]
verbose('a_ell = %s' % aell)
pp = T.charpoly().change_ring(hord.parent().base_ring())
verbose('deg charpoly(T_ell) = %s' % pp.degree())
x = pp.parent().gen()
this_is_zero = pp.subs({x: R(aell)})
if this_is_zero.valuation() < 8: # DEBUG this value is arbitrary...
verbose(
'!!! Should we skip ell = %s (because %s != 0 (val = %s))?????'
% (ell, this_is_zero, this_is_zero.valuation()))
if pp.derivative(derivative_order).subs({
x: R(aell)
}).valuation() >= 8: # DEBUG this value is arbitrary...
verbose('pp.derivative(derivative_order).subs({x:R(aell)}) = %s' %
pp.derivative().subs({x: R(aell)}))
verbose(
'... Skipping ell = %s because polynomial has repeated roots' %
ell)
continue
qq = pp.quo_rem((x - R(aell))**ZZ(derivative_order))[0]
break
qqT = qq(T).apply_map(lambda x: x.lift())
qq_aell = R(qq.subs({x: R(aell)}))
ord_basis_small = ord_basis.submatrix(0, 0, ncols=len(hord))
try:
ord_basis_small = ord_basis_small.apply_map(lambda x: x.lift())
except AttributeError:
pass
hord_in_katz = solve_xAb_echelon(ord_basis_small, hord)
qT_hord_in_katz = hord_in_katz * qqT
qT_hord = qT_hord_in_katz * ord_basis
return ell, (qq_aell.lift()**-1 *
qT_hord.apply_map(lambda x: x.lift())).change_ring(R)
The authors can be reached at: [email protected] and [email protected]. ==================================================================== """ from sage.all import QQ, prime_range from sage.arith.misc import next_prime, gcd from sage.rings.number_field.number_field import NumberField from sage.rings.polynomial.polynomial_ring import polygen x = polygen(QQ) K = NumberField(x**2 - 11 * 17 * 9011 * 23629, "b") test_cases_true = [(K, int(q), q, False) for q in prime_range(1000)] test_cases_false = [(K, int(q), next_prime(q), True) for q in prime_range(1000)] def test_warmup(): for K, q, p, result in test_cases_true + test_cases_false: qq = (q * K).factor()[0][0] assert qq.is_coprime(p) == (not qq.divides(p)) == result def test_is_coprime(): for K, q, p, result in test_cases_true + test_cases_false: qq = (q * K).factor()[0][0] is_coprime = qq.is_coprime(p) assert is_coprime == result
def Lpvalue(f,
g,
h,
p,
prec,
N=None,
modformsring=False,
weightbound=6,
eps=None,
orthogonal_form=None,
magma_args=None,
force_computation=False,
algorithm='twostage'):
if magma_args is None:
magma_args = {}
if algorithm not in ['twostage', 'threestage']:
raise ValueError(
'Algorithm should be one of "twostage" (default) or "threestage"')
from sage.interfaces.magma import Magma
magma = Magma(**magma_args)
ll, mm = g.weight(), h.weight()
t = 0 # Assume t = 0 here
kk = ll + mm - 2 * (1 + t) # Is this correct?
p = ZZ(p)
if N is None:
N = LCM([ZZ(f.level()), ZZ(g.level()), ZZ(h.level())])
N = N.prime_to_m_part(p)
print("Tame level N = %s, prime p = %s" % (N, p))
prec = ZZ(prec)
print("Step 1: Compute the Up matrix")
computation_name = '%s_%s_%s_%s_%s_%s' % (
p, N, kk, prec, 'triv' if eps is None else 'char', algorithm)
tmp_filename = '/tmp/magma_mtx_%s.tmp' % computation_name
import os.path
from sage.misc.persist import db, db_save
try:
if force_computation:
raise IOError
V = db('Lpvalue_Apow_ordbasis_eimat_%s' % computation_name)
ord_basis, eimat, zetapm, elldash, mdash = V[:5]
Apow_data = V[5:]
except IOError:
if force_computation or not os.path.exists(tmp_filename):
if eps is not None:
eps_magma = sage_character_to_magma(eps, magma=magma)
Am, zetapm, eimatm, elldash, mdash = magma.UpOperatorData(
p, eps_magma, kk, prec, nvals=5)
else:
Am, zetapm, eimatm, elldash, mdash = magma.UpOperatorData(
p, N, kk, prec, nvals=5)
print(" ..Converting to Sage...")
Amodulus = Am[1, 1].Parent().Modulus().sage()
Arows = Am.NumberOfRows().sage()
Acols = Am.NumberOfColumns().sage()
Emodulus = eimatm[1, 1].Parent().Modulus().sage()
Erows = eimatm.NumberOfRows().sage()
Ecols = eimatm.NumberOfColumns().sage()
magma.eval('F := Open("%s", "w");' % tmp_filename)
magma.eval('fprintf F, "Matrix(Zmod(%s),%s, %s, "' %
(Amodulus, Arows, Acols))
magma.eval('fprintf F, "%%o", ElementToSequence(%s)' % Am.name())
magma.eval('fprintf F, ") \\n"')
magma.eval('fprintf F, "Matrix(Zmod(%s),%s, %s, "' %
(Emodulus, Erows, Ecols))
magma.eval('fprintf F, "%%o", ElementToSequence(%s)' %
eimatm.name())
magma.eval('fprintf F, ") \\n"')
magma.eval('fprintf F, "%%o\\n", %s' % zetapm.name())
magma.eval('fprintf F, "%%o\\n", %s' % elldash.name())
magma.eval('fprintf F, "%%o\\n", %s' % mdash.name())
magma.eval('delete F;')
magma.quit()
# Read A and eimat from file
from sage.structure.sage_object import load
from sage.misc.sage_eval import sage_eval
with open(tmp_filename, 'r') as fmagma:
A = sage_eval(fmagma.readline(), preparse=False)
eimat = sage_eval(fmagma.readline(), preparse=False)
zetapm = sage_eval(fmagma.readline())
elldash = sage_eval(fmagma.readline())
mdash = sage_eval(fmagma.readline())
print("Step 3b: Apply Up^(r-1) to H")
if algorithm == 'twostage':
V0 = list(find_Apow_and_ord(A, eimat, p, prec))
else:
V0 = list(find_Apow_and_ord_three_stage(A, eimat, p, prec))
ord_basis = V0[0]
Apow_data = V0[1:]
V = [ord_basis]
V.extend([eimat, zetapm, elldash, mdash])
V.extend(Apow_data)
db_save(V, 'Lpvalue_Apow_ordbasis_eimat_%s' % computation_name)
from posix import remove
remove(tmp_filename)
print("Step 2: p-depletion, Coleman primitive, and multiply")
H = depletion_coleman_multiply(g, h, p, p * elldash, t=0)
print("Step 3a: Compute Up(H)")
UpH = vector([H(p * n) for n in range(elldash)])
if len(Apow_data) == 1:
Hord = compute_ordinary_projection_two_stage(UpH, Apow_data, eimat,
elldash)
else:
Hord = compute_ordinary_projection_three_stage(UpH,
[ord_basis] + Apow_data,
eimat, elldash)
Hord = Hord.change_ring(Apow_data[0].parent().base_ring())
print("Step 4: Project onto f-component")
R = Qp(p, prec)
if orthogonal_form is None:
ell, piHord = project_onto_eigenspace(f, ord_basis,
Hord.change_ring(R), kk, N * p,
eps)
n = 1
while f[n] == 0:
n += 1
Lpa = R(piHord[n]) / R(f[n])
else:
ell, piHord = project_onto_eigenspace(f,
ord_basis,
Hord.change_ring(R),
kk,
N * p,
eps,
derivative_order=2)
gplus, gminus = f, orthogonal_form
l1 = 2
while N * p * ell % l1 == 0 or gplus[l1] == 0:
l1 = next_prime(l1)
proj_mat = matrix([[gplus[l1], gplus[p]], [gminus[l1], gminus[p]]])
Lpalist = (matrix([piHord[l1], piHord[p]]) * proj_mat**-1).list()
Lpa = Lpalist[0]
if Lpa.valuation() > prec / 2: # this is quite arbitrary!
Lpa = Lpalist[1]
n = 1
while f[n] == 0:
n += 1
Lpa = Lpa / f[n]
return Lpa, ell