def get_magma_qexpansions(filename, i1, prec, base_ring, magma=None):
if magma is None:
from sage.interfaces.magma import Magma
magma = Magma()
magma.set('prec',prec)
magma.load("get_qexpansions.m")
if i1 is None:
for line in filename:
print line
magma.eval(line)
magma.eval("f := g") # In case text is reversed
f = 'f'
eps_data_f = 'eps_on_gens'
else:
magma.load(filename)
f = 'eigenforms_list[%s][1]'%i1
eps_data_f = 'eigenforms_list[%s][2]'%i1
qexpm = magma.extend_qexpansion(f, eps_data_f, prec)
F0 = [0] * qexpm.Valuation().sage() + [o.sage() for o in qexpm.ElementToSequence()]
K = F0[-1].parent()
a = K.gen()
phi = find_embeddings(K,base_ring)[0]
F = [phi(o) for o in F0]
eps_f = magma.get_character(f, eps_data_f).ValueList().sage()
eps_f_full = magma.get_character_full(f, eps_data_f).ValueList().sage()
N = len(eps_f)
Geps = DirichletGroup(N, base_ring = K)
eps_f = Geps([eps_f[i - 1] for i in Geps.unit_gens()])
Geps_full = DirichletGroup(N)
psi = eps_f_full[0].parent().embeddings(Geps_full.base_ring())[0]
eps_f_full = Geps_full([psi(eps_f_full[i - 1]) for i in Geps_full.unit_gens()])
if a == 1:
sigma = lambda x:x
else:
try:
sigma = next((s for s in K.automorphisms() if s(a)*a == 1))
except StopIteration:
raise NotImplementedError
G = [phi(sigma(o)) for o in F0]
eps_g = eps_f**-1
eps_g_full = eps_f_full**-1
F = ModFormqExp(F, base_ring, weight=1, level = N, character = eps_f, character_full = eps_f_full)
G = ModFormqExp(G, base_ring, weight=1, level = N, character = eps_g, character_full = eps_g_full)
return F, G
def Lpvalue(f,g,h,p,prec,N = None,modformsring = False, weightbound = False, eps = None, orthogonal_form = None, data_idx=None, magma_args = None,force_computation=False, algorithm='threestage', derivative_order=1, lauders_advice = False, use_magma = True, magma = None, num_coeffs_qexpansion = 20000, max_primes=5, outfile = None):
if magma_args is None:
magma_args = {}
if algorithm not in ['twostage','threestage']:
raise ValueError('Algorithm should be one of "twostage" (default) or "threestage"')
if magma is None:
from sage.interfaces.magma import Magma
magma = Magma(**magma_args)
if hasattr(g,'j_invariant'):
elliptic_curve = g
g = g.modular_form()
else:
elliptic_curve = None
data = None
if h is None:
if hasattr(f, 'modulus'):
# Assume we need to create f and h from Dirichlet character
kronecker_character = f
f, _, h = define_qexpansions_from_dirichlet_character(p, prec, kronecker_character, num_coeffs_qexpansion, magma)
else:
kronecker_character = None
# Assume that f contains a list of lines of text to initialize both f and h
data = f
f, h = get_magma_qexpansions(data, data_idx, max(prec,200), Qp(p,prec), magma=magma)
eps = f.character_full()
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())])
nu = N.valuation(p)
N = N.prime_to_m_part(p)
else:
N = ZZ(N)
nu = N.valuation(p)
if outfile is None:
outfile = "output_iterated_integral_%s_%s_%s_%s.txt"%(p,g.level(), h.level(), prec)
print("Writing output to file %s"%outfile)
fwrite("######### STARTING COMPUTATION OF Lp ###########", outfile)
if elliptic_curve is not None:
fwrite("E = EllipticCurve(%s)"%list(elliptic_curve.ainvs()), outfile)
fwrite(" cond(E) = %s"%elliptic_curve.conductor(), outfile)
if kronecker_character is not None:
fwrite("kronecker_character = %s"%kronecker_character, outfile)
fwrite(" conductor = %s"%kronecker_character.conductor(), outfile)
if data is not None:
fwrite("Data for weight-1 forms:", outfile)
for line in data:
fwrite(line, outfile)
fwrite("Tame level N = %s, prime p = %s, nu = %s"%(N,p,nu), outfile)
fwrite("precision = %s"%prec, outfile)
fwrite("------ parameters --------------------", outfile)
fwrite("modformsring = %s"%modformsring, outfile)
fwrite("weightbound = %s"%weightbound, outfile)
fwrite("eps = %s"%eps, outfile)
fwrite("orthogonal_form = %s"%orthogonal_form, outfile)
fwrite("magma_args = %s"%magma_args, outfile)
fwrite("force_computation = %s"%force_computation, outfile)
fwrite("algorithm = %s"%algorithm, outfile)
fwrite("derivative_order = %s"%derivative_order, outfile)
fwrite("lauders_advice = %s"%lauders_advice, outfile)
fwrite("use_magma = %s"%use_magma, outfile)
fwrite("num_coeffs_qexpansion = %s"%num_coeffs_qexpansion, outfile)
fwrite("##########################################", outfile)
prec = ZZ(prec)
fwrite("Step 1: Compute the Up matrix", outfile)
if algorithm == "twostage":
computation_name = '%s_%s_%s_%s_%s_%s_%s'%(p,N,nu,kk,prec,'triv' if eps is None else 'char',algorithm)
else:
computation_name = '%s_%s_%s_%s_%s_%s'%(p,N,nu,kk,prec,'triv' if eps is None else 'char')
if use_magma:
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,N,magma=magma)
magma.load("overconvergent_alan.m")
# Am, zetapm, eimatm, elldash, mdash = magma.UpOperatorData(p, eps_magma, kk, prec,WeightBound=weightbound,nvals=5)
Am, zetapm, eimatm, elldash, mdash = magma.HigherLevelUpGj(p, kk, prec, weightbound, eps_magma,'"B"',nvals=5)
else:
# Am, zetapm, eimatm, elldash, mdash = magma.UpOperatorData(p, N, kk, prec,WeightBound=weightbound,nvals=5)
magma.load("overconvergent_alan.m")
Am, zetapm, eimatm, elldash, mdash = magma.HigherLevelUpGj(p, kk, prec, weightbound, N,'"B"',nvals=5)
fwrite(" ..Converting to Sage...", outfile)
Amodulus = Am[1,1].Parent().Modulus().sage()
Aprec = Amodulus.valuation(p)
Arows = Am.NumberOfRows().sage()
Acols = Am.NumberOfColumns().sage()
Emodulus = eimatm[1,1].Parent().Modulus().sage()
Eprec = Emodulus.valuation(p)
Erows = eimatm.NumberOfRows().sage()
Ecols = eimatm.NumberOfColumns().sage()
magma.load("get_qexpansions.m")
magma.eval('F := Open("%s", "w");'%tmp_filename)
magma.eval('fprintf F, "%s, %s, %s, %s \\n"'%(p,Aprec,Arows,Acols)) # parameters
magma.eval('save_matrix(%s, F)'%(Am.name()))
# for i in range(1,Arows+1):
# magma.eval('fprintf F, "%%o\\n", %s[%s]'%(Am.name(),i))
magma.eval('fprintf F, "%s, %s, %s, %s \\n"'%(p,Eprec,Erows,Ecols)) # parameters
magma.eval('save_matrix(%s, F)'%(eimatm.name()))
# for i in range(1,Erows+1):
# magma.eval('fprintf F, "%%o\\n", %s[%s]'%(eimatm.name(),i))
magma.eval('fprintf F, "%s\\n"'%zetapm)
magma.eval('fprintf F, "%s\\n"'%elldash)
magma.eval('fprintf F, "%s\\n"'%mdash)
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 = read_matrix_from_file(fmagma)
eimat = read_matrix_from_file(fmagma)
zetapm= sage_eval(fmagma.readline())
elldash = sage_eval(fmagma.readline())
mdash = sage_eval(fmagma.readline())
fwrite("Step 3b: Apply Up^(r-1) to H", outfile)
if algorithm == 'twostage':
V0 = list(find_Apow_and_ord_two_stage(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)
else:
A, eimat, elldash, mdash = UpOperator(p,N,kk,prec, modformsring = False, weightbound = 6)
fwrite("Step 2: p-depletion, Coleman primitive, and multiply", outfile)
fwrite(".. Need %s coefficients of the q-expansion..."%(p**(nu+1) * elldash), outfile)
if data is not None:
f, h = get_magma_qexpansions(data, data_idx, (p**(nu+1) * elldash) + 200, Qp(p,prec), magma=magma)
H = depletion_coleman_multiply(g, h, p, p**(nu+1) * elldash, t=0)
fwrite("Step 3a: Compute ordinary projection", outfile)
if len(Apow_data) == 1:
Hord = compute_ordinary_projection_two_stage(H, Apow_data, eimat, elldash,p)
else:
Hord = compute_ordinary_projection_three_stage(H, [ord_basis] + Apow_data, eimat, elldash,p,nu)
fwrite('Changing Hord to ring %s'%g[1].parent(), outfile)
Hord = Hord.change_ring(h[1].parent())
print [Hord[i] for i in range(30)]
fwrite("Step 4: Project onto f-component", outfile)
while True:
try:
ell, piHord, epstwist = project_onto_eigenspace(f, ord_basis, Hord, kk, N * p, p = p, derivative_order=derivative_order, max_primes=max_primes)
break
except RuntimeError:
derivative_order += 1
verbose("Increasing experimental derivative order to %s"%derivative_order)
except ValueError:
verbose("Experimental derivative order (%s) seems too high"%derivative_order)
fwrite("Experimental derivative_order = %s"%derivative_order, outfile)
fwrite("Seems too high...", outfile)
fwrite("######################################", outfile)
assert 0
n = 1
while f[n] == 0:
n = next_prime(n)
if lauders_advice == True or orthogonal_form is None:
Lpa = piHord[n] / (f[n] * epstwist(n))
fwrite("Experimental derivative_order = %s"%derivative_order, outfile)
fwrite("Checking Lauder's coincidence... (following should be a bunch of 'large' valuations)", outfile)
fwrite(str([(i,(Lpa * f[i] * epstwist(i) - piHord[i]).valuation(p)) for i in prime_range(50)]), outfile)
fwrite("Done", outfile)
else:
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]
Lpa = Lpa / f[n]
fwrite("ell = %s"%ell, outfile)
fwrite("######### FINISHED COMPUTATION ###########", outfile)
fwrite("Lp = %s"%Lpa, outfile)
fwrite("##########################################", outfile)
return Lpa, ell