def irreps(p, q):
"""
Returns the irreducible representations of the cyclic group C_p
over the field F_q, where p and q are distinct primes.
Each representation is given by a matrix over F_q giving the
action of the preferred generator of C_p.
sage: [M.nrows() for M in irreps(3, 7)]
[1, 1, 1]
sage: [M.nrows() for M in irreps(7, 11)]
[1, 3, 3]
sage: sum(M.nrows() for M in irreps(157, 13))
157
"""
p, q = ZZ(p), ZZ(q)
assert p.is_prime() and q.is_prime() and p != q
R = PolynomialRing(GF(q), 'x')
x = R.gen()
polys = [f for f, e in (x**p - 1).factor()]
polys.sort(key=lambda f: (f.degree(), -f.constant_coefficient()))
reps = [poly_to_rep(f) for f in polys]
assert all(A**p == 1 for A in reps)
assert reps[0] == 1
return reps
def get_AB_primes(q, epsilons, q_class_group_order):
output_dict_AB = {}
alphas = (q ** q_class_group_order).gens_reduced()
assert len(alphas) == 1, "q^q_class_group_order not principal, which is very bad"
alpha = alphas[0]
rat_q = ZZ(q.norm())
assert rat_q.is_prime(), "somehow the degree 1 prime is not prime"
for eps in epsilons:
alpha_to_eps = group_ring_exp(alpha, eps)
A = (alpha_to_eps - 1).norm()
B = (alpha_to_eps - (rat_q ** (12 * q_class_group_order))).norm()
output_dict_AB[eps] = lcm(A, B)
return output_dict_AB
def includes_composite(s):
s = s.replace(' ', '').replace('..', '-')
for interval in s.split(','):
if '-' in interval[1:]:
ix = interval.index('-', 1)
a, b = int(interval[:ix]), int(interval[ix + 1:])
if b == a:
if a != 1 and not a.is_prime():
return True
if b > a and b > 3:
return True
else:
a = ZZ(interval)
if a != 1 and not a.is_prime():
return True
def includes_composite(s):
s = s.replace(" ", "").replace("..", "-")
for interval in s.split(","):
if "-" in interval[1:]:
ix = interval.index("-", 1)
a, b = int(interval[:ix]), int(interval[ix + 1 :])
if b == a:
if a != 1 and not a.is_prime():
return True
if b > a and b > 3:
return True
else:
a = ZZ(interval)
if a != 1 and not a.is_prime():
return True
def includes_composite(s):
s = s.replace(' ','').replace('..','-')
for interval in s.split(','):
if '-' in interval[1:]:
ix = interval.index('-',1)
a,b = int(interval[:ix]), int(interval[ix+1:])
if b == a:
if a != 1 and not a.is_prime():
return True
if b > a and b > 3:
return True
else:
a = ZZ(interval)
if a != 1 and not a.is_prime():
return True
def get_the_lcm(C_K, gen_list):
K = C_K.number_field()
epsilons = {(6, 6): "type-2"}
running_lcm = 1
for q in gen_list:
q_class_group_order = C_K(q).multiplicative_order()
AB_lcm = get_AB_primes(q, epsilons, q_class_group_order)[(6, 6)]
C_o = get_C_primes(K, q, epsilons, q_class_group_order, ordinary=True)[(6, 6)]
rat_q = ZZ(q.norm())
assert (
rat_q.is_prime()
), "Somehow there is a split prime ideal whose norm is not prime!"
running_lcm = lcm([running_lcm, AB_lcm, C_o, rat_q])
return running_lcm
def find_curve(P, DB, NE, prec, sign_ap = None, magma = None, return_all = False, initial_data = None, ramification_at_infinity = None, **kwargs):
r'''
EXAMPLES:
First example::
sage: from darmonpoints.findcurve import find_curve
sage: find_curve(5,6,30,20) # long time # optional - magma
# B = F<i,j,k>, with i^2 = -1 and j^2 = 3
...
'(1, 0, 1, -289, 1862)'
A second example, now over a real quadratic::
sage: from darmonpoints.findcurve import find_curve
sage: F.<r> = QuadraticField(5)
sage: P = F.ideal(3/2*r + 1/2)
sage: D = F.ideal(3)
sage: find_curve(P,D,P*D,30,ramification_at_infinity = F.real_places()[:1]) # long time # optional - magma
...
Now over a cubic of mixed signature::
sage: from darmonpoints.findcurve import find_curve
sage: F.<r> = NumberField(x^3 -3)
sage: P = F.ideal(r-2)
sage: D = F.ideal(r-1)
sage: find_curve(P,D,P*D,30) # long time # optional - magma
...
'''
config = ConfigParser.ConfigParser()
config.read('config.ini')
param_dict = config_section_map(config, 'General')
param_dict.update(config_section_map(config, 'FindCurve'))
param_dict.update(kwargs)
param = Bunch(**param_dict)
# Get general parameters
outfile = param.get('outfile')
use_ps_dists = param.get('use_ps_dists',False)
use_shapiro = param.get('use_shapiro',True)
use_sage_db = param.get('use_sage_db',False)
magma_seed = param.get('magma_seed',1515316)
parallelize = param.get('parallelize',False)
Up_method = param.get('up_method','naive')
use_magma = param.get('use_magma',True)
progress_bar = param.get('progress_bar',True)
sign_at_infinity = param.get('sign_at_infinity',ZZ(1))
# Get find_curve specific parameters
grouptype = param.get('grouptype')
hecke_bound = param.get('hecke_bound',3)
timeout = param.get('timeout',0)
check_conductor = param.get('check_conductor',True)
if initial_data is None:
page_path = os.path.dirname(__file__) + '/KleinianGroups-1.0/klngpspec'
if magma is None:
from sage.interfaces.magma import Magma
quit_when_done = True
magma = Magma()
else:
quit_when_done = False
if magma_seed is not None:
magma.eval('SetSeed(%s)'%magma_seed)
magma.attach_spec(page_path)
magma.eval('Page_initialized := true')
else:
quit_when_done = False
sys.setrecursionlimit(10**6)
# global qE, Linv, G, Coh, phiE, xgen, xi1, xi2, Phi
try:
F = P.ring()
Fdisc = F.discriminant()
if not (P*DB).divides(NE):
raise ValueError,'Conductor (NE) should be divisible by P*DB'
p = ZZ(P.norm()).abs()
except AttributeError:
F = QQ
P = ZZ(P)
p = ZZ(P)
Fdisc = ZZ(1)
if NE % (P*DB) != 0:
raise ValueError,'Conductor (NE) should be divisible by P*DB'
Ncartan = kwargs.get('Ncartan',None)
Np = NE / (P * DB)
if Ncartan is not None:
Np = Np / Ncartan**2
if use_ps_dists is None:
use_ps_dists = False # More efficient our own implementation
if not p.is_prime():
raise ValueError,'P (= %s) should be a prime, of inertia degree 1'%P
working_prec = max([2 * prec + 10, 100])
sgninfty = 'plus' if sign_at_infinity == 1 else 'minus'
fname = 'moments_%s_%s_%s_%s_%s_%s.sobj'%(Fdisc,p,DB,NE,sgninfty,prec)
if outfile == 'log':
outfile = '%s_%s_%s_%s_%s.log'%(P,NE,sgninfty,prec,datetime.datetime.now().strftime("%Y%m%d-%H%M%S"))
outfile = outfile.replace('/','div')
outfile = '/tmp/findcurve_' + outfile
if F != QQ and ramification_at_infinity is None:
if F.signature()[0] > 1:
if F.signature()[1] == 1:
ramification_at_infinity = F.real_places(prec = Infinity) # Totally 'definite'
else:
raise ValueError,'Please specify the ramification at infinity'
elif F.signature()[0] == 1:
if len(F.ideal(DB).factor()) % 2 == 0:
ramification_at_infinity = [] # Split at infinity
else:
ramification_at_infinity = F.real_places(prec = Infinity) # Ramified at infinity
else:
ramification_at_infinity = None
if outfile is not None:
print("Partial results will be saved in %s"%outfile)
if initial_data is not None:
G,phiE = initial_data
else:
# Define the S-arithmetic group
try:
if F == QQ:
abtuple = QuaternionAlgebra(DB).invariants()
else:
abtuple = quaternion_algebra_invariants_from_ramification(F,DB,ramification_at_infinity)
G = BigArithGroup(P, abtuple, Np, use_sage_db = use_sage_db, grouptype = grouptype, magma = magma, seed = magma_seed, timeout = timeout, use_shapiro = use_shapiro, nscartan = Ncartan)
except RuntimeError as e:
if quit_when_done:
magma.quit()
mystr = str(e)
if len(mystr) > 30:
mystr = mystr[:14] + ' ... ' + mystr[-14:]
if return_all:
return ['Error when computing G: ' + mystr]
else:
return 'Error when computing G: ' + mystr
# Define phiE, the cohomology class associated to the system of eigenvalues.
Coh = ArithCoh(G)
try:
phiE = Coh.get_rational_cocycle(sign = sign_at_infinity,bound = hecke_bound,return_all = return_all,use_magma = True)
except Exception as e:
if quit_when_done:
magma.quit()
if return_all:
return ['Error when finding cohomology class: ' + str(e)]
else:
return 'Error when finding cohomology class: ' + str(e)
if use_sage_db:
G.save_to_db()
fwrite('Cohomology class found', outfile)
try:
ker = [G.inverse_shapiro(o) for o in G.get_homology_kernel()]
except Exception as e:
if quit_when_done:
magma.quit()
if return_all:
return ['Problem calculating homology kernel: ' + str(e)]
else:
return 'Problem calculating homology kernel: ' + str(e)
if not return_all:
phiE = [phiE]
ret_vals = []
for phi in phiE:
try:
Phi = get_overconvergent_class_quaternionic(P,phi,G,prec,sign_at_infinity,sign_ap,use_ps_dists,method = Up_method, progress_bar = progress_bar)
except ValueError as e:
ret_vals.append('Problem when getting overconvergent class: ' + str(e))
continue
fwrite('Done overconvergent lift', outfile)
# Find an element x of Gpn for not in the kernel of phi,
# and such that both x and wp^-1 * x * wp are trivial in the abelianization of Gn.
try:
found = False
for o in ker:
phi_o = sum( [phi.evaluate(t**a) for t, a in o], 0 )
if use_shapiro:
phi_o = phi_o.evaluate_at_identity()
if phi_o != 0:
found = True
break
if not found:
raise RuntimeError('Cocycle evaluates always to zero')
except Exception as e:
ret_vals.append('Problem when choosing element in kernel: ' + str(e))
continue
xgenlist = o
found = False
while not found:
try:
xi1, xi2 = lattice_homology_cycle(G,xgenlist,working_prec,outfile = outfile)
found = True
except PrecisionError:
working_prec = 2 * working_prec
verbose('Setting working_prec to %s'%working_prec)
except Exception as e:
ret_vals.append('Problem when computing homology cycle: ' + str(e))
verbose('Exception occurred: ' + str(e))
break
if not found:
continue
try:
qE1 = integrate_H1(G,xi1,Phi,1,method = 'moments',prec = working_prec, twist = False,progress_bar = progress_bar)
qE2 = integrate_H1(G,xi2,Phi,1,method = 'moments',prec = working_prec, twist = True,progress_bar = progress_bar)
except Exception as e:
ret_vals.append('Problem with integration: %s'%str(e))
continue
qE = qE1/qE2
qE = qE.add_bigoh(prec + qE.valuation())
Linv = qE.log(p_branch = 0)/qE.valuation()
fwrite('Integral done. Now trying to recognize the curve', outfile)
fwrite('F.<r> = NumberField(%s)'%(F.gen(0).minpoly()),outfile)
fwrite('N_E = %s = %s'%(NE,factor(NE)),outfile)
fwrite('D_B = %s = %s'%(DB,factor(DB)),outfile)
fwrite('Np = %s = %s'%(Np,factor(Np)),outfile)
if Ncartan is not None:
fwrite('Ncartan = %s'%(Ncartan),outfile)
fwrite('Calculation with p = %s and prec = %s+%s'%(P,prec,working_prec-prec),outfile)
fwrite('qE = %s'%qE,outfile)
fwrite('Linv = %s'%Linv,outfile)
curve = discover_equation(qE,G._F_to_local,NE,prec,check_conductor = check_conductor)
if curve is None:
if quit_when_done:
magma.quit()
ret_vals.append('None')
else:
try:
curve = curve.global_minimal_model()
except AttributeError,NotImplementedError:
pass
fwrite('EllipticCurve(F, %s )'%(list(curve.a_invariants())), outfile)
fwrite('=' * 60, outfile)
ret_vals.append(str(curve.a_invariants()))
def find_curve(P,
DB,
NE,
prec,
sign_ap=None,
magma=None,
return_all=False,
initial_data=None,
ramification_at_infinity=None,
implementation=None,
**kwargs):
r'''
EXAMPLES:
First example::
sage: from darmonpoints.findcurve import find_curve
sage: find_curve(5,6,30,20) # long time # optional - magma
# B = F<i,j,k>, with i^2 = -1 and j^2 = 3
...
'(1, 0, 1, -289, 1862)'
A second example, now over a real quadratic::
sage: from darmonpoints.findcurve import find_curve
sage: F.<r> = QuadraticField(5)
sage: P = F.ideal(3/2*r + 1/2)
sage: D = F.ideal(3)
sage: find_curve(P,D,P*D,30,ramification_at_infinity = F.real_places()[:1]) # long time # optional - magma
...
Now over a cubic of mixed signature::
sage: from darmonpoints.findcurve import find_curve
sage: F.<r> = NumberField(x^3 -3)
sage: P = F.ideal(r-2)
sage: D = F.ideal(r-1)
sage: find_curve(P,D,P*D,30) # long time # optional - magma
...
'''
config = ConfigParser.ConfigParser()
config.read('config.ini')
param_dict = config_section_map(config, 'General')
param_dict.update(config_section_map(config, 'FindCurve'))
param_dict.update(kwargs)
param = Bunch(**param_dict)
# Get general parameters
outfile = param.get('outfile')
use_ps_dists = param.get('use_ps_dists', False)
use_shapiro = param.get('use_shapiro', False)
use_sage_db = param.get('use_sage_db', False)
magma_seed = param.get('magma_seed', 1515316)
parallelize = param.get('parallelize', False)
Up_method = param.get('up_method', 'naive')
use_magma = param.get('use_magma', True)
progress_bar = param.get('progress_bar', True)
sign_at_infinity = param.get('sign_at_infinity', ZZ(1))
# Get find_curve specific parameters
grouptype = param.get('grouptype')
hecke_bound = param.get('hecke_bound', 3)
timeout = param.get('timeout', 0)
check_conductor = param.get('check_conductor', True)
if initial_data is None:
page_path = os.path.dirname(__file__) + '/KleinianGroups-1.0/klngpspec'
if magma is None:
from sage.interfaces.magma import Magma
quit_when_done = True
magma = Magma()
else:
quit_when_done = False
if magma_seed is not None:
magma.eval('SetSeed(%s)' % magma_seed)
magma.attach_spec(page_path)
magma.eval('Page_initialized := true')
else:
quit_when_done = False
sys.setrecursionlimit(10**6)
# global qE, Linv, G, Coh, phiE, xgen, xi1, xi2, Phi
try:
F = P.ring()
Fdisc = F.discriminant()
if not (P * DB).divides(NE):
raise ValueError('Conductor (NE) should be divisible by P*DB')
p = ZZ(P.norm()).abs()
except AttributeError:
F = QQ
P = ZZ(P)
p = ZZ(P)
Fdisc = ZZ(1)
if NE % (P * DB) != 0:
raise ValueError('Conductor (NE) should be divisible by P*DB')
Ncartan = kwargs.get('Ncartan', None)
param_dict['nscartan'] = Ncartan
Np = NE / (P * DB)
if Ncartan is not None:
Np = Np / Ncartan**2
if use_ps_dists is None:
use_ps_dists = False # More efficient our own implementation
if not p.is_prime():
raise ValueError('P (= %s) should be a prime, of inertia degree 1' % P)
working_prec = max([2 * prec + 10, 100])
sgninfty = 'plus' if sign_at_infinity == 1 else 'minus'
fname = 'moments_%s_%s_%s_%s_%s_%s.sobj' % (Fdisc, p, DB, NE, sgninfty,
prec)
if outfile == 'log':
outfile = '%s_%s_%s_%s_%s.log' % (
P, NE, sgninfty, prec,
datetime.datetime.now().strftime("%Y%m%d-%H%M%S"))
outfile = outfile.replace('/', 'div')
outfile = '/tmp/findcurve_' + outfile
if F != QQ and ramification_at_infinity is None:
if F.signature()[0] > 1:
if F.signature()[1] == 1:
ramification_at_infinity = F.real_places(
prec=Infinity) # Totally 'definite'
else:
raise ValueError('Please specify the ramification at infinity')
elif F.signature()[0] == 1:
if len(F.ideal(DB).factor()) % 2 == 0:
ramification_at_infinity = [] # Split at infinity
else:
ramification_at_infinity = F.real_places(
prec=Infinity) # Ramified at infinity
else:
ramification_at_infinity = None
if outfile is not None:
print("Partial results will be saved in %s" % outfile)
if initial_data is not None:
G, phiE = initial_data
else:
# Define the S-arithmetic group
try:
if F == QQ:
abtuple = QuaternionAlgebra(DB).invariants()
else:
abtuple = quaternion_algebra_invariants_from_ramification(
F, DB, ramification_at_infinity, magma=magma)
G = BigArithGroup(P,
abtuple,
Np,
magma=magma,
seed=magma_seed,
**param_dict)
except RuntimeError as e:
if quit_when_done:
magma.quit()
mystr = str(e)
if len(mystr) > 30:
mystr = mystr[:14] + ' ... ' + mystr[-14:]
if return_all:
return ['Error when computing G: ' + mystr]
else:
return 'Error when computing G: ' + mystr
# Define phiE, the cohomology class associated to the system of eigenvalues.
Coh = ArithCoh(G)
try:
phiE = Coh.get_rational_cocycle(sign=sign_at_infinity,
bound=hecke_bound,
return_all=return_all,
use_magma=True)
except Exception as e:
if quit_when_done:
magma.quit()
if return_all:
return ['Error when finding cohomology class: ' + str(e)]
else:
return 'Error when finding cohomology class: ' + str(e)
if use_sage_db:
G.save_to_db()
fwrite('Cohomology class found', outfile)
try:
ker = [G.inverse_shapiro(o) for o in G.get_homology_kernel()]
except Exception as e:
if quit_when_done:
magma.quit()
if return_all:
return ['Problem calculating homology kernel: ' + str(e)]
else:
return 'Problem calculating homology kernel: ' + str(e)
if not return_all:
phiE = [phiE]
ret_vals = []
for phi in phiE:
try:
Phi = get_overconvergent_class_quaternionic(
P,
phi,
G,
prec,
sign_at_infinity,
sign_ap,
use_ps_dists,
method=Up_method,
progress_bar=progress_bar)
except ValueError as e:
ret_vals.append('Problem when getting overconvergent class: ' +
str(e))
continue
fwrite('Done overconvergent lift', outfile)
# Find an element x of Gpn for not in the kernel of phi,
# and such that both x and wp^-1 * x * wp are trivial in the abelianization of Gn.
try:
found = False
for o in ker:
phi_o = sum([phi.evaluate(t**a) for t, a in o], 0)
if use_shapiro:
phi_o = phi_o.evaluate_at_identity()
if phi_o != 0:
found = True
break
if not found:
raise RuntimeError('Cocycle evaluates always to zero')
except Exception as e:
ret_vals.append('Problem when choosing element in kernel: ' +
str(e))
continue
xgenlist = o
found = False
while not found:
try:
xi1, xi2 = lattice_homology_cycle(p,
G.Gn,
G.wp(),
xgenlist,
working_prec,
outfile=outfile)
found = True
except PrecisionError:
working_prec = 2 * working_prec
verbose('Setting working_prec to %s' % working_prec)
except Exception as e:
ret_vals.append('Problem when computing homology cycle: ' +
str(e))
verbose('Exception occurred: ' + str(e))
break
if not found:
continue
try:
qE1 = integrate_H1(G,
xi1,
Phi,
1,
method='moments',
prec=working_prec,
twist=False,
progress_bar=progress_bar)
qE2 = integrate_H1(G,
xi2,
Phi,
1,
method='moments',
prec=working_prec,
twist=True,
progress_bar=progress_bar)
except Exception as e:
ret_vals.append('Problem with integration: %s' % str(e))
continue
qE = qE1 / qE2
qE = qE.add_bigoh(prec + qE.valuation())
Linv = qE.log(p_branch=0) / qE.valuation()
fwrite('Integral done. Now trying to recognize the curve', outfile)
fwrite('F.<r> = NumberField(%s)' % (F.gen(0).minpoly()), outfile)
fwrite('N_E = %s = %s' % (NE, factor(NE)), outfile)
fwrite('D_B = %s = %s' % (DB, factor(DB)), outfile)
fwrite('Np = %s = %s' % (Np, factor(Np)), outfile)
if Ncartan is not None:
fwrite('Ncartan = %s' % (Ncartan), outfile)
fwrite(
'Calculation with p = %s and prec = %s+%s' %
(P, prec, working_prec - prec), outfile)
fwrite('qE = %s' % qE, outfile)
fwrite('Linv = %s' % Linv, outfile)
curve = discover_equation(qE,
G._F_to_local,
NE,
prec,
check_conductor=check_conductor)
if curve is None:
if quit_when_done:
magma.quit()
ret_vals.append('None')
else:
try:
curve = curve.global_minimal_model()
except AttributeError, NotImplementedError:
pass
fwrite('EllipticCurve(F, %s )' % (list(curve.a_invariants())),
outfile)
fwrite('=' * 60, outfile)
ret_vals.append(str(curve.a_invariants()))
