def anlist_over_sqrt5(E, bound):
"""
Compute the Dirichlet L-series coefficients, up to and including
a_bound. The i-th element of the return list is a[i].
INPUT:
- E -- elliptic curve over Q(sqrt(5)), which must have
defining polynomial `x^2-x-1`.
- ``bound`` -- integer
OUTPUT:
- list of integers of length bound + 1
EXAMPLES::
sage: from psage.ellcurve.lseries.lseries_nf import anlist_over_sqrt5
sage: K.<a> = NumberField(x^2-x-1); E = EllipticCurve([0,-a,a,0,0])
sage: v = anlist_over_sqrt5(E, 50); v
[0, 1, 0, 0, -2, -1, 0, 0, 0, -4, 0, 3, 0, 0, 0, 0, 0, 0, 0, 5, 2, 0, 0, 0, 0, -4, 0, 0, 0, 11, 0, -6, 0, 0, 0, 0, 8, 0, 0, 0, 0, -1, 0, 0, -6, 4, 0, 0, 0, -6, 0]
sage: len(v)
51
This function isn't super fast, but at least it will work in a few
seconds up to `10^4`::
sage: t = cputime()
sage: v = anlist_over_sqrt5(E, 10^4)
sage: assert cputime(t) < 5
"""
from . import aplist_sqrt5
from psage.number_fields.sqrt5.prime import primes_of_bounded_norm, Prime
# Compute all of the prime ideals of the ring of integers up to the given bound
primes = primes_of_bounded_norm(bound + 1)
# Compute the traces of Frobenius: this is supposed to be the hard part
v = aplist_sqrt5.aplist(E, bound + 1)
# Compute information about the primes of bad reduction, in
# particular the integers i such that primes[i] is a prime of bad
# reduction.
bad_primes = set([Prime(a.prime()) for a in E.local_data()])
# We compute the local factors of the L-series as power series in ZZ[T].
P = PowerSeriesRing(ZZ, 'T')
T = P.gen()
# Table of powers of T, so we don't have to compute T^4 (say) thousands of times.
Tp = [T**i for i in range(5)]
# For each prime, we write down the local factor.
L_P = []
for i, P in enumerate(primes):
inertial_deg = 2 if P.is_inert() else 1
a_p = v[i]
if P in bad_primes:
# bad reduction
f = 1 - a_p * Tp[inertial_deg]
else:
# good reduction
q = P.norm()
f = 1 - a_p * Tp[inertial_deg] + q * Tp[2 * inertial_deg]
L_P.append(f)
# Use the local factors of the L-series to compute the Dirichlet
# series coefficients of prime-power index.
coefficients = [0, 1] + [0] * (bound - 1)
i = 0
while i < len(primes):
P = primes[i]
if P.is_split():
s = L_P[i] * L_P[i + 1]
i += 2
else:
s = L_P[i]
i += 1
p = P.p
# We need enough terms t so that p^t > bound
accuracy_p = int(math.floor(old_div(math.log(bound), math.log(p)))) + 1
series_p = s.add_bigoh(accuracy_p)**(-1)
for j in range(1, accuracy_p):
coefficients[p**j] = series_p[j]
# Using multiplicativity, fill in the non-prime power Dirichlet
# series coefficients.
extend_multiplicatively_generic(coefficients)
return coefficients
def anlist_over_sqrt5(E, bound):
"""
Compute the Dirichlet L-series coefficients, up to and including
a_bound. The i-th element of the return list is a[i].
INPUT:
- E -- elliptic curve over Q(sqrt(5)), which must have
defining polynomial `x^2-x-1`.
- ``bound`` -- integer
OUTPUT:
- list of integers of length bound + 1
EXAMPLES::
sage: from psage.ellcurve.lseries.lseries_nf import anlist_over_sqrt5
sage: K.<a> = NumberField(x^2-x-1); E = EllipticCurve([0,-a,a,0,0])
sage: v = anlist_over_sqrt5(E, 50); v
[0, 1, 0, 0, -2, -1, 0, 0, 0, -4, 0, 3, 0, 0, 0, 0, 0, 0, 0, 5, 2, 0, 0, 0, 0, -4, 0, 0, 0, 11, 0, -6, 0, 0, 0, 0, 8, 0, 0, 0, 0, -1, 0, 0, -6, 4, 0, 0, 0, -6, 0]
sage: len(v)
51
This function isn't super fast, but at least it will work in a few
seconds up to `10^4`::
sage: t = cputime()
sage: v = anlist_over_sqrt5(E, 10^4)
sage: assert cputime(t) < 5
"""
import aplist_sqrt5
from psage.number_fields.sqrt5.prime import primes_of_bounded_norm, Prime
# Compute all of the prime ideals of the ring of integers up to the given bound
primes = primes_of_bounded_norm(bound+1)
# Compute the traces of Frobenius: this is supposed to be the hard part
v = aplist_sqrt5.aplist(E, bound+1)
# Compute information about the primes of bad reduction, in
# particular the integers i such that primes[i] is a prime of bad
# reduction.
bad_primes = set([Prime(a.prime()) for a in E.local_data()])
# We compute the local factors of the L-series as power series in ZZ[T].
P = PowerSeriesRing(ZZ, 'T')
T = P.gen()
# Table of powers of T, so we don't have to compute T^4 (say) thousands of times.
Tp = [T**i for i in range(5)]
# For each prime, we write down the local factor.
L_P = []
for i, P in enumerate(primes):
inertial_deg = 2 if P.is_inert() else 1
a_p = v[i]
if P in bad_primes:
# bad reduction
f = 1 - a_p*Tp[inertial_deg]
else:
# good reduction
q = P.norm()
f = 1 - a_p*Tp[inertial_deg] + q*Tp[2*inertial_deg]
L_P.append(f)
# Use the local factors of the L-series to compute the Dirichlet
# series coefficients of prime-power index.
coefficients = [0,1] + [0]*(bound-1)
i = 0
while i < len(primes):
P = primes[i]
if P.is_split():
s = L_P[i] * L_P[i+1]
i += 2
else:
s = L_P[i]
i += 1
p = P.p
# We need enough terms t so that p^t > bound
accuracy_p = int(math.floor(math.log(bound)/math.log(p))) + 1
series_p = s.add_bigoh(accuracy_p)**(-1)
for j in range(1, accuracy_p):
coefficients[p**j] = series_p[j]
# Using multiplicativity, fill in the non-prime power Dirichlet
# series coefficients.
extend_multiplicatively_generic(coefficients)
return coefficients
