def zeta_function(self, var_name='T'):
r""" Return the Zeta function of the curve.
For any scheme `X` of finite type over `\mathbb{Z}`, the **arithmetic
zeta funtion** of `X` is defined as the product
.. MATH::
\zeta(X,s) := \prod_x \frac{1}{1-N(x)^{-s}},
where `x` runs over over all closed points of `X` and `N(x)`
denotes the cardinality of the residue field of `x`.
If `X` is a smooth projective curve over a field with
`q` elements, then `\zeta(X,s) = Z(X,q^{-s})`,
where `Z(X,T)` is a rational function in `T` of the form
.. MATH::
Z(X,T) = \frac{P(T)}{(1-T)(1-qT)},
for a polynomial `P` of degree `2g`, with some extra properties
reflecting the Weil conjectures. See:
- Hartshorn, *Algebraic Geometry*, Appendix C, Section 1.
Note that that this makes only sense if the constant base
field of self is finite, and that `Z(X,T)` depends on the
choice of the constant base field (unlike the function
`\zeta(X,s)`!).
"""
if hasattr(self,"_zeta_function"):
return self._zeta_function
K = self._constant_base_field
q = K.order()
g = self.genus()
S = PowerSeriesRing(ZZ, var_name, g+1)
N = self.count_points(g)
Z_series = S(1)
for k in range(1,g+1):
Z_series *= (1-S.gen()**k)**(-N[k])
P = (Z_series*(1-S.gen())*(1-q*S.gen())).polynomial()
c = range(2*g+1)
for k in range(g+1):
c[k] = P[k]
for k in range(g+1,2*g+1):
c[k] = c[2*g-k]*q**(k-g)
R = P.parent()
zeta = R(c)/(1-R.gen())/(1-q*R.gen())
self._zeta_function = zeta
return zeta
def get_factor_over_nf(curve, prime_ideal, prime_number, conductor, accuracy):
"""
Returns the inverse of the factor corresponding to the given prime
ideal in the Euler product expansion of the L-function at
prime_ideal. Unless the accuracy doesn't need this expansion, and
then returns 1 in power series ring.
"""
P = PowerSeriesRing(ZZ, 'T')
T = P.gen()
q = prime_ideal.norm()
inertial_deg = Integer(q).ord(prime_number)
if inertial_deg > accuracy:
return P(1)
if prime_ideal.divides(conductor):
a = curve.local_data(prime_ideal).bad_reduction_type()
L = 1 - a * (T**inertial_deg)
else:
discriminant = curve.discriminant()
if prime_ideal.divides(discriminant):
a = q + 1 - curve.local_minimal_model(prime_ideal).reduction(
prime_ideal).count_points()
else:
a = q + 1 - curve.reduction(prime_ideal).count_points()
L = 1 - a * (T**inertial_deg) + q * (T**(2 * inertial_deg))
return L
def set_from_coefficients(self, coeffs):
QR = PowerSeriesRing(QQ,name='q',order='neglex')
q = QR.gen()
res = 0*q**0
for n, c in coeffs.iteritems():
res += c*q**n
res = res.add_bigoh(len(coeffs.keys())+1)
self.set_value(res)
def set_from_coefficients(self, coeffs):
if not len(coeffs) == 0:
QR = PowerSeriesRing(coeffs.values()[0].parent(),name='q',order='neglex')
q = QR.gen()
res = 0*q**0
for n, c in coeffs.iteritems():
res += c*q**n
res = res.add_bigoh(len(coeffs.keys())+1)
self.set_value(res)
def euler_p_factor(root_list, PREC):
''' computes the coefficients of the pth Euler factor expanded as a geometric series
ax^n is the Dirichlet series coefficient p^(-ns)
'''
PREC = floor(PREC)
# return satake_list
R = PowerSeriesRing(CF, 'x')
x = R.gen()
ep = ~R.prod([1 - a * x for a in root_list])
return ep.O(PREC + 1)
def hilbert_series_using_cand_wts(i, parity, prec=None):
wts = load_cand_wts(i, parity)
if prec is None:
prec = max(wts[0])
ps = PowerSeriesRing(QQ, names='t', default_prec=prec + 1)
t = ps.gen()
num = sum(t**a for a in wts[0])
if len(wts) > 1:
num -= sum(t**a for a in wts[1])
dnm = (1 - t**2) * (1 - t**4) * (1 - t**6)
return num / dnm
def hilbert_series_using_dimension_formula(i, prec=10):
'''
This assumes dimension formula is true for weight (2, k) where k > 2
and the dimesion for (1, k) k > 2 is zero.
If one believes magma, the assumption for (2, k) can be checked for some cases.
And the assumption for (1, k) can be checked by the construction.
'''
ps = PowerSeriesRing(QQ, names='t', default_prec=prec + 1)
t = ps.gen()
return (sum(dimension_cuspforms_sqrt5(a, a + 2 * i) * t**a for a in range(2, prec + 1)) +
O(t**(prec + 1)))
def hilbert_series_maybe(j, parity=None, prec=30):
'''
Returns a hilbert series which is equal to
sum_{k > 0} M_{k, j}(Gamma_{2}) t^k
modulo a polynomial of degree < 5.
'''
R = PowerSeriesRing(QQ, names="t", default_prec=prec)
t = R.gen()
dnm = R(t_dnm())
nm = hilbert_series_num_maybe(j, parity=parity)
return (nm + O(t ** prec)) / dnm
def t_delete_terms_of_small_degrees(f):
'''
f is a polynomial of t.
Returns a polynomial g which is congruent to f modulo t_dnm
so that g/t_dnm does not have terms with degree < 4.
'''
R = PowerSeriesRing(QQ, names="t")
S = PolynomialRing(QQ, names="t")
t = R.gen()
dnm = R(t_dnm())
g = R(f / dnm) + O(t ** 4)
a = S(sum([t ** i * g[i] for i in range(4)]))
return f - t_dnm() * a
def _dimension_Gamma0_4(wt):
"""
Return the dimensions of subspaces of Siegel modular forms on $Gamma0(4)$.
OUTPUT
("Total",)
REMARK
Not completely implemented
"""
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(wt + 1)
H_all = (1 + x**4) * (1 + x**11) / (1 - x**2)**3 / (1 - x**6)
return (H_all[wt], )
def _dimension_Gamma0_3(wt):
"""
Return the dimensions of subspaces of Siegel modular forms on $Gamma0(3)$.
OUTPUT
("Total")
REMARK
Only total dimension implemented.
"""
R = PowerSeriesRing(ZZ, 'x')
x = R.gen().O(wt + 1)
H_all = (1 + 2 * x**4 + x**6 + x**15 *
(1 + 2 * x**2 + x**6)) / (1 - x**2) / (1 - x**4) / (1 - x**6)**2
return (H_all[wt], )
def get_factor_over_nf(curve, prime_ideal, prime_number, conductor, accuracy):
"""
Returns the inverse of the factor corresponding to the given prime
ideal in the Euler product expansion of the L-function at
prime_ideal. Unless the accuracy doesn't need this expansion, and
then returns 1 in power series ring.
"""
P = PowerSeriesRing(ZZ, 'T')
T = P.gen()
q = prime_ideal.norm()
inertial_deg = Integer(q).ord(prime_number)
if inertial_deg > accuracy:
return P(1)
if prime_ideal.divides(conductor):
a = curve.local_data(prime_ideal).bad_reduction_type()
L = 1 - a*(T**inertial_deg)
else:
discriminant = curve.discriminant()
if prime_ideal.divides(discriminant):
a = q + 1 - curve.local_minimal_model(prime_ideal).reduction(prime_ideal).count_points()
else:
a = q + 1 - curve.reduction(prime_ideal).count_points()
L = 1 - a*(T**inertial_deg) + q*(T**(2*inertial_deg))
return L
def hilbert_series_using_dimension_formula(i, parity, prec=10):
ps = PowerSeriesRing(QQ, names='t', default_prec=prec + 1)
t = ps.gen()
return (sum(
cuspforms_dimension((a, a + 2 * i)) * t**a
for a in range(2, prec + 1) if is_even(a + parity)) + O(t**(prec + 1)))
def gen_func_maybe_cusp_num_t_power_srs(parity=None, prec=10):
R = PolynomialRing(QQ, names="t")
S = PowerSeriesRing(R, names="s", default_prec=prec)
s = S.gen()
num = gen_func_maybe_cusp_num_t(parity=parity)
return S(num) + O(s ** prec)
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
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 verify_algebraically_PS(g, P0, alpha, trace_and_norm, verbose=True):
# input:
# * P0 (only necessary to shift the series)
# * [trace_numerator, trace_denominator, norm_numerator, norm_denominator]
# output:
# a boolean
if verbose:
print "verify_algebraically()"
L = P0.base_ring()
assert alpha.base_ring() is L
L_poly = PolynomialRing(L, "xL")
xL = L_poly.gen()
# shifting the series makes our life easier
trace_numerator, trace_denominator, norm_numerator, norm_denominator = [
L_poly(coeff)(L_poly.gen() + P0[0]) for coeff in trace_and_norm
]
L_fpoly = L_poly.fraction_field()
trace = L_fpoly(trace_numerator) / L_fpoly(trace_denominator)
norm = L_fpoly(norm_numerator) / L_fpoly(norm_denominator)
Xpoly = L_poly([norm(0), -trace(0), 1])
if verbose:
print "xpoly = %s" % Xpoly
if Xpoly.is_irreducible():
M = Xpoly.root_field("c")
else:
# this avoids bifurcation later on in the code
M = NumberField(xL, "c")
if verbose:
print M
xi_degree = max(
[elt.degree() for elt in [trace_denominator, norm_denominator]])
D = 2 * xi_degree
hard_bound = D + (4 + 2)
soft_bound = hard_bound + 5
M_ps = PowerSeriesRing(M, "T", default_prec=soft_bound)
T = M_ps.gen()
Tsub = T + P0[0]
trace_M = M_ps(trace)
norm_M = M_ps(norm)
sqrtdisc = sqrt(trace_M**2 - 4 * norm_M)
x1 = (trace_M - sqrtdisc) / 2
x2 = (trace_M + sqrtdisc) / 2
y1 = sqrt(g(x1))
y2 = sqrt(g(x2))
iy = 1 / sqrt(g(Tsub))
dx1 = x1.derivative(T)
dx2 = x2.derivative(T)
dx1_y1 = dx1 / y1
dx2_y2 = dx2 / y2
eq1 = Matrix([[-2 * M_ps(alpha.row(0).list())(Tsub) * iy, dx1_y1, dx2_y2]])
eq2 = Matrix(
[[-2 * M_ps(alpha.row(1).list())(Tsub) * iy, x1 * dx1_y1,
x2 * dx2_y2]])
branches = Matrix([[1, 1, 1], [1, 1, -1], [1, -1, 1], [1, -1,
-1]]).transpose()
meq1 = eq1 * branches
meq2 = eq2 * branches
algzero = False
for j in range(4):
if meq1[0, j] == 0 and meq2[0, j] == 0:
algzero = True
break
if verbose:
print "Done, verify_algebraically() = %s" % algzero
return algzero