def _iter_positive_forms_with_content_and_discriminant(self) :
if self.__disc is infinity :
raise ValueError, "infinity is not a true filter index"
if self.__reduced :
for (l, (u,x)) in enumerate(self.__p1list) :
if u == 0 :
for a in xrange(self.__level, isqrt(self.__disc // 4) + 1, self.__level) :
frpa = 4 * a
for b in xrange(a+1) :
g = gcd(a // self.__level,b)
bsq = b**2
for c in xrange(a, (b**2 + (self.__disc - 1))//(4*a) + 1) :
yield (((a,b,c), l), gcd(g,c), frpa*c - bsq)
else :
for a in xrange(1, isqrt(self.__disc // 3) + 1) :
frpa = 4 * a
for b in xrange(a+1) :
g = gcd(a, b)
bsq = b**2
## We need x**2 * a + x * b + c % N == 0
h = (-((x**2 + 1) * a + x * b)) % self.__level
for c in xrange( a + h,
(b**2 + (self.__disc - 1))//(4*a) + 1, self.__level ) :
yield (((a,b,c), l), gcd(g,(x**2 * a + x * b + c) // self.__level), frpa*c - bsq)
#! if self.__reduced
else :
raise NotImplementedError
raise StopIteration
def iter_positive_forms_with_content(self) :
if self.__disc is infinity :
raise ValueError, "infinity is not a true filter index"
if self.__reduced :
for a in xrange(1,isqrt(self.__disc // 3) + 1) :
for b in xrange(a+1) :
g = gcd(a, b)
for c in xrange(a, (b**2 + (self.__disc - 1))//(4*a) + 1) :
yield (a,b,c), gcd(g,c)
else :
maxtrace = floor(5*self.__disc / 15 + sqrt(self.__disc)/2)
for a in xrange(1, maxtrace + 1) :
for c in xrange(1, maxtrace - a + 1) :
g = gcd(a,c)
Bu = isqrt(4*a*c - 1)
di = 4*a*c - self.__disc
if di >= 0 :
Bl = isqrt(di) + 1
else :
Bl = 0
for b in xrange(-Bu, -Bl + 1) :
yield (a,b,c), gcd(g,b)
for b in xrange(Bl, Bu + 1) :
yield (a,b,c), gcd(g,b)
#! if self.__reduced
raise StopIteration
def _calc_iter_reduced_sub4(self) :
try :
return self.__iter_reduced_sub4
except AttributeError :
pass
sub3 = self._calc_iter_reduced_sub3()
sub4 = list()
for (a0,a1,b01,sub3s) in sub3 :
sub4s = list()
for (a2, b02, b12) in sub3s :
sub4ss = list()
for a3 in xrange(2, 2 * self.index() + 1, 2) :
# obstruction for t[0,3]
B1 = isqrt(a0 * a3 - 1)
for b03 in xrange(-B1, B1 + 1) :
# obstruction for t[1,3]
B3 = isqrt(a1 * a3 - 1)
for b13 in xrange(-B3, B3 + 1) :
# obstruction for the minor [0,1,3] of t
if a0*a1*a3 - a0*b12**2 + 2*b01*b13*b03 - b01**2*a3 - a1*b03**2 <= 0 :
continue
# obstruction for t[2,3]
B3 = isqrt(a2 * a3 - 1)
for b23 in xrange(-B3, B3 + 1) :
# obstruction for the minor [0,2,3] of t
if a0*a2*a3 - a0*b23**2 + 2*b02*b23*b03 - b02**2*a3 - a2*b03**2 <= 0 :
continue
# obstruction for the minor [1,2,3] of t
if a1*a2*a3 - a1*b23**2 + 2*b12*b23*b13 - b12**2*a3 - a2*b13**2 <= 0 :
continue
t = matrix(ZZ, 4, [a0, b01, b02, b03, b01, a1, b12, b13, b02, b12, a2, b23, b03, b13, b23, a3], check = False)
if t.det() <= 0 :
continue
u = t.LLL_gram()
if u.transpose() * t * u != t :
continue
sub4ss.append((a3, b03, b13, b23))
sub4s.append((a2, b02, b12, sub4ss))
sub4.append((a0, a1, b01, sub4s))
self.__iter_reduced_sub4 = sub4
return sub4
def decompositions(self, s) :
(a, b, c) = s
for a1 in xrange(a + 1) :
a2 = a - a1
for c1 in xrange(c + 1) :
c2 = c - c1
B1 = isqrt(4*a1*c1)
B2 = isqrt(4*a2*c2)
for b1 in xrange(max(-B1, b - B2), min(B1 + 1, b + B2 + 1)) :
yield ((a1, b1, c1), (a2,b - b1, c2))
raise StopIteration
def decompositions(self, s) :
(n, r) = s
fm = 4 * self.__m
if self.__weak_forms :
yield ((0,0), (n,r))
yield ((n,r), (0,0))
msq = self.__m**2
for n1 in xrange(1, n) :
n2 = n - n1
for r1 in xrange( max(r - isqrt(fm * n2 + msq),
isqrt(fm * n1 + msq - 1) + 1),
min( r + isqrt(fm * n2 + msq) + 1,
isqrt(fm * n1 + msq) + 1 ) ) :
yield ((n1, r1), (n2, r - r1))
else :
yield ((0,0), (n,r))
yield ((n,r), (0,0))
for n1 in xrange(1, n) :
n2 = n - n1
##r = r1 + r2
##r1**2 <= 4 n1 m
## (r - r1)**2 <= 4 n2 m
## r1**2 - 2*r1*r + r**2 - 4 m n2 <= 0
## r1 <-> r \pm \sqrt{r**2 - r**2 + 4 m n2}
for r1 in xrange( max(r - isqrt(fm * n2),
isqrt(fm * n1 - 1) + 1),
min( r + isqrt(fm * n2) + 1,
isqrt(fm * n1) + 1 ) ) :
yield ((n1, r1), (n2, r - r1))
raise StopIteration
def duke_imamoglu_lift(self, f, f_k, precision, half_integral_weight = False) :
"""
INPUT:
- ``half_integral_weight`` -- If ``False`` we assume that `f` is the Fourier expansion of a
Jacobi form. Otherwise we assume it is the Fourier expansion
of a half integral weight elliptic modular form.
"""
if half_integral_weight :
coeff_index = lambda d : d
else :
coeff_index = lambda d : ((d + (-d % 4)) // 4, (-d) % 4)
coeffs = dict()
for t in precision.iter_positive_forms() :
dt = (-1)**(precision.genus() // 2) * t.det()
d = fundamental_discriminant(dt)
eps = Integer(isqrt(dt / d))
coeffs[t] = 0
for a in eps.divisors() :
d_a = abs(d * (eps // a)**2)
coeffs[t] = coeffs[t] + a**(f_k - 1) * self._kohnen_phi(a, t) \
* f[coeff_index(d_a)]
return coeffs
def _kohnen_phi(self, a, t) :
## We use a modified power of a, namely for each prime factor p of a
## we use p**floor(v_p(a)/2). This is compensated for in the routine
## of \rho
a_modif = 1
for (p,e) in a.factor() :
a_modif = a_modif * p**(e // 2)
res = 0
for dsq in filter(lambda d: d.is_square(), a.divisors()) :
d = isqrt(dsq)
for g_diag in itertools.ifilter( lambda diag: prod(diag) == d,
itertools.product(*[d.divisors() for _ in xrange(t.nrows())]) ) :
for subents in itertools.product(*[xrange(r) for (j,r) in enumerate(g_diag) for _ in xrange(j)]) :
columns = [subents[(j * (j - 1)) // 2:(j * (j + 1)) // 2] for j in xrange(t.nrows())]
g = diagonal_matrix(list(g_diag))
for j in xrange(t.nrows()) :
for i in xrange(j) :
g[i,j] = columns[j][i]
ginv = g.inverse()
tg = ginv.transpose() * t * ginv
try :
tg= matrix(ZZ, tg)
except :
continue
if any(tg[i,i] % 2 == 1 for i in xrange(tg.nrows())) :
continue
tg.set_immutable()
res = res + self._kohnen_rho(tg, a // dsq)
return a_modif * res
def decompositions(self, t) :
for diag in itertools.product(*[xrange(t[i,i] + 1) for i in xrange(self.__n)]) :
for subents in itertools.product( *[ xrange(-2 * isqrt(diag[i] * diag[j]), 2 * isqrt(diag[i] * diag[j]) + 1)
for i in xrange(self.__n - 1)
for j in xrange(i + 1, self.__n) ] ) :
t1 = matrix(ZZ, [[ 2 * diag[i]
if i == j else
(subents[self.__n * i - (i * (i + 1)) // 2 + j - i - 1]
if i < j else
subents[self.__n * j - (j * (j + 1)) // 2 + i - j - 1])
for i in xrange(self.__n) ]
for j in xrange(self.__n) ] )
if self._check_definiteness(t1) == -1 :
continue
t2 = t - t1
if self._check_definiteness(t2) == -1 :
continue
t1.set_immutable()
t2.set_immutable()
yield (t1, t2)
raise StopIteration
def _by_taylor_expansion_m1(self, f_divs, k, is_integral=False) :
r"""
This provides special, faster code in the Jacobi index `1` case.
"""
if is_integral :
PS = self.integral_power_series_ring()
else :
PS = self.power_series_ring()
qexp_prec = self._qexp_precision()
fderiv = f_divs[(0,0)].derivative().shift(1)
f = f_divs[(0,0)] * Integer(k/2)
gfderiv = f_divs[(1,0)] - fderiv
ab_prec = isqrt(qexp_prec + 1)
a1dict = dict(); a0dict = dict()
b1dict = dict(); b0dict = dict()
for t in xrange(1, ab_prec + 1) :
tmp = t**2
a1dict[tmp] = -8*tmp
b1dict[tmp] = -2
tmp += t
a0dict[tmp] = 8*tmp + 2
b0dict[tmp] = 2
b1dict[0] = -1
a0dict[0] = 2; b0dict[0] = 2
a1 = PS(a1dict); b1 = PS(b1dict)
a0 = PS(a0dict); b0 = PS(b0dict)
Ifg0 = (self._eta_factor() * (f*a0 + gfderiv*b0)).list()
Ifg1 = (self._eta_factor() * (f*a1 + gfderiv*b1)).list()
if len(Ifg0) < qexp_prec :
Ifg0 += [0]*(qexp_prec - len(Ifg0))
if len(Ifg1) < qexp_prec :
Ifg1 += [0]*(qexp_prec - len(Ifg1))
Cphi = dict([(0,0)])
for i in xrange(qexp_prec) :
Cphi[-4*i] = Ifg0[i]
Cphi[1-4*i] = Ifg1[i]
del Ifg0[:], Ifg1[:]
phi_coeffs = dict()
m = self.__precision.jacobi_index()
for r in xrange(2 * self.__precision.jacobi_index()) :
for n in xrange(qexp_prec) :
k = 4 * m * n - r**2
if k >= 0 :
phi_coeffs[(n, r)] = Cphi[-k]
return phi_coeffs
def __iter__(self) :
if self.__bound is infinity :
raise ValueError, "infinity is not a true filter index"
if self.__reduced :
##TODO: This is really primitive. We barely reduce arbitrary forms.
for diag in itertools.product(*[xrange(self.__bound) for _ in xrange(self.__n)]) :
for subents in itertools.product( *[ xrange(-2 * isqrt(diag[i] * diag[j]), 2 * isqrt(diag[i] * diag[j]) + 1)
for i in xrange(self.__n - 1)
for j in xrange(i + 1, self.__n) ] ) :
t = matrix(ZZ, [[ 2 * diag[i]
if i == j else
(subents[self.__n * i - (i * (i + 1)) // 2 + j - i - 1]
if i < j else
subents[self.__n * j - (j * (j + 1)) // 2 + i - j - 1])
for i in xrange(self.__n) ]
for j in xrange(self.__n) ] )
t = self.__ambient.reduce(t)
if t is None : continue
t = t[0]
t.set_immutable()
yield t
else :
for diag in itertools.product(*[xrange(self.__bound) for _ in xrange(self.__n)]) :
for subents in itertools.product( *[ xrange(-2 * isqrt(diag[i] * diag[j]), 2 * isqrt(diag[i] * diag[j]) + 1)
for i in xrange(self.__n - 1)
for j in xrange(i + 1, self.__n) ] ) :
t = matrix(ZZ, [[ 2 * diag[i]
if i == j else
(subents[self.__n * i - (i * (i + 1)) // 2 + j - i - 1]
if i < j else
subents[self.__n * j - (j * (j + 1)) // 2 + i - j - 1])
for i in xrange(self.__n) ]
for j in xrange(self.__n) ] )
if self._check_definiteness(t) == -1 :
continue
t.set_immutable()
yield t
raise StopIteration
def decompositions(self, s) :
((a, b, c), l) = s
# r t r^tr = t_1 + t_2
# \Rightleftarrow t = r t_1 r^tr + r t_2 r^tr
for a1 in xrange(a + 1) :
a2 = a - a1
for c1 in xrange(c + 1) :
c2 = c - c1
B1 = isqrt(4*a1*c1)
B2 = isqrt(4*a2*c2)
for b1 in xrange(max(-B1, b - B2), min(B1 + 1, b + B2 + 1)) :
h1 = apply_GL_to_form(self.__p1list[l], (a1, b1, c1))
h2 = apply_GL_to_form(self.__p1list[l], (a2, b - b1, c2))
if h1[2] % self.__level == 0 and h2[2] % self.__level == 0:
yield ((h1, 1), (h2,1))
raise StopIteration
def iter_indefinite_forms(self) :
r"""
Iterate over indices with non-positive discriminant.
TESTS::
sage: from psage.modform.jacobiforms.jacobiformd1nn_fourierexpansion import *
sage: list(JacobiFormD1NNFilter(2, 2, reduced = False, weak_forms = True).iter_indefinite_forms())
[(0, -1), (1, 3), (0, 0), (1, -3), (0, 1), (0, -2), (0, 2)]
sage: list(JacobiFormD1NNFilter(3, 2, reduced = False, weak_forms = True).iter_indefinite_forms())
[(0, -1), (1, 3), (2, -4), (0, 0), (2, 4), (1, -3), (0, 1), (0, -2), (0, 2)]
sage: list(JacobiFormD1NNFilter(10, 2, reduced = True, weak_forms = True).iter_indefinite_forms())
[(0, 0), (0, 1), (0, 2)]
sage: list(JacobiFormD1NNFilter(10, 3, reduced = True, weak_forms = True).iter_indefinite_forms())
[(0, 0), (0, 1), (0, 2), (0, 3)]
sage: list(JacobiFormD1NNFilter(10, 10, reduced = True, weak_forms = True).iter_indefinite_forms())
[(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0, 6), (0, 7), (0, 8), (0, 9), (0, 10), (1, 7), (1, 8), (1, 9), (1, 10), (2, 9), (2, 10)]
"""
B = self.__bound
m = self.__m
fm = Integer(4 * self.__m)
if self.__reduced :
if self.__weak_forms :
for n in xrange(0, min(self.__m // 4 + 1, self.__bound)) :
for r in xrange( isqrt(fm * n - 1) + 1 if n != 0 else 0, self.__m + 1 ) :
yield (n, r)
else :
for r in xrange(0, min(self.__m + 1,
isqrt((self.__bound - 1) * fm) + 1) ) :
if fm.divides(r**2) :
yield (r**2 // fm, r)
else :
if self.__weak_forms :
## We first determine the reduced indices.
for n in xrange(0, min(m // 4 + 1, B)) :
if n == 0 :
r_iteration = range(-m + 1, m + 1)
else :
r_iteration = range( -m + 1, -isqrt(fm * n - 1) ) \
+ range( isqrt(fm * n - 1) + 1, m + 1 )
for r in r_iteration :
for l in range( (- r - isqrt(r**2 - 4 * m * (n - (B - 1))) - 1) // (2 * m) + 1,
(- r + isqrt(r**2 - 4 * m * (n - (B - 1)))) // (2 * m) + 1 ) :
if n + l * r + m * l**2 >= B :
print l, n, r
yield (n + l * r + m * l**2, r + 2 * m * l)
else :
if self.__bound > 0 :
yield (0,0)
for n in xrange(1, self.__bound) :
if (fm * n).is_square() :
rt_fmm = isqrt(fm * n)
yield(n, rt_fmm)
yield(n, -rt_fmm)
raise StopIteration
def decompositions(self, s):
((a, b, c), l) = s
# r t r^tr = t_1 + t_2
# \Rightleftarrow t = r t_1 r^tr + r t_2 r^tr
for a1 in xrange(a + 1):
a2 = a - a1
for c1 in xrange(c + 1):
c2 = c - c1
B1 = isqrt(4 * a1 * c1)
B2 = isqrt(4 * a2 * c2)
for b1 in xrange(max(-B1, b - B2), min(B1 + 1, b + B2 + 1)):
h1 = apply_GL_to_form(self.__p1list[l], (a1, b1, c1))
h2 = apply_GL_to_form(self.__p1list[l], (a2, b - b1, c2))
if h1[2] % self.__level == 0 and h2[2] % self.__level == 0:
yield ((h1, 1), (h2, 1))
raise StopIteration
def iter_positive_forms(self) :
if self.__reduced :
##TODO: This is really primitive. We barely reduce arbitrary forms.
for diag in itertools.product(*[xrange(self.__bound) for _ in xrange(self.__n)]) :
for subents in itertools.product( *[ xrange(-2 * isqrt(diag[i] * diag[j]), 2 * isqrt(diag[i] * diag[j]) + 1)
for i in xrange(self.__n - 1)
for j in xrange(i + 1, self.__n) ] ) :
t = matrix(ZZ, [[ 2 * diag[i]
if i == j else
(subents[self.__n * i - (i * (i + 1)) // 2 + j - i - 1]
if i < j else
subents[self.__n * j - (j * (j + 1)) // 2 + i - j - 1])
for i in xrange(self.__n) ]
for j in xrange(self.__n) ] )
t = self.__ambient.reduce(t)
if t is None : continue
t = t[0]
if t[0,0] == 0 : continue
t.set_immutable()
yield t
else :
for diag in itertools.product(*[xrange(self.__bound) for _ in xrange(self.__n)]) :
for subents in itertools.product( *[ xrange(-2 * isqrt(diag[i] * diag[j]), 2 * isqrt(diag[i] * diag[j]) + 1)
for i in xrange(self.__n - 1)
for j in xrange(i + 1, self.__n) ] ) :
t = matrix(ZZ, [[ 2 * diag[i]
if i == j else
(subents[self.__n * i - (i * (i + 1)) // 2 + j - i - 1]
if i < j else
subents[self.__n * j - (j * (j + 1)) // 2 + i - j - 1])
for i in xrange(self.__n) ]
for j in xrange(self.__n) ] )
if self.__ambient._check_definiteness(t) != 1 :
continue
t.set_immutable()
yield t
def _kohnen_rho(self, t, a):
dt = (-1)**(t.nrows() // 2) * t.det()
d = fundamental_discriminant(dt)
eps = isqrt(dt // d)
res = 1
for (p, e) in gcd(a, eps).factor():
pol = self._kohnen_rho_polynomial(t, p).coefficients()
if e < len(pol):
res = res * pol[e]
return res
def _kohnen_rho(self, t, a) :
dt = (-1)**(t.nrows() // 2) * t.det()
d = fundamental_discriminant(dt)
eps = isqrt(dt // d)
res = 1
for (p,e) in gcd(a, eps).factor() :
pol = self._kohnen_rho_polynomial(t, p).coefficients()
if e < len(pol) :
res = res * pol[e]
return res
def iter_indefinite_forms(self) :
fm = Integer(4 * self.__m)
if self.__reduced :
if self.__weak_forms :
msq = self.__m**2
for n in xrange(0, min(self.__m // 4 + 1, self.__bound)) :
for r in xrange( isqrt(fm * n - 1) + 1 if n != 0 else 0,
isqrt(fm * n + msq + 1) ) :
yield (n, r)
else :
for r in xrange(0, min(self.__m + 1,
isqrt((self.__bound - 1) * fm) + 1) ) :
if fm.divides(r**2) :
yield (r**2 // fm, r)
else :
if self.__weak_forms :
msq = self.__m**2
for n in xrange(0, self.__bound) :
for r in xrange( isqrt(fm * n - 1) + 1 if n != 0 else 0,
isqrt(fm * n + msq + 1) ) :
yield (n, r)
else :
for n in xrange(0, self.__bound) :
if (fm * n).is_square() :
yield(n, isqrt(fm * n))
raise StopIteration
def iter_indefinite_forms(self) :
fm = Integer(4 * self.__m)
if self.__reduced :
if self.__weak_forms :
msq = self.__m**2
for n in range(0, min(self.__m // 4 + 1, self.__bound)) :
for r in range( isqrt(fm * n - 1) + 1 if n != 0 else 0,
isqrt(fm * n + msq + 1) ) :
yield (n, r)
else :
for r in range(0, min(self.__m + 1,
isqrt((self.__bound - 1) * fm) + 1) ) :
if fm.divides(r**2) :
yield (r**2 // fm, r)
else :
if self.__weak_forms :
msq = self.__m**2
for n in range(0, self.__bound) :
for r in range( isqrt(fm * n - 1) + 1 if n != 0 else 0,
isqrt(fm * n + msq + 1) ) :
yield (n, r)
else :
for n in range(0, self.__bound) :
if (fm * n).is_square() :
yield(n, isqrt(fm * n))
raise StopIteration
def hz_pullback(self, mu):
r"""
Compute the pullbacks to Hirzebruch--Zagier curves.
This computes the pullback f(\tau * \mu, \tau * \mu') of f to the embedded half-plane H * (\mu, \mu') where \mu' is the conjugate of \mu. The result is a modular form of level equal to the norm of \mu.
INPUT:
- ``mu`` -- a totally-positive integer in the base-field K.
OUTPUT: an OrthogonalModularForm for a signature (2, 1) lattice
WARNING: the output's weyl vector is not implemented
EXAMPLES::
sage: from weilrep import *
sage: x = var('x')
sage: K.<sqrt13> = NumberField(x * x - 13)
sage: HMF(K).eisenstein_series(2, 15).hz_pullback(4 - sqrt13)
1 + 24*q + O(q^2)
"""
K = self.base_field()
mu = K(mu)
nn = mu.norm()
tt = mu.trace()
if tt <= 0 or nn <= 0:
raise ValueError(
'You called hz_pullback with a number that is not totally-positive!'
)
d = K.discriminant()
a = isqrt((tt * tt - 4 * nn) / d)
h = self.fourier_expansion()
t, = PowerSeriesRing(QQ, 't').gens()
d = K.discriminant()
prec = floor(h.prec() / (tt / 2 + 2 * a / sqrt(d)))
if d % 4:
f = sum([
p[n] * t**((i * tt + n * a) / 2)
for i, p in enumerate(h.list()) for n in p.exponents()
]) + O(t**prec)
else:
f = sum([
p[n] * t**((i * tt + 2 * n * a) / 2)
for i, p in enumerate(h.list()) for n in p.exponents()
]) + O(t**prec)
return OrthogonalModularFormLorentzian(self.weight(),
WeilRep(matrix([[-2 * nn]])),
f,
scale=self.scale(),
weylvec=vector([0]),
qexp_representation='shimura')
def __iter__(self):
if self.__disc is infinity:
raise ValueError("infinity is not a true filter index")
if self.__reduced:
for c in range(0, self._indefinite_content_bound()):
yield (0, 0, c)
for a in range(1, isqrt(self.__disc // 3) + 1):
for b in range(a + 1):
for c in range(a,
(b**2 + (self.__disc - 1)) // (4 * a) + 1):
yield (a, b, c)
else:
##FIXME: These are not all matrices
for a in range(0, self._indefinite_content_bound()):
yield (a, 0, 0)
for c in range(1, self._indefinite_content_bound()):
yield (0, 0, c)
maxtrace = floor(5 * self.__disc / 15 + sqrt(self.__disc) / 2)
for a in range(1, maxtrace + 1):
for c in range(1, maxtrace - a + 1):
Bu = isqrt(4 * a * c - 1)
di = 4 * a * c - self.__disc
if di >= 0:
Bl = isqrt(di) + 1
else:
Bl = 0
for b in range(-Bu, -Bl + 1):
yield (a, b, c)
for b in range(Bl, Bu + 1):
yield (a, b, c)
#! if self.__reduced
raise StopIteration
def _iter_positive_forms_with_content_and_discriminant(self):
if self.__disc is infinity:
raise ValueError, "infinity is not a true filter index"
if self.__reduced:
for (l, (u, x)) in enumerate(self.__p1list):
if u == 0:
for a in xrange(self.__level,
isqrt(self.__disc // 4) + 1, self.__level):
frpa = 4 * a
for b in xrange(a + 1):
g = gcd(a // self.__level, b)
bsq = b**2
for c in xrange(a, (b**2 + (self.__disc - 1)) //
(4 * a) + 1):
yield (((a, b, c), l), gcd(g,
c), frpa * c - bsq)
else:
for a in xrange(1, isqrt(self.__disc // 3) + 1):
frpa = 4 * a
for b in xrange(a + 1):
g = gcd(a, b)
bsq = b**2
## We need x**2 * a + x * b + c % N == 0
h = (-((x**2 + 1) * a + x * b)) % self.__level
for c in xrange(a + h,
(b**2 + (self.__disc - 1)) //
(4 * a) + 1, self.__level):
yield (((a, b, c), l),
gcd(g, (x**2 * a + x * b + c) //
self.__level), frpa * c - bsq)
#! if self.__reduced
else:
raise NotImplementedError
raise StopIteration
def __init__(self, precision, level, reduced = True) :
self.__level = level
if isinstance(precision, ParamodularFormD2Filter_trace) :
precision = precision.index()
elif isinstance(precision, ParamodularFormD2Filter_discriminant) :
if precision.index() is infinity :
precision = infinity
else :
precision = isqrt(precision.index() - 1) + 1
self.__trace = precision
self.__reduced = reduced
self.__p1list = P1List(level)
def iter_positive_forms(self) :
r"""
TESTS::
sage: from psage.modform.jacobiforms.jacobiformd1nn_fourierexpansion import *
sage: list(JacobiFormD1NNFilter(3, 2, reduced = True).iter_positive_forms())
[(1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
sage: list(JacobiFormD1NNFilter(3, 2, reduced = False).iter_positive_forms())
[(1, 0), (1, 1), (1, -1), (1, 2), (1, -2), (2, 0), (2, 1), (2, -1), (2, 2), (2, -2), (2, 3), (2, -3)]
"""
fm = 4 * self.__m
if self.__reduced :
for n in xrange(1, self.__bound) :
for r in xrange(min(self.__m + 1, isqrt(fm * n - 1) + 1)) :
yield (n, r)
else :
for n in xrange(1, self.__bound) :
yield(n, 0)
for r in xrange(1, isqrt(fm * n - 1) + 1) :
yield (n, r)
yield (n, -r)
raise StopIteration
def iter_positive_forms(self):
r"""
TESTS::
sage: from psage.modform.jacobiforms.jacobiformd1nn_fourierexpansion import *
sage: list(JacobiFormD1NNFilter(3, 2, reduced = True).iter_positive_forms())
[(1, 0), (1, 1), (1, 2), (2, 0), (2, 1), (2, 2)]
sage: list(JacobiFormD1NNFilter(3, 2, reduced = False).iter_positive_forms())
[(1, 0), (1, 1), (1, -1), (1, 2), (1, -2), (2, 0), (2, 1), (2, -1), (2, 2), (2, -2), (2, 3), (2, -3)]
"""
fm = 4 * self.__m
if self.__reduced:
for n in xrange(1, self.__bound):
for r in xrange(min(self.__m + 1, isqrt(fm * n - 1) + 1)):
yield (n, r)
else:
for n in xrange(1, self.__bound):
yield (n, 0)
for r in xrange(1, isqrt(fm * n - 1) + 1):
yield (n, r)
yield (n, -r)
raise StopIteration
def _calc_iter_reduced_sub3(self):
try:
return self.__iter_reduced_sub3
except AttributeError:
pass
sub2 = self._calc_iter_reduced_sub2()
sub3 = list()
for (a0, a1, b01) in sub2:
sub3s = list()
for a2 in range(2, 2 * self.index(), 2):
# obstruction for t[0,2]
B1 = isqrt(a0 * a2 - 1)
for b02 in range(-B1, B1 + 1):
# obstruction for t[1,2]
B3 = isqrt(a1 * a2 - 1)
for b12 in range(-B3, B3 + 1):
# obstruction for the minor [0,1,2] of t
if a0 * a1 * a2 - a0 * b12**2 + 2 * b01 * b12 * b02 - a2 * b01**2 - a1 * b02**2 <= 0:
continue
t = matrix(ZZ, 3,
[a0, b01, b02, b01, a1, b12, b02, b12, a2])
u = t.LLL_gram()
if u.transpose() * t * u != t:
continue
sub3s.append((a2, b02, b12))
sub3.append((a0, a1, b01, sub3s))
self.__iter_reduced_sub3 = sub3
return sub3
def __iter__(self) :
if self.__disc is infinity :
raise ValueError, "infinity is not a true filter index"
if self.__reduced :
for c in xrange(0, self._indefinite_content_bound()) :
yield (0,0,c)
for a in xrange(1,isqrt(self.__disc // 3) + 1) :
for b in xrange(a+1) :
for c in xrange(a, (b**2 + (self.__disc - 1))//(4*a) + 1) :
yield (a,b,c)
else :
##FIXME: These are not all matrices
for a in xrange(0, self._indefinite_content_bound()) :
yield (a,0,0)
for c in xrange(1, self._indefinite_content_bound()) :
yield (0,0,c)
maxtrace = floor(5*self.__disc / 15 + sqrt(self.__disc)/2)
for a in xrange(1, maxtrace + 1) :
for c in xrange(1, maxtrace - a + 1) :
Bu = isqrt(4*a*c - 1)
di = 4*a*c - self.__disc
if di >= 0 :
Bl = isqrt(di) + 1
else :
Bl = 0
for b in xrange(-Bu, -Bl + 1) :
yield (a,b,c)
for b in xrange(Bl, Bu + 1) :
yield (a,b,c)
#! if self.__reduced
raise StopIteration
def iter_positive_forms(self) :
fm = 4 * self.__m
if self.__reduced :
if self.__weak_forms :
msq = self.__m**2
for n in range(1, self.__bound) :
for r in range(min(self.__m + 1, isqrt(fm * n + msq - 1) + 1)) :
yield (n, r)
else :
for n in range(1, self.__bound) :
for r in range(min(self.__m + 1, isqrt(fm * n - 1) + 1)) :
yield (n, r)
else :
if self.__weak_forms :
msq = self.__m**2
for n in range(1, self.__bound) :
for r in range(isqrt(fm * n + msq - 1) + 1) :
yield (n, r)
else :
for n in range(1, self.__bound) :
for r in range(isqrt(fm * n - 1) + 1) :
yield (n, r)
raise StopIteration
def iter_positive_forms(self) :
fm = 4 * self.__m
if self.__reduced :
if self.__weak_forms :
msq = self.__m**2
for n in xrange(1, self.__bound) :
for r in xrange(min(self.__m + 1, isqrt(fm * n + msq - 1) + 1)) :
yield (n, r)
else :
for n in xrange(1, self.__bound) :
for r in xrange(min(self.__m + 1, isqrt(fm * n - 1) + 1)) :
yield (n, r)
else :
if self.__weak_forms :
msq = self.__m**2
for n in xrange(1, self.__bound) :
for r in xrange(isqrt(fm * n + msq - 1) + 1) :
yield (n, r)
else :
for n in xrange(1, self.__bound) :
for r in xrange(isqrt(fm * n - 1) + 1) :
yield (n, r)
raise StopIteration
def _calc_iter_reduced_sub3(self) :
try :
return self.__iter_reduced_sub3
except AttributeError :
pass
sub2 = self._calc_iter_reduced_sub2()
sub3 = list()
for (a0, a1, b01) in sub2 :
sub3s = list()
for a2 in xrange(2, 2 * self.index(), 2) :
# obstruction for t[0,2]
B1 = isqrt(a0 * a2 - 1)
for b02 in xrange(-B1, B1 + 1) :
# obstruction for t[1,2]
B3 = isqrt(a1 * a2 - 1)
for b12 in xrange(-B3, B3 + 1) :
# obstruction for the minor [0,1,2] of t
if a0*a1*a2 - a0*b12**2 + 2*b01*b12*b02 - a2*b01**2 - a1*b02**2 <= 0 :
continue
t = matrix(ZZ, 3, [a0, b01, b02, b01, a1, b12, b02, b12, a2])
u = t.LLL_gram()
if u.transpose() * t * u != t :
continue
sub3s.append((a2, b02, b12))
sub3.append((a0, a1, b01, sub3s))
self.__iter_reduced_sub3 = sub3
return sub3
def init_ecdb_stats(self):
if self._stats:
return
logger.debug("Computing elliptic curve stats...")
ecdb = self.ecdb
counts = self._counts
stats = {}
rank_counts = []
for r in range(counts['max_rank']+1):
ncu = ecdb.find({'rank': r}).count()
ncl = ecdb.find({'rank': r, 'number': 1}).count()
prop = format_percentage(ncl,counts['nclasses'])
rank_counts.append({'r': r, 'ncurves': ncu, 'nclasses': ncl, 'prop': prop})
stats['rank_counts'] = rank_counts
tor_counts = []
tor_counts2 = []
ncurves = counts['ncurves']
for t in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16]:
ncu = ecdb.find({'torsion': t}).count()
if t in [4,8,12]: # two possible structures
ncyc = ecdb.find({'torsion_structure': [str(t)]}).count()
gp = "\(C_{%s}\)"%t
prop = format_percentage(ncyc,ncurves)
tor_counts.append({'t': t, 'gp': gp, 'ncurves': ncyc, 'prop': prop})
nncyc = ncu-ncyc
gp = "\(C_{2}\\times C_{%s}\)"%(t//2)
prop = format_percentage(nncyc,ncurves)
tor_counts2.append({'t': t, 'gp': gp, 'ncurves': nncyc, 'prop': prop})
elif t==16: # all C_2 x C_8
gp = "\(C_{2}\\times C_{8}\)"
prop = format_percentage(ncu,ncurves)
tor_counts2.append({'t': t, 'gp': gp, 'ncurves': ncu, 'prop': prop})
else: # all cyclic
gp = "\(C_{%s}\)"%t
prop = format_percentage(ncu,ncurves)
tor_counts.append({'t': t, 'gp': gp, 'ncurves': ncu, 'prop': prop})
stats['tor_counts'] = tor_counts+tor_counts2
stats['max_sha'] = ecdb.find().sort('sha', DESCENDING).limit(1)[0]['sha']
sha_counts = []
from sage.misc.functional import isqrt
for s in range(1,1+isqrt(stats['max_sha'])):
s2 = s*s
nc = ecdb.find({'sha': s2}).count()
if nc:
sha_counts.append({'s': s, 'ncurves': nc})
stats['sha_counts'] = sha_counts
self._stats = stats
logger.debug("... finished computing elliptic curve stats.")
def __iter__(self):
r"""
Iterate over all GL(2,Z)-reduced semi positive forms
which are within the bounds of this precision.
EXAMPLES::
sage: from sage.modular.siegel.siegel_modular_form_prec import SiegelModularFormPrecision
sage: prec = SiegelModularFormPrecision(11)
sage: for k in prec.__iter__(): print k
(0, 0, 0)
(0, 0, 1)
(0, 0, 2)
(1, 0, 1)
(1, 0, 2)
(1, 1, 1)
(1, 1, 2)
NOTE
The forms are enumerated in lexicographic order.
"""
if self.__type == 'disc':
bound = self.get_contents_bound_for_semi_definite_forms()
for c in range(0, bound):
yield (0,0,c)
atop = isqrt(self.__prec // 3)
if 3*atop*atop == self.__prec: atop -= 1
for a in range(1,atop + 1):
for b in range(a+1):
for c in range(a, ceil((b**2 + self.__prec)/(4*a))):
yield (a,b,c)
elif 'box' == self.__type:
(am, bm, cm) = self.__prec
for a in range(am):
for b in range(min(bm,a+1)):
for c in range(a, cm):
yield (a,b,c)
else:
raise RuntimeError("Unexpected value of self.__type")
raise StopIteration
def __iter__(self):
r"""
Iterate over all GL(2,Z)-reduced semi positive forms
which are within the bounds of this precision.
EXAMPLES::
sage: from sage.modular.siegel.siegel_modular_form_prec import SiegelModularFormPrecision
sage: prec = SiegelModularFormPrecision(11)
sage: for k in prec.__iter__(): print k
(0, 0, 0)
(0, 0, 1)
(0, 0, 2)
(1, 0, 1)
(1, 0, 2)
(1, 1, 1)
(1, 1, 2)
NOTE
The forms are enumerated in lexicographic order.
"""
if self.__type == 'disc':
bound = self.get_contents_bound_for_semi_definite_forms()
for c in xrange(0, bound):
yield (0,0,c)
atop = isqrt(self.__prec // 3)
if 3*atop*atop == self.__prec: atop -= 1
for a in xrange(1,atop + 1):
for b in xrange(a+1):
for c in xrange(a, ceil((b**2 + self.__prec)/(4*a))):
yield (a,b,c)
elif 'box' == self.__type:
(am, bm, cm) = self.__prec
for a in xrange(am):
for b in xrange( min(bm,a+1)):
for c in xrange(a, cm):
yield (a,b,c)
else:
raise RuntimeError, "Unexpected value of self.__type"
raise StopIteration
def _calc_iter_reduced_sub2(self):
try:
return self.__iter_reduced_sub2
except AttributeError:
pass
sub2 = list()
for a0 in range(2, 2 * self.index(), 2):
for a1 in range(a0, 2 * self.index(), 2):
# obstruction for t[0,1]
B1 = isqrt(a0 * a1 - 1)
for b01 in range(0, min(B1, a0 // 2) + 1):
sub2.append((a0, a1, -b01))
self.__iter_reduced_sub2 = sub2
return sub2
def _jacobi_forms_by_taylor_expansion_coords(index, weight, precision) :
global _jacobi_forms_by_taylor_expansion_coords_cache
key = (index, weight)
try :
return _jacobi_forms_by_taylor_expansion_coords_cache[key]
except KeyError :
if precision < (index - 1) // 4 + 1 :
precision = (index - 1) // 4 + 1
weak_forms = _all_weak_jacobi_forms_by_taylor_expansion(index, weight, precision)
weak_index_matrix = matrix(ZZ, [ [ f[(n,r)] for n in xrange((index - 1) // 4 + 1)
for r in xrange(isqrt(4 * n * index) + 1, index + 1) ]
for f in weak_forms] )
_jacobi_forms_by_taylor_expansion_coords_cache[key] = \
weak_index_matrix.left_kernel().echelonized_basis()
return _jacobi_forms_by_taylor_expansion_coords_cache[key]
def _calc_iter_reduced_sub2(self) :
try :
return self.__iter_reduced_sub2
except AttributeError :
pass
sub2 = list()
for a0 in xrange(2, 2 * self.index(), 2) :
for a1 in xrange(a0, 2 * self.index(), 2) :
# obstruction for t[0,1]
B1 = isqrt(a0 * a1 - 1)
for b01 in xrange(0, min(B1, a0 // 2) + 1) :
sub2.append((a0,a1,-b01))
self.__iter_reduced_sub2 = sub2
return sub2
def __init__( self, q):
"""
We initialize by a list of integers $[a_1,...,a_N]$. The
lattice self is then $L = (L,\beta) = (G^{-1}\ZZ^n/\ZZ^n, G[x])$,
where $G$ is the symmetric matrix
$G=[a_1,...,a_n;*,a_{n+1},....;..;* ... * a_N]$,
i.e.~the $a_j$ denote the elements above the diagonal of $G$.
"""
self.__form = q
# We compute the rank
N = len(q)
assert is_square( 1+8*N)
n = Integer( (-1+isqrt(1+8*N))/2)
self.__rank = n
# We set up the Gram matrix
self.__G = matrix( IntegerRing(), n, n)
i = j = 0
for a in q:
self.__G[i,j] = self.__G[j,i] = Integer(a)
if j < n-1:
j += 1
else:
i += 1
j = i
# We compute the level
Gi = self.__G**-1
self.__Gi = Gi
a = lcm( map( lambda x: x.denominator(), Gi.list()))
I = Gi.diagonal()
b = lcm( map( lambda x: (x/2).denominator(), I))
self.__level = lcm( a, b)
# We define the undelying module and the ambient space
self.__module = FreeModule( IntegerRing(), n)
self.__space = self.__module.ambient_vector_space()
# We compute a shadow vector
self.__shadow_vector = self.__space([(a%2)/2 for a in self.__G.diagonal()])*Gi
# We define a basis
M = Matrix( IntegerRing(), n, n, 1)
self.__basis = self.__module.basis()
# We prepare a cache
self.__dual_vectors = None
self.__values = None
self.__chi = {}
def _kohnen_phi(self, a, t):
## We use a modified power of a, namely for each prime factor p of a
## we use p**floor(v_p(a)/2). This is compensated for in the routine
## of \rho
a_modif = 1
for (p, e) in a.factor():
a_modif = a_modif * p**(e // 2)
res = 0
for dsq in [d for d in a.divisors() if d.is_square()]:
d = isqrt(dsq)
for g_diag in filter(
lambda diag: prod(diag) == d,
itertools.product(
*[d.divisors() for _ in range(t.nrows())])):
for subents in itertools.product(*[
range(r) for (j, r) in enumerate(g_diag)
for _ in range(j)
]):
columns = [
subents[(j * (j - 1)) // 2:(j * (j + 1)) // 2]
for j in range(t.nrows())
]
g = diagonal_matrix(list(g_diag))
for j in range(t.nrows()):
for i in range(j):
g[i, j] = columns[j][i]
ginv = g.inverse()
tg = ginv.transpose() * t * ginv
try:
tg = matrix(ZZ, tg)
except:
continue
if any(tg[i, i] % 2 == 1 for i in range(tg.nrows())):
continue
tg.set_immutable()
res = res + self._kohnen_rho(tg, a // dsq)
return a_modif * res
def _jacobi_forms_by_taylor_expansion_coords(index, weight, precision):
global _jacobi_forms_by_taylor_expansion_coords_cache
key = (index, weight)
try:
return _jacobi_forms_by_taylor_expansion_coords_cache[key]
except KeyError:
if precision < (index - 1) // 4 + 1:
precision = (index - 1) // 4 + 1
weak_forms = _all_weak_jacobi_forms_by_taylor_expansion(
index, weight, precision)
weak_index_matrix = matrix(ZZ, [[
f[(n, r)] for n in xrange((index - 1) // 4 + 1)
for r in xrange(isqrt(4 * n * index) + 1, index + 1)
] for f in weak_forms])
_jacobi_forms_by_taylor_expansion_coords_cache[key] = \
weak_index_matrix.left_kernel().echelonized_basis()
return _jacobi_forms_by_taylor_expansion_coords_cache[key]
def decompositions(self, s):
(n, r) = s
fm = 4 * self.__m
if self.__weak_forms:
yield ((0, 0), (n, r))
yield ((n, r), (0, 0))
msq = self.__m**2
for n1 in xrange(1, n):
n2 = n - n1
for r1 in xrange(
max(r - isqrt(fm * n2 + msq),
isqrt(fm * n1 + msq - 1) + 1),
min(r + isqrt(fm * n2 + msq) + 1,
isqrt(fm * n1 + msq) + 1)):
yield ((n1, r1), (n2, r - r1))
else:
yield ((0, 0), (n, r))
yield ((n, r), (0, 0))
for n1 in xrange(1, n):
n2 = n - n1
##r = r1 + r2
##r1**2 <= 4 n1 m
## (r - r1)**2 <= 4 n2 m
## r1**2 - 2*r1*r + r**2 - 4 m n2 <= 0
## r1 <-> r \pm \sqrt{r**2 - r**2 + 4 m n2}
for r1 in xrange(
max(r - isqrt(fm * n2),
isqrt(fm * n1 - 1) + 1),
min(r + isqrt(fm * n2) + 1,
isqrt(fm * n1) + 1)):
yield ((n1, r1), (n2, r - r1))
raise StopIteration
def iter_positive_forms(self):
if self.__disc is infinity:
raise ValueError("infinity is not a true filter index")
if self.__reduced:
for (l, (u, x)) in enumerate(self.__p1list):
if u == 0:
for a in range(self.__level, self.__trace, self.__level):
for c in range(a, self.__trace - a):
for b in range(
2 * isqrt(a * c - 1) +
2 if a * c != 1 else 1, 2 * isqrt(a * c)):
yield ((a, b, c), l)
else:
for a in range(1, self.__trace):
for b in range(a + 1):
## We need x**2 * a + x * b + c % N == 0
h = ((x**2 + 1) * a + x * b) % self.__level
if x == 0 and h == 0: h = 1
for c in range(
h, self.__trace - x * b - (x**2 + 1) * a,
self.__level):
yield ((a, b, c), l)
#! if self.__reduced
else:
for (l, (u, x)) in enumerate(self.__p1list):
if u == 0:
for a in range(self.__level, self.__trace, self.__level):
for c in range(1, self.__trace - a):
for b in range(
2 * isqrt(a * c - 1) +
2 if a * c != 1 else 1, 2 * isqrt(a * c)):
yield ((a, b, c), l)
else:
for a in range(1, self.__trace):
for c in range(self.__level, self.__trace - a,
self.__level):
for b in range(
2 * isqrt(a * c - 1) +
2 if a * c != 1 else 1, 2 * isqrt(a * c)):
yield ((a, b - 2 * x * a,
c - x * b - x**2 * a), l)
#! else self.__reduced
raise StopIteration
def iter_positive_forms(self) :
if self.__disc is infinity :
raise ValueError, "infinity is not a true filter index"
if self.__reduced :
for (l, (u,x)) in enumerate(self.__p1list) :
if u == 0 :
for a in xrange(self.__level, self.__trace, self.__level) :
for c in xrange(a, self.__trace - a) :
for b in xrange( 2 * isqrt(a * c - 1) + 2 if a*c != 1 else 1,
2 * isqrt(a * c) ) :
yield ((a,b,c), l)
else :
for a in xrange(1, self.__trace) :
for b in xrange(a+1) :
## We need x**2 * a + x * b + c % N == 0
h = ((x**2 + 1) * a + x * b) % self.__level
if x == 0 and h == 0 : h = 1
for c in xrange( h, self.__trace - x * b - (x**2 + 1) * a, self.__level ) :
yield ((a,b,c), l)
#! if self.__reduced
else :
for (l, (u,x)) in enumerate(self.__p1list) :
if u == 0 :
for a in xrange(self.__level, self.__trace, self.__level) :
for c in xrange(1, self.__trace - a) :
for b in xrange( 2 * isqrt(a * c - 1) + 2 if a*c != 1 else 1,
2 * isqrt(a * c) ) :
yield ((a,b,c), l)
else :
for a in xrange(1, self.__trace) :
for c in xrange(self.__level, self.__trace - a, self.__level) :
for b in xrange( 2 * isqrt(a * c - 1) + 2 if a*c != 1 else 1,
2 * isqrt(a * c) ) :
yield ((a, b - 2 * x * a, c - x * b - x**2 * a), l)
#! else self.__reduced
raise StopIteration
def init_ecdb_stats(self):
if self._stats:
return
logger.debug("Computing elliptic curve stats...")
ecdbstats = db.ec_curves.stats
counts = self._counts
stats = {}
rank_counts = []
rdict = dict(ecdbstats.get_oldstat('rank')['counts'])
crdict = dict(ecdbstats.get_oldstat('class/rank')['counts'])
for r in range(counts['max_rank']+1):
try:
ncu = rdict[str(r)]
ncl = crdict[str(r)]
except KeyError:
ncu = rdict[r]
ncl = crdict[r]
prop = format_percentage(ncl,counts['nclasses'])
rank_counts.append({'r': r, 'ncurves': ncu, 'nclasses': ncl, 'prop': prop})
stats['rank_counts'] = rank_counts
tor_counts = []
tor_counts2 = []
ncurves = counts['ncurves']
tdict = dict(ecdbstats.get_oldstat('torsion')['counts'])
tsdict = dict(ecdbstats.get_oldstat('torsion_structure')['counts'])
for t in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16]:
try:
ncu = tdict[t]
except KeyError:
ncu = tdict[str(t)]
if t in [4,8,12]: # two possible structures
ncyc = tsdict[str(t)]
gp = "\(C_{%s}\)"%t
prop = format_percentage(ncyc,ncurves)
tor_counts.append({'t': t, 'gp': gp, 'ncurves': ncyc, 'prop': prop})
nncyc = ncu-ncyc
gp = "\(C_{2}\\times C_{%s}\)"%(t//2)
prop = format_percentage(nncyc,ncurves)
tor_counts2.append({'t': t, 'gp': gp, 'ncurves': nncyc, 'prop': prop})
elif t==16: # all C_2 x C_8
gp = "\(C_{2}\\times C_{8}\)"
prop = format_percentage(ncu,ncurves)
tor_counts2.append({'t': t, 'gp': gp, 'ncurves': ncu, 'prop': prop})
else: # all cyclic
gp = "\(C_{%s}\)"%t
prop = format_percentage(ncu,ncurves)
tor_counts.append({'t': t, 'gp': gp, 'ncurves': ncu, 'prop': prop})
stats['tor_counts'] = tor_counts+tor_counts2
shadict = dict(ecdbstats.get_oldstat('sha')['counts'])
stats['max_sha'] = max([int(s) for s in shadict])
sha_counts = []
from sage.misc.functional import isqrt
sha_is_int = True
try:
nc = shadict[1]
except KeyError:
sha_is_int = False
for s in range(1,1+isqrt(stats['max_sha'])):
s2 = s*s
if sha_is_int:
nc = shadict.get(s2,0)
else:
nc = shadict.get(str(s2),0)
if nc:
sha_counts.append({'s': s, 'ncurves': nc})
stats['sha_counts'] = sha_counts
self._stats = stats
logger.debug("... finished computing elliptic curve stats.")
def maass_form(self, f, g, k=None, is_integral=False):
r"""
Return the Siegel modular form `I(f,g)` (Notation as in [Sko]).
INPUT:
- `f` -- modular form of level `1`
- `g` -- cusp form of level `1` and weight = ``weight of f + 2``
- ``is_integral`` -- ``True`` if the result is garanteed to have integer
coefficients
"""
## we introduce an abbreviations
if is_integral:
PS = self.integral_power_series_ring()
else:
PS = self.power_series_ring()
fismodular = isinstance(f, ModularFormElement)
gismodular = isinstance(g, ModularFormElement)
## We only check the arguments if f and g are ModularFormElements.
## Otherwise we trust in the user
if fismodular and gismodular:
assert( f.weight() + 2 == g.weight() | (f==0) | (g==0)), \
"incorrect weights!"
assert (
g.q_expansion(1) == 0), "second argument is not a cusp form"
qexp_prec = self._get_maass_form_qexp_prec()
if qexp_prec is None: # there are no forms below prec
return dict()
if fismodular:
k = f.weight()
if f == f.parent()(0):
f = PS(0, qexp_prec)
else:
f = PS(f.qexp(qexp_prec), qexp_prec)
elif f == 0:
f = PS(0, qexp_prec)
else:
f = PS(f(qexp_prec), qexp_prec)
if gismodular:
k = g.weight() - 2
if g == g.parent()(0):
g = PS(0, qexp_prec)
else:
g = PS(g.qexp(qexp_prec), qexp_prec)
elif g == 0:
g = PS(0, qexp_prec)
else:
g = PS(g(qexp_prec), qexp_prec)
if k is None:
raise ValueError, "if neither f nor g are not ModularFormElements " + \
"you must pass k"
fderiv = f.derivative().shift(1)
f *= Integer(k / 2)
gfderiv = g - fderiv
## Form A and B - the Jacobi forms used in [Sko]'s I map.
## This is not necessary if we multiply Ifg0 and Ifg1 by etapow
# (A0,A1,B0,B1) = (a0*etapow, a1*etapow, b0*etapow, b1*etapow)
## Calculate the image of the pair of modular forms (f,g) under
## [Sko]'s isomorphism I : M_{k} \oplus S_{k+2} -> J_{k,1}.
# Multiplication of big polynomials may take > 60 GB, so wie have
# to do it in small parts; This is only implemented for integral
# coefficients.
"""
Create the Jacobi form I(f,g) as in [Sko].
It suffices to construct for all Jacobi forms phi only the part
sum_{r=0,1;n} c_phi(r^2-4n) q^n zeta^r.
When, in this code part, we speak of Jacobi form we only mean this part.
We need to compute Ifg = \sum_{r=0,1; n} c(r^2-4n) q^n zeta^r up to
4n-r^2 <= Dtop, i.e. n < precision
"""
## Create the Jacobi forms A=a*etapow and B=b*etapow in stages.
## Recall a = sum_{s != r mod 2} s^2*(-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## b = sum_{s != r mod 2} (-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## r, s run over ZZ but with opposite parities.
## For r=0, we need s odd, (s^2-1)/4 < precision, with s=2t+1 hence t^2+t < precision.
## For r=1, we need s even, s^2/4 < precision, with s=2t hence t^2 < precision.
## we use a slightly overestimated ab_prec
ab_prec = isqrt(qexp_prec + 1)
a1dict = dict()
a0dict = dict()
b1dict = dict()
b0dict = dict()
for t in xrange(1, ab_prec + 1):
tmp = t**2
a1dict[tmp] = -8 * tmp
b1dict[tmp] = -2
tmp += t
a0dict[tmp] = 8 * tmp + 2
b0dict[tmp] = 2
b1dict[0] = -1
a0dict[0] = 2
b0dict[0] = 2
a1 = PS(a1dict)
b1 = PS(b1dict)
a0 = PS(a0dict)
b0 = PS(b0dict)
## Finally: I(f,g) is given by the formula below:
## We multiply by etapow explecitely and save two multiplications
# Ifg0 = k/2*f*A0 - fderiv*B0 + g*B0 + O(q^precision)
# Ifg1 = k/2*f*A1 - fderiv*B1 + g*B1 + O(q^precision)
Ifg0 = (self._eta_power() * (f * a0 + gfderiv * b0)).list()
Ifg1 = (self._eta_power() * (f * a1 + gfderiv * b1)).list()
if len(Ifg0) < qexp_prec:
Ifg0 += [0] * (qexp_prec - len(Ifg0))
if len(Ifg1) < qexp_prec:
Ifg1 += [0] * (qexp_prec - len(Ifg1))
## For applying the Maass' lifting to genus 2 modular forms.
## we put the coefficients of Ifg into a dictionary Chi
## so that we can access the coefficient corresponding to
## discriminant D by going Chi[D].
Cphi = dict([(0, 0)])
for i in xrange(qexp_prec):
Cphi[-4 * i] = Ifg0[i]
Cphi[1 - 4 * i] = Ifg1[i]
del Ifg0[:], Ifg1[:]
"""
Create the Maas lift F := VI(f,g) as in [Sko].
"""
## The constant term is given by -Cphi[0]*B_{2k}/(4*k)
## (note in [Sko] this coeff has typos).
## For nonconstant terms,
## The Siegel coefficient of q^n * zeta^r * qdash^m is given
## by the formula \sum_{ a | gcd(n,r,m) } Cphi[D/a^2] where
## D = r^2-4*n*m is the discriminant.
## Hence in either case the coefficient
## is fully deterimined by the pair (D,gcd(n,r,m)).
## Put (D,t) -> \sum_{ a | t } Cphi[D/a^2]
## in a dictionary (hash table) maassc.
maass_coeffs = dict()
divisor_dict = self._divisor_dict()
## First calculate maass coefficients corresponding to strictly positive definite matrices:
for disc in self._negative_fundamental_discriminants():
for s in xrange(
1,
isqrt((-self.__precision.discriminant()) // disc) + 1):
## add (disc*s^2,t) as a hash key, for each t that divides s
for t in divisor_dict[s]:
maass_coeffs[(disc * s**2,t)] = \
sum( a**(k-1) * Cphi[disc * s**2 / a**2]
for a in divisor_dict[t] )
## Compute the coefficients of the Siegel form $F$:
siegel_coeffs = dict()
for (n, r,
m), g in self.__precision.iter_positive_forms_with_content():
siegel_coeffs[(n, r, m)] = maass_coeffs[(r**2 - 4 * m * n, g)]
## Secondly, deal with the singular part.
## Include the coeff corresponding to (0,0,0):
## maass_coeffs = {(0,0): -bernoulli(k)/(2*k)*Cphi[0]}
siegel_coeffs[(0, 0, 0)] = -bernoulli(k) / (2 * k) * Cphi[0]
if is_integral:
siegel_coeffs[(0, 0, 0)] = Integer(siegel_coeffs[(0, 0, 0)])
## Calculate the other discriminant-zero maass coefficients.
## Since sigma is quite cheap it is faster to estimate the bound and
## save the time for repeated calculation
for i in xrange(1, self.__precision._indefinite_content_bound()):
## maass_coeffs[(0,i)] = sigma(i, k-1) * Cphi[0]
siegel_coeffs[(0, 0, i)] = sigma(i, k - 1) * Cphi[0]
return siegel_coeffs
def init_ecdb_stats(self):
if self._stats:
return
logger.debug("Computing elliptic curve stats...")
ecdbstats = db.ec_curves.stats
counts = self._counts
stats = {}
rank_counts = []
rdict = dict(ecdbstats.get_oldstat('rank')['counts'])
crdict = dict(ecdbstats.get_oldstat('class/rank')['counts'])
for r in range(counts['max_rank'] + 1):
try:
ncu = rdict[str(r)]
ncl = crdict[str(r)]
except KeyError:
ncu = rdict[r]
ncl = crdict[r]
prop = format_percentage(ncl, counts['nclasses'])
rank_counts.append({
'r': r,
'ncurves': ncu,
'nclasses': ncl,
'prop': prop
})
stats['rank_counts'] = rank_counts
tor_counts = []
tor_counts2 = []
ncurves = counts['ncurves']
tdict = dict(ecdbstats.get_oldstat('torsion')['counts'])
tsdict = dict(ecdbstats.get_oldstat('torsion_structure')['counts'])
for t in [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 16]:
try:
ncu = tdict[t]
except KeyError:
ncu = tdict[str(t)]
if t in [4, 8, 12]: # two possible structures
ncyc = tsdict[str(t)]
gp = "\(C_{%s}\)" % t
prop = format_percentage(ncyc, ncurves)
tor_counts.append({
't': t,
'gp': gp,
'ncurves': ncyc,
'prop': prop
})
nncyc = ncu - ncyc
gp = "\(C_{2}\\times C_{%s}\)" % (t // 2)
prop = format_percentage(nncyc, ncurves)
tor_counts2.append({
't': t,
'gp': gp,
'ncurves': nncyc,
'prop': prop
})
elif t == 16: # all C_2 x C_8
gp = "\(C_{2}\\times C_{8}\)"
prop = format_percentage(ncu, ncurves)
tor_counts2.append({
't': t,
'gp': gp,
'ncurves': ncu,
'prop': prop
})
else: # all cyclic
gp = "\(C_{%s}\)" % t
prop = format_percentage(ncu, ncurves)
tor_counts.append({
't': t,
'gp': gp,
'ncurves': ncu,
'prop': prop
})
stats['tor_counts'] = tor_counts + tor_counts2
shadict = dict(ecdbstats.get_oldstat('sha')['counts'])
stats['max_sha'] = max([int(s) for s in shadict])
sha_counts = []
from sage.misc.functional import isqrt
sha_is_int = True
try:
nc = shadict[1]
except KeyError:
sha_is_int = False
for s in range(1, 1 + isqrt(stats['max_sha'])):
s2 = s * s
if sha_is_int:
nc = shadict.get(s2, 0)
else:
nc = shadict.get(str(s2), 0)
if nc:
sha_counts.append({'s': s, 'ncurves': nc})
stats['sha_counts'] = sha_counts
self._stats = stats
logger.debug("... finished computing elliptic curve stats.")
def maass_form( self, f, g, k = None, is_integral = False) :
r"""
Return the Siegel modular form `I(f,g)` (Notation as in [Sko]).
INPUT:
- `f` -- modular form of level `1`
- `g` -- cusp form of level `1` and weight = ``weight of f + 2``
- ``is_integral`` -- ``True`` if the result is garanteed to have integer
coefficients
"""
## we introduce an abbreviations
if is_integral :
PS = self.integral_power_series_ring()
else :
PS = self.power_series_ring()
fismodular = isinstance(f, ModularFormElement)
gismodular = isinstance(g, ModularFormElement)
## We only check the arguments if f and g are ModularFormElements.
## Otherwise we trust in the user
if fismodular and gismodular :
assert( f.weight() + 2 == g.weight() | (f==0) | (g==0)), \
"incorrect weights!"
assert( g.q_expansion(1) == 0), "second argument is not a cusp form"
qexp_prec = self._get_maass_form_qexp_prec()
if qexp_prec is None : # there are no forms below prec
return dict()
if fismodular :
k = f.weight()
if f == f.parent()(0) :
f = PS(0, qexp_prec)
else :
f = PS(f.qexp(qexp_prec), qexp_prec)
elif f == 0 :
f = PS(0, qexp_prec)
else :
f = PS(f(qexp_prec), qexp_prec)
if gismodular :
k = g.weight() - 2
if g == g.parent()(0) :
g = PS(0, qexp_prec)
else :
g = PS(g.qexp(qexp_prec), qexp_prec)
elif g == 0 :
g = PS(0, qexp_prec)
else :
g = PS(g(qexp_prec), qexp_prec)
if k is None :
raise ValueError, "if neither f nor g are not ModularFormElements " + \
"you must pass k"
fderiv = f.derivative().shift(1)
f *= Integer(k/2)
gfderiv = g - fderiv
## Form A and B - the Jacobi forms used in [Sko]'s I map.
## This is not necessary if we multiply Ifg0 and Ifg1 by etapow
# (A0,A1,B0,B1) = (a0*etapow, a1*etapow, b0*etapow, b1*etapow)
## Calculate the image of the pair of modular forms (f,g) under
## [Sko]'s isomorphism I : M_{k} \oplus S_{k+2} -> J_{k,1}.
# Multiplication of big polynomials may take > 60 GB, so wie have
# to do it in small parts; This is only implemented for integral
# coefficients.
"""
Create the Jacobi form I(f,g) as in [Sko].
It suffices to construct for all Jacobi forms phi only the part
sum_{r=0,1;n} c_phi(r^2-4n) q^n zeta^r.
When, in this code part, we speak of Jacobi form we only mean this part.
We need to compute Ifg = \sum_{r=0,1; n} c(r^2-4n) q^n zeta^r up to
4n-r^2 <= Dtop, i.e. n < precision
"""
## Create the Jacobi forms A=a*etapow and B=b*etapow in stages.
## Recall a = sum_{s != r mod 2} s^2*(-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## b = sum_{s != r mod 2} (-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## r, s run over ZZ but with opposite parities.
## For r=0, we need s odd, (s^2-1)/4 < precision, with s=2t+1 hence t^2+t < precision.
## For r=1, we need s even, s^2/4 < precision, with s=2t hence t^2 < precision.
## we use a slightly overestimated ab_prec
ab_prec = isqrt(qexp_prec + 1)
a1dict = dict(); a0dict = dict()
b1dict = dict(); b0dict = dict()
for t in xrange(1, ab_prec + 1) :
tmp = t**2
a1dict[tmp] = -8*tmp
b1dict[tmp] = -2
tmp += t
a0dict[tmp] = 8*tmp + 2
b0dict[tmp] = 2
b1dict[0] = -1
a0dict[0] = 2; b0dict[0] = 2
a1 = PS(a1dict); b1 = PS(b1dict)
a0 = PS(a0dict); b0 = PS(b0dict)
## Finally: I(f,g) is given by the formula below:
## We multiply by etapow explecitely and save two multiplications
# Ifg0 = k/2*f*A0 - fderiv*B0 + g*B0 + O(q^precision)
# Ifg1 = k/2*f*A1 - fderiv*B1 + g*B1 + O(q^precision)
Ifg0 = (self._eta_power() * (f*a0 + gfderiv*b0)).list()
Ifg1 = (self._eta_power() * (f*a1 + gfderiv*b1)).list()
if len(Ifg0) < qexp_prec :
Ifg0 += [0]*(qexp_prec - len(Ifg0))
if len(Ifg1) < qexp_prec :
Ifg1 += [0]*(qexp_prec - len(Ifg1))
## For applying the Maass' lifting to genus 2 modular forms.
## we put the coefficients of Ifg into a dictionary Chi
## so that we can access the coefficient corresponding to
## discriminant D by going Chi[D].
Cphi = dict([(0,0)])
for i in xrange(qexp_prec) :
Cphi[-4*i] = Ifg0[i]
Cphi[1-4*i] = Ifg1[i]
del Ifg0[:], Ifg1[:]
"""
Create the Maas lift F := VI(f,g) as in [Sko].
"""
## The constant term is given by -Cphi[0]*B_{2k}/(4*k)
## (note in [Sko] this coeff has typos).
## For nonconstant terms,
## The Siegel coefficient of q^n * zeta^r * qdash^m is given
## by the formula \sum_{ a | gcd(n,r,m) } Cphi[D/a^2] where
## D = r^2-4*n*m is the discriminant.
## Hence in either case the coefficient
## is fully deterimined by the pair (D,gcd(n,r,m)).
## Put (D,t) -> \sum_{ a | t } Cphi[D/a^2]
## in a dictionary (hash table) maassc.
maass_coeffs = dict()
divisor_dict = self._divisor_dict()
## First calculate maass coefficients corresponding to strictly positive definite matrices:
for disc in self._negative_fundamental_discriminants() :
for s in xrange(1, isqrt((-self.__precision.discriminant()) // disc) + 1) :
## add (disc*s^2,t) as a hash key, for each t that divides s
for t in divisor_dict[s] :
maass_coeffs[(disc * s**2,t)] = \
sum( a**(k-1) * Cphi[disc * s**2 / a**2]
for a in divisor_dict[t] )
## Compute the coefficients of the Siegel form $F$:
siegel_coeffs = dict()
for (n,r,m), g in self.__precision.iter_positive_forms_with_content() :
siegel_coeffs[(n,r,m)] = maass_coeffs[(r**2 - 4*m*n, g)]
## Secondly, deal with the singular part.
## Include the coeff corresponding to (0,0,0):
## maass_coeffs = {(0,0): -bernoulli(k)/(2*k)*Cphi[0]}
siegel_coeffs[(0,0,0)] = -bernoulli(k)/(2*k)*Cphi[0]
if is_integral :
siegel_coeffs[(0,0,0)] = Integer(siegel_coeffs[(0,0,0)])
## Calculate the other discriminant-zero maass coefficients.
## Since sigma is quite cheap it is faster to estimate the bound and
## save the time for repeated calculation
for i in xrange(1, self.__precision._indefinite_content_bound()) :
## maass_coeffs[(0,i)] = sigma(i, k-1) * Cphi[0]
siegel_coeffs[(0,0,i)] = sigma(i, k-1) * Cphi[0]
return siegel_coeffs
def iter_positive_forms(self) :
if self.__disc is infinity :
raise ValueError, "infinity is not a true filter index"
if self.__reduced :
for (l, (u,x)) in enumerate(self.__p1list) :
if u == 0 :
for a in xrange(self.__level, isqrt(self.__disc // 4) + 1, self.__level) :
for b in xrange(a+1) :
for c in xrange(a, (b**2 + (self.__disc - 1))//(4*a) + 1) :
yield ((a,b,c), l)
else :
for a in xrange(1, isqrt(self.__disc // 3) + 1) :
for b in xrange(a+1) :
## We need x**2 * a + x * b + c % N == 0
h = (-((x**2 + 1) * a + x * b)) % self.__level
for c in xrange( a + h,
(b**2 + (self.__disc - 1))//(4*a) + 1, self.__level ) :
yield ((a,b,c), l)
#! if self.__reduced
else :
maxtrace = floor(self.__disc / Integer(3) + sqrt(self.__disc / Integer(3)))
for (l, (u,x)) in enumerate(self.__p1list) :
if u == 0 :
for a in xrange(self.__level, maxtrace + 1, self.__level) :
for c in xrange(1, maxtrace - a + 1) :
Bu = isqrt(4*a*c - 1)
di = 4*a*c - self.__disc
if di >= 0 :
Bl = isqrt(di) + 1
else :
Bl = 0
for b in xrange(-Bu, -Bl + 1) :
yield ((a,b,c), l)
for b in xrange(Bl, Bu + 1) :
yield ((a,b,c), l)
else :
for a in xrange(1, maxtrace + 1) :
for c in xrange(1, maxtrace - a + 1) :
Bu = isqrt(4*a*c - 1)
di = 4*a*c - self.__disc
if di >= 0 :
Bl = isqrt(di) + 1
else :
Bl = 0
h = (-x * a - int(Mod(x, self.__level)**-1) * c - Bu) % self.__level \
if x != 0 else \
(-Bu) % self.__level
for b in xrange(-Bu + self.__level - h, -Bl + 1, self.__level) :
yield ((a,b,c), l)
h = (-x * a - int(Mod(x, self.__level)**-1) * c + Bl) % self.__level \
if x !=0 else \
Bl % self.__level
for b in xrange(Bl + self.__level - h, Bu + 1, self.__level) :
yield ((a,b,c), l)
#! else self.__reduced
raise StopIteration
def _by_taylor_expansion_m1(self, f_divs, k, is_integral=False):
r"""
This provides faster implementation of by_taylor_expansion in the case
of Jacobi index `1` (and even weight). It avoids the computation of the
Wronskian by providing an explicit formula.
"""
raise RuntimeError(
"This code is known to have a bug. Use, for example,JacobiFormD1NNFactory_class._test__by_taylor_expansion(200, 10, 1) to check."
)
if is_integral:
PS = self.integral_power_series_ring()
else:
PS = self.power_series_ring()
qexp_prec = self._qexp_precision()
fderiv = f_divs[(0, 0)].derivative().shift(1)
f = f_divs[(0, 0)] * Integer(k / 2)
gfderiv = f_divs[(1, 0)] - fderiv
ab_prec = isqrt(qexp_prec + 1)
a1dict = dict()
a0dict = dict()
b1dict = dict()
b0dict = dict()
for t in xrange(1, ab_prec + 1):
tmp = t**2
a1dict[tmp] = -8 * tmp
b1dict[tmp] = -2
tmp += t
a0dict[tmp] = 8 * tmp + 2
b0dict[tmp] = 2
b1dict[0] = -1
a0dict[0] = 2
b0dict[0] = 2
a1 = PS(a1dict)
b1 = PS(b1dict)
a0 = PS(a0dict)
b0 = PS(b0dict)
Ifg0 = (self._wronskian_invdeterminant() *
(f * a0 + gfderiv * b0)).list()
Ifg1 = (self._wronskian_invdeterminant() *
(f * a1 + gfderiv * b1)).list()
if len(Ifg0) < qexp_prec:
Ifg0 += [0] * (qexp_prec - len(Ifg0))
if len(Ifg1) < qexp_prec:
Ifg1 += [0] * (qexp_prec - len(Ifg1))
Cphi = dict([(0, 0)])
for i in xrange(qexp_prec):
Cphi[-4 * i] = Ifg0[i]
Cphi[1 - 4 * i] = Ifg1[i]
del Ifg0[:], Ifg1[:]
phi_coeffs = dict()
for r in xrange(2):
for n in xrange(qexp_prec):
k = 4 * n - r**2
if k >= 0:
phi_coeffs[(n, r)] = Cphi[-k]
return phi_coeffs
def _theta_factors(self, p = None) :
r"""
Return the factor `W^\# (\theta_0, .., \theta_{2m - 1})^{\mathrm{T}}` as a list.
The `q`-expansion is shifted by `-(m + 1)(2*m + 1) / 24` which will be compensated
for by the eta factor.
"""
try :
if p is None :
return self.__theta_factors
else :
P = PowerSeriesRing(GF(p), 'q')
return [map(P, facs) for facs in self.__theta_factors]
except AttributeError :
qexp_prec = self._qexp_precision()
if p is None :
PS = self.integral_power_series_ring()
else :
PS = PowerSeriesRing(GF(p), 'q')
m = self.__precision.jacobi_index()
twom = 2 * m
frmsq = twom ** 2
thetas = dict( ((i, j), dict())
for i in xrange(m + 1) for j in xrange(m + 1) )
## We want to calculate \hat \theta_{j,l} = sum_r (2 m r + j)**2l q**(m r**2 + j r).
for r in xrange(0, isqrt((qexp_prec - 1 + m)//m) + 2) :
for j in [0,m] :
fact = (twom*r + j)**2
coeff = 2
for l in xrange(0, m + 1) :
thetas[(j,l)][m*r**2 + r*j] = coeff
coeff = coeff * fact
thetas[(0,0)][0] = 1
for r in xrange(0, isqrt((qexp_prec - 1 + m)//m) + 2) :
for j in xrange(1, m) :
fact_p = (twom*r + j)**2
fact_m = (twom*r - j)**2
coeff_p = 2
coeff_m = 2
for l in xrange(0, m + 1) :
thetas[(j,l)][m*r**2 + r*j] = coeff_p
thetas[(j,l)][m*r**2 - r*j] = coeff_m
coeff_p = coeff_p * fact_p
coeff_m = coeff_m * fact_m
thetas = dict( ( k, PS(th).add_bigoh(qexp_prec) )
for (k,th) in thetas.iteritems() )
W = matrix(PS, m + 1, [ thetas[(j, l)]
for j in xrange(m + 1) for l in xrange(m + 1) ])
## Since the adjoint of matrices with entries in a general ring
## is extremely slow for matrices of small size, we hard code the
## the cases `m = 2` and `m = 3`. The expressions are obtained by
## computing the adjoint of a matrix with entries `w_{i,j}` in a
## polynomial algebra.
if m == 2 and qexp_prec > 10**5 :
adj00 = W[1,1] * W[2,2] - W[2,1] * W[1,2]
adj01 = - W[1,0] * W[2,2] + W[2,0] * W[1,2]
adj02 = W[1,0] * W[2,1] - W[2,0] * W[1,1]
adj10 = - W[0,1] * W[2,2] + W[2,1] * W[0,2]
adj11 = W[0,0] * W[2,2] - W[2,0] * W[0,2]
adj12 = - W[0,0] * W[2,1] + W[2,0] * W[0,1]
adj20 = W[0,1] * W[1,2] - W[1,1] * W[0,2]
adj21 = - W[0,0] * W[1,2] + W[1,0] * W[0,2]
adj22 = W[0,0] * W[1,1] - W[1,0] * W[0,1]
Wadj = matrix(PS, [ [adj00, adj01, adj02],
[adj10, adj11, adj12],
[adj20, adj21, adj22] ])
elif m == 3 and qexp_prec > 10**5 :
adj00 = -W[0,2]*W[1,1]*W[2,0] + W[0,1]*W[1,2]*W[2,0] + W[0,2]*W[1,0]*W[2,1] - W[0,0]*W[1,2]*W[2,1] - W[0,1]*W[1,0]*W[2,2] + W[0,0]*W[1,1]*W[2,2]
adj01 = -W[0,3]*W[1,1]*W[2,0] + W[0,1]*W[1,3]*W[2,0] + W[0,3]*W[1,0]*W[2,1] - W[0,0]*W[1,3]*W[2,1] - W[0,1]*W[1,0]*W[2,3] + W[0,0]*W[1,1]*W[2,3]
adj02 = -W[0,3]*W[1,2]*W[2,0] + W[0,2]*W[1,3]*W[2,0] + W[0,3]*W[1,0]*W[2,2] - W[0,0]*W[1,3]*W[2,2] - W[0,2]*W[1,0]*W[2,3] + W[0,0]*W[1,2]*W[2,3]
adj03 = -W[0,3]*W[1,2]*W[2,1] + W[0,2]*W[1,3]*W[2,1] + W[0,3]*W[1,1]*W[2,2] - W[0,1]*W[1,3]*W[2,2] - W[0,2]*W[1,1]*W[2,3] + W[0,1]*W[1,2]*W[2,3]
adj10 = -W[0,2]*W[1,1]*W[3,0] + W[0,1]*W[1,2]*W[3,0] + W[0,2]*W[1,0]*W[3,1] - W[0,0]*W[1,2]*W[3,1] - W[0,1]*W[1,0]*W[3,2] + W[0,0]*W[1,1]*W[3,2]
adj11 = -W[0,3]*W[1,1]*W[3,0] + W[0,1]*W[1,3]*W[3,0] + W[0,3]*W[1,0]*W[3,1] - W[0,0]*W[1,3]*W[3,1] - W[0,1]*W[1,0]*W[3,3] + W[0,0]*W[1,1]*W[3,3]
adj12 = -W[0,3]*W[1,2]*W[3,0] + W[0,2]*W[1,3]*W[3,0] + W[0,3]*W[1,0]*W[3,2] - W[0,0]*W[1,3]*W[3,2] - W[0,2]*W[1,0]*W[3,3] + W[0,0]*W[1,2]*W[3,3]
adj13 = -W[0,3]*W[1,2]*W[3,1] + W[0,2]*W[1,3]*W[3,1] + W[0,3]*W[1,1]*W[3,2] - W[0,1]*W[1,3]*W[3,2] - W[0,2]*W[1,1]*W[3,3] + W[0,1]*W[1,2]*W[3,3]
adj20 = -W[0,2]*W[2,1]*W[3,0] + W[0,1]*W[2,2]*W[3,0] + W[0,2]*W[2,0]*W[3,1] - W[0,0]*W[2,2]*W[3,1] - W[0,1]*W[2,0]*W[3,2] + W[0,0]*W[2,1]*W[3,2]
adj21 = -W[0,3]*W[2,1]*W[3,0] + W[0,1]*W[2,3]*W[3,0] + W[0,3]*W[2,0]*W[3,1] - W[0,0]*W[2,3]*W[3,1] - W[0,1]*W[2,0]*W[3,3] + W[0,0]*W[2,1]*W[3,3]
adj22 = -W[0,3]*W[2,2]*W[3,0] + W[0,2]*W[2,3]*W[3,0] + W[0,3]*W[2,0]*W[3,2] - W[0,0]*W[2,3]*W[3,2] - W[0,2]*W[2,0]*W[3,3] + W[0,0]*W[2,2]*W[3,3]
adj23 = -W[0,3]*W[2,2]*W[3,1] + W[0,2]*W[2,3]*W[3,1] + W[0,3]*W[2,1]*W[3,2] - W[0,1]*W[2,3]*W[3,2] - W[0,2]*W[2,1]*W[3,3] + W[0,1]*W[2,2]*W[3,3]
adj30 = -W[1,2]*W[2,1]*W[3,0] + W[1,1]*W[2,2]*W[3,0] + W[1,2]*W[2,0]*W[3,1] - W[1,0]*W[2,2]*W[3,1] - W[1,1]*W[2,0]*W[3,2] + W[1,0]*W[2,1]*W[3,2]
adj31 = -W[1,3]*W[2,1]*W[3,0] + W[1,1]*W[2,3]*W[3,0] + W[1,3]*W[2,0]*W[3,1] - W[1,0]*W[2,3]*W[3,1] - W[1,1]*W[2,0]*W[3,3] + W[1,0]*W[2,1]*W[3,3]
adj32 = -W[1,3]*W[2,2]*W[3,0] + W[1,2]*W[2,3]*W[3,0] + W[1,3]*W[2,0]*W[3,2] - W[1,0]*W[2,3]*W[3,2] - W[1,2]*W[2,0]*W[3,3] + W[1,0]*W[2,2]*W[3,3]
adj33 = -W[1,3]*W[2,2]*W[3,1] + W[1,2]*W[2,3]*W[3,1] + W[1,3]*W[2,1]*W[3,2] - W[1,1]*W[2,3]*W[3,2] - W[1,2]*W[2,1]*W[3,3] + W[1,1]*W[2,2]*W[3,3]
Wadj = matrix(PS, [ [adj00, adj01, adj02, adj03],
[adj10, adj11, adj12, adj13],
[adj20, adj21, adj22, adj23],
[adj30, adj31, adj32, adj33] ])
else :
Wadj = W.adjoint()
theta_factors = [ [ Wadj[i,r] for i in xrange(m + 1) ]
for r in xrange(m + 1) ]
if p is None :
self.__theta_factors = theta_factors
return theta_factors
def _by_taylor_expansion_m1(self, f_divs, k, is_integral=False):
r"""
This provides special, faster code in the Jacobi index `1` case.
"""
if is_integral:
PS = self.integral_power_series_ring()
else:
PS = self.power_series_ring()
qexp_prec = self._qexp_precision()
fderiv = f_divs[(0, 0)].derivative().shift(1)
f = f_divs[(0, 0)] * Integer(k / 2)
gfderiv = f_divs[(1, 0)] - fderiv
ab_prec = isqrt(qexp_prec + 1)
a1dict = dict()
a0dict = dict()
b1dict = dict()
b0dict = dict()
for t in xrange(1, ab_prec + 1):
tmp = t**2
a1dict[tmp] = -8 * tmp
b1dict[tmp] = -2
tmp += t
a0dict[tmp] = 8 * tmp + 2
b0dict[tmp] = 2
b1dict[0] = -1
a0dict[0] = 2
b0dict[0] = 2
a1 = PS(a1dict)
b1 = PS(b1dict)
a0 = PS(a0dict)
b0 = PS(b0dict)
Ifg0 = (self._eta_factor() * (f * a0 + gfderiv * b0)).list()
Ifg1 = (self._eta_factor() * (f * a1 + gfderiv * b1)).list()
if len(Ifg0) < qexp_prec:
Ifg0 += [0] * (qexp_prec - len(Ifg0))
if len(Ifg1) < qexp_prec:
Ifg1 += [0] * (qexp_prec - len(Ifg1))
Cphi = dict([(0, 0)])
for i in xrange(qexp_prec):
Cphi[-4 * i] = Ifg0[i]
Cphi[1 - 4 * i] = Ifg1[i]
del Ifg0[:], Ifg1[:]
phi_coeffs = dict()
m = self.__precision.jacobi_index()
for r in xrange(2 * self.__precision.jacobi_index()):
for n in xrange(qexp_prec):
k = 4 * m * n - r**2
if k >= 0:
phi_coeffs[(n, r)] = Cphi[-k]
return phi_coeffs
def __iter__(self):
if self.index() is infinity:
raise ValueError("infinity is not a true filter index")
if self.is_reduced():
## We only iterate positive definite matrices
## and later build the semidefinite ones
## We first find possible upper left matrices
sub2 = self._calc_iter_reduced_sub2()
sub3 = self._calc_iter_reduced_sub3()
sub4 = self._calc_iter_reduced_sub4()
t = zero_matrix(ZZ, 4)
t.set_immutable()
yield t
for a0 in range(2, 2 * self.index(), 2):
t = zero_matrix(ZZ, 4)
t[3, 3] = a0
t.set_immutable()
yield t
for (a0, a1, b01) in sub2:
t = zero_matrix(ZZ, 4)
t[2, 2] = a0
t[3, 3] = a1
t[2, 3] = b01
t[3, 2] = b01
t.set_immutable()
yield t
for (a0, a1, b01, sub3s) in sub3:
t = zero_matrix(ZZ, 4)
t[1, 1] = a0
t[2, 2] = a1
t[1, 2] = b01
t[2, 1] = b01
for (a2, b02, b12) in sub3s:
ts = copy(t)
ts[3, 3] = a2
ts[1, 3] = b02
ts[3, 1] = b02
ts[2, 3] = b12
ts[3, 2] = b12
ts.set_immutable()
yield ts
for (a0, a1, b01, sub4s) in sub4:
t = zero_matrix(ZZ, 4)
t[0, 0] = a0
t[1, 1] = a1
t[0, 1] = b01
t[1, 0] = b01
for (a2, b02, b12, sub4ss) in sub4s:
ts = copy(t)
ts[2, 2] = a2
ts[0, 2] = b02
ts[2, 0] = b02
ts[1, 2] = b12
ts[2, 1] = b12
for (a3, b03, b13, b23) in sub4ss:
tss = copy(ts)
tss[3, 3] = a3
tss[0, 1] = b01
tss[1, 0] = b01
tss[0, 2] = b02
tss[2, 0] = b02
tss[0, 3] = b03
tss[3, 0] = b03
tss.set_immutable()
yield tss
#! if self.is_reduced()
else:
## We first find possible upper left matrices
sub2 = list()
for a0 in range(2 * self.index(), 2):
for a1 in range(2 * self.index(), 2):
# obstruction for t[0,1]
B1 = isqrt(a0 * a1)
for b01 in range(-B1, B1 + 1):
sub2.append((a0, a1, b01))
sub3 = list()
for (a0, a1, b01) in sub2:
sub3s = list()
for a2 in range(2 * self.index(), 2):
# obstruction for t[0,2]
B1 = isqrt(a0 * a2)
for b02 in range(-B1, B1 + 1):
# obstruction for t[1,2]
B3 = isqrt(a1 * a2)
for b12 in range(-B3, B3 + 1):
# obstruction for the minor [0,1,2] of t
if a0 * a1 * a2 - a0 * b12**2 + 2 * b01 * b12 * b02 - b01**2 * a2 - a1 * b02**2 < 0:
continue
sub3s.append((a2, b02, b12))
sub3.append((a0, a1, b01, sub3s))
for (a0, a1, b01, sub3s) in sub3:
for (a2, b02, b12) in sub3s:
for a3 in range(2 * self.index(), 2):
# obstruction for t[0,3]
B1 = isqrt(a0 * a3)
for b03 in range(-B1, B1 + 1):
# obstruction for t[1,3]
B3 = isqrt(a1 * a3)
for b13 in range(-B3, B3 + 1):
# obstruction for the minor [0,1,3] of t
if a0 * a1 * a3 - a0 * b13**2 + 2 * b01 * b13 * b03 - b01**2 * a3 - a1 * b03**2 < 0:
continue
# obstruction for t[2,3]
B3 = isqrt(a2 * a3)
for b23 in range(-B3, B3 + 1):
# obstruction for the minor [0,2,3] of t
if a0 * a2 * a3 - a0 * b23**2 + 2 * b02 * b23 * b03 - b02**2 * a3 - a2 * b03**2 < 0:
continue
# obstruction for the minor [1,2,3] of t
if a1 * a2 * a3 - a1 * b23**2 + 2 * b12 * b23 * b13 - b12**2 * a3 - a2 * b13**2 < 0:
continue
t = matrix(ZZ,
4, [
a0, b01, b02, b03, b01, a1,
b12, b13, b02, b12, a2, b23,
b03, b13, b23, a3
],
check=False)
if t.det() < 0:
continue
t.set_immutable()
yield t
raise StopIteration
def _theta_factors(self, p=None):
r"""
Return the factor `W^\# (\theta_0, .., \theta_{2m - 1})^{\mathrm{T}}` as a list.
The `q`-expansion is shifted by `-(m + 1)(2*m + 1) / 24` which will be compensated
for by the eta factor.
"""
try:
if p is None:
return self.__theta_factors
else:
P = PowerSeriesRing(GF(p), 'q')
return [map(P, facs) for facs in self.__theta_factors]
except AttributeError:
qexp_prec = self._qexp_precision()
if p is None:
PS = self.integral_power_series_ring()
else:
PS = PowerSeriesRing(GF(p), 'q')
m = self.__precision.jacobi_index()
twom = 2 * m
frmsq = twom**2
thetas = dict(
((i, j), dict()) for i in xrange(m + 1) for j in xrange(m + 1))
## We want to calculate \hat \theta_{j,l} = sum_r (2 m r + j)**2l q**(m r**2 + j r).
for r in xrange(0, isqrt((qexp_prec - 1 + m) // m) + 2):
for j in [0, m]:
fact = (twom * r + j)**2
coeff = 2
for l in xrange(0, m + 1):
thetas[(j, l)][m * r**2 + r * j] = coeff
coeff = coeff * fact
thetas[(0, 0)][0] = 1
for r in xrange(0, isqrt((qexp_prec - 1 + m) // m) + 2):
for j in xrange(1, m):
fact_p = (twom * r + j)**2
fact_m = (twom * r - j)**2
coeff_p = 2
coeff_m = 2
for l in xrange(0, m + 1):
thetas[(j, l)][m * r**2 + r * j] = coeff_p
thetas[(j, l)][m * r**2 - r * j] = coeff_m
coeff_p = coeff_p * fact_p
coeff_m = coeff_m * fact_m
thetas = dict((k, PS(th).add_bigoh(qexp_prec))
for (k, th) in thetas.iteritems())
W = matrix(
PS, m + 1,
[thetas[(j, l)] for j in xrange(m + 1) for l in xrange(m + 1)])
## Since the adjoint of matrices with entries in a general ring
## is extremely slow for matrices of small size, we hard code the
## the cases `m = 2` and `m = 3`. The expressions are obtained by
## computing the adjoint of a matrix with entries `w_{i,j}` in a
## polynomial algebra.
if m == 2 and qexp_prec > 10**5:
adj00 = W[1, 1] * W[2, 2] - W[2, 1] * W[1, 2]
adj01 = -W[1, 0] * W[2, 2] + W[2, 0] * W[1, 2]
adj02 = W[1, 0] * W[2, 1] - W[2, 0] * W[1, 1]
adj10 = -W[0, 1] * W[2, 2] + W[2, 1] * W[0, 2]
adj11 = W[0, 0] * W[2, 2] - W[2, 0] * W[0, 2]
adj12 = -W[0, 0] * W[2, 1] + W[2, 0] * W[0, 1]
adj20 = W[0, 1] * W[1, 2] - W[1, 1] * W[0, 2]
adj21 = -W[0, 0] * W[1, 2] + W[1, 0] * W[0, 2]
adj22 = W[0, 0] * W[1, 1] - W[1, 0] * W[0, 1]
Wadj = matrix(PS,
[[adj00, adj01, adj02], [adj10, adj11, adj12],
[adj20, adj21, adj22]])
elif m == 3 and qexp_prec > 10**5:
adj00 = -W[0, 2] * W[1, 1] * W[2, 0] + W[0, 1] * W[1, 2] * W[
2, 0] + W[0, 2] * W[1, 0] * W[2, 1] - W[0, 0] * W[
1, 2] * W[2, 1] - W[0, 1] * W[1, 0] * W[2, 2] + W[
0, 0] * W[1, 1] * W[2, 2]
adj01 = -W[0, 3] * W[1, 1] * W[2, 0] + W[0, 1] * W[1, 3] * W[
2, 0] + W[0, 3] * W[1, 0] * W[2, 1] - W[0, 0] * W[
1, 3] * W[2, 1] - W[0, 1] * W[1, 0] * W[2, 3] + W[
0, 0] * W[1, 1] * W[2, 3]
adj02 = -W[0, 3] * W[1, 2] * W[2, 0] + W[0, 2] * W[1, 3] * W[
2, 0] + W[0, 3] * W[1, 0] * W[2, 2] - W[0, 0] * W[
1, 3] * W[2, 2] - W[0, 2] * W[1, 0] * W[2, 3] + W[
0, 0] * W[1, 2] * W[2, 3]
adj03 = -W[0, 3] * W[1, 2] * W[2, 1] + W[0, 2] * W[1, 3] * W[
2, 1] + W[0, 3] * W[1, 1] * W[2, 2] - W[0, 1] * W[
1, 3] * W[2, 2] - W[0, 2] * W[1, 1] * W[2, 3] + W[
0, 1] * W[1, 2] * W[2, 3]
adj10 = -W[0, 2] * W[1, 1] * W[3, 0] + W[0, 1] * W[1, 2] * W[
3, 0] + W[0, 2] * W[1, 0] * W[3, 1] - W[0, 0] * W[
1, 2] * W[3, 1] - W[0, 1] * W[1, 0] * W[3, 2] + W[
0, 0] * W[1, 1] * W[3, 2]
adj11 = -W[0, 3] * W[1, 1] * W[3, 0] + W[0, 1] * W[1, 3] * W[
3, 0] + W[0, 3] * W[1, 0] * W[3, 1] - W[0, 0] * W[
1, 3] * W[3, 1] - W[0, 1] * W[1, 0] * W[3, 3] + W[
0, 0] * W[1, 1] * W[3, 3]
adj12 = -W[0, 3] * W[1, 2] * W[3, 0] + W[0, 2] * W[1, 3] * W[
3, 0] + W[0, 3] * W[1, 0] * W[3, 2] - W[0, 0] * W[
1, 3] * W[3, 2] - W[0, 2] * W[1, 0] * W[3, 3] + W[
0, 0] * W[1, 2] * W[3, 3]
adj13 = -W[0, 3] * W[1, 2] * W[3, 1] + W[0, 2] * W[1, 3] * W[
3, 1] + W[0, 3] * W[1, 1] * W[3, 2] - W[0, 1] * W[
1, 3] * W[3, 2] - W[0, 2] * W[1, 1] * W[3, 3] + W[
0, 1] * W[1, 2] * W[3, 3]
adj20 = -W[0, 2] * W[2, 1] * W[3, 0] + W[0, 1] * W[2, 2] * W[
3, 0] + W[0, 2] * W[2, 0] * W[3, 1] - W[0, 0] * W[
2, 2] * W[3, 1] - W[0, 1] * W[2, 0] * W[3, 2] + W[
0, 0] * W[2, 1] * W[3, 2]
adj21 = -W[0, 3] * W[2, 1] * W[3, 0] + W[0, 1] * W[2, 3] * W[
3, 0] + W[0, 3] * W[2, 0] * W[3, 1] - W[0, 0] * W[
2, 3] * W[3, 1] - W[0, 1] * W[2, 0] * W[3, 3] + W[
0, 0] * W[2, 1] * W[3, 3]
adj22 = -W[0, 3] * W[2, 2] * W[3, 0] + W[0, 2] * W[2, 3] * W[
3, 0] + W[0, 3] * W[2, 0] * W[3, 2] - W[0, 0] * W[
2, 3] * W[3, 2] - W[0, 2] * W[2, 0] * W[3, 3] + W[
0, 0] * W[2, 2] * W[3, 3]
adj23 = -W[0, 3] * W[2, 2] * W[3, 1] + W[0, 2] * W[2, 3] * W[
3, 1] + W[0, 3] * W[2, 1] * W[3, 2] - W[0, 1] * W[
2, 3] * W[3, 2] - W[0, 2] * W[2, 1] * W[3, 3] + W[
0, 1] * W[2, 2] * W[3, 3]
adj30 = -W[1, 2] * W[2, 1] * W[3, 0] + W[1, 1] * W[2, 2] * W[
3, 0] + W[1, 2] * W[2, 0] * W[3, 1] - W[1, 0] * W[
2, 2] * W[3, 1] - W[1, 1] * W[2, 0] * W[3, 2] + W[
1, 0] * W[2, 1] * W[3, 2]
adj31 = -W[1, 3] * W[2, 1] * W[3, 0] + W[1, 1] * W[2, 3] * W[
3, 0] + W[1, 3] * W[2, 0] * W[3, 1] - W[1, 0] * W[
2, 3] * W[3, 1] - W[1, 1] * W[2, 0] * W[3, 3] + W[
1, 0] * W[2, 1] * W[3, 3]
adj32 = -W[1, 3] * W[2, 2] * W[3, 0] + W[1, 2] * W[2, 3] * W[
3, 0] + W[1, 3] * W[2, 0] * W[3, 2] - W[1, 0] * W[
2, 3] * W[3, 2] - W[1, 2] * W[2, 0] * W[3, 3] + W[
1, 0] * W[2, 2] * W[3, 3]
adj33 = -W[1, 3] * W[2, 2] * W[3, 1] + W[1, 2] * W[2, 3] * W[
3, 1] + W[1, 3] * W[2, 1] * W[3, 2] - W[1, 1] * W[
2, 3] * W[3, 2] - W[1, 2] * W[2, 1] * W[3, 3] + W[
1, 1] * W[2, 2] * W[3, 3]
Wadj = matrix(PS, [[adj00, adj01, adj02, adj03],
[adj10, adj11, adj12, adj13],
[adj20, adj21, adj22, adj23],
[adj30, adj31, adj32, adj33]])
else:
Wadj = W.adjoint()
theta_factors = [[Wadj[i, r] for i in xrange(m + 1)]
for r in xrange(m + 1)]
if p is None:
self.__theta_factors = theta_factors
return theta_factors
def decompositions(self, t):
## We find all decompositions t1 + t2 = t
## We first find possible upper left matrices
sub2 = list()
for a0 in range(0, t[0, 0] + 1, 2):
for a1 in range(0, t[1, 1] + 1, 2):
# obstruction for t1[0,1]
B1 = isqrt(a0 * a1)
# obstruction for t2[0,1]
B2 = isqrt((t[0, 0] - a0) * (t[1, 1] - a1))
for b01 in range(max(-B1, t[0, 1] - B2),
min(B1, t[0, 1] + B2) + 1):
sub2.append((a0, a1, b01))
sub3 = list()
for (a0, a1, b01) in sub2:
sub3s = list()
for a2 in range(0, t[2, 2] + 1, 2):
# obstruction for t1[0,2]
B1 = isqrt(a0 * a2)
# obstruction for t2[0,2]
B2 = isqrt((t[0, 0] - a0) * (t[2, 2] - a2))
for b02 in range(max(-B1, t[0, 2] - B2),
min(B1, t[0, 2] + B2) + 1):
# obstruction for t1[1,2]
B3 = isqrt(a1 * a2)
# obstruction for t2[1,2]
B4 = isqrt((t[1, 1] - a1) * (t[2, 2] - a2))
for b12 in range(max(-B3, t[1, 2] - B4),
min(B3, t[1, 2] + B4) + 1):
# obstruction for the minor [0,1,2] of t1
if a0 * a1 * a2 - a0 * b12**2 + 2 * b01 * b12 * b02 - b01**2 * a2 - a1 * b02**2 < 0:
continue
# obstruction for the minor [0,1,2] of t2
if (t[0,0] - a0)*(t[1,1] - a1)*(t[2,2] - a2) - (t[0,0] - a0)*(t[1,2] - b12)**2 \
+ 2*(t[0,1] - b01)*(t[1,2] - b12)*(t[0,2] - b02) - (t[0,1] - b01)**2*(t[2,2] - a2) \
- (t[1,1] - a1)*(t[0,2] - b02)**2 < 0:
continue
sub3s.append((a2, b02, b12))
sub3.append((a0, a1, b01, sub3s))
for (a0, a1, b01, sub3s) in sub3:
for (a2, b02, b12) in sub3s:
for a3 in range(0, t[3, 3] + 1, 2):
# obstruction for t1[0,3]
B1 = isqrt(a0 * a3)
# obstruction for t2[0,3]
B2 = isqrt((t[0, 0] - a0) * (t[3, 3] - a3))
for b03 in range(max(-B1, t[0, 3] - B2),
min(B1, t[0, 3] + B2) + 1):
# obstruction for t1[1,3]
B3 = isqrt(a1 * a3)
# obstruction for t2[1,3]
B4 = isqrt((t[1, 1] - a1) * (t[3, 3] - a3))
for b13 in range(max(-B3, t[1, 3] - B4),
min(B3, t[1, 3] + B4) + 1):
# obstruction for the minor [0,1,3] of t1
if a0 * a1 * a3 - a0 * b13**2 + 2 * b01 * b13 * b03 - b01**2 * a3 - a1 * b03**2 < 0:
continue
# obstruction for the minor [0,1,3] of t2
if (t[0,0] - a0)*(t[1,1] - a1)*(t[3,3] - a3) - (t[0,0] - a0)*(t[1,3] - b13)**2 \
+ 2*(t[0,1] - b01)*(t[1,3] - b13)*(t[0,3] - b03) - (t[0,1] - b01)**2*(t[3,3] - a3) \
- (t[1,1] - a1)*(t[0,3] - b03)**2 < 0:
continue
# obstruction for t1[2,3]
B3 = isqrt(a2 * a3)
# obstruction for t2[2,3]
B4 = isqrt((t[2, 2] - a2) * (t[3, 3] - a3))
for b23 in range(max(-B3, t[2, 3] - B4),
min(B3, t[2, 3] + B4) + 1):
# obstruction for the minor [0,2,3] of t1
if a0 * a2 * a3 - a0 * b23**2 + 2 * b02 * b23 * b03 - b02**2 * a3 - a2 * b03**2 < 0:
continue
# obstruction for the minor [0,2,3] of t2
if (t[0,0] - a0)*(t[2,2] - a2)*(t[3,3] - a3) - (t[0,0] - a0)*(t[2,3] - b23)**2 \
+ 2*(t[0,2] - b02)*(t[2,3] - b23)*(t[0,3] - b03) - (t[0,2] - b02)**2*(t[3,3] - a3) \
- (t[2,2] - a2)*(t[0,3] - b03)**2 < 0:
continue
# obstruction for the minor [1,2,3] of t1
if a1 * a2 * a3 - a1 * b23**2 + 2 * b12 * b23 * b13 - b12**2 * a3 - a2 * b13**2 < 0:
continue
# obstruction for the minor [1,2,3] of t2
if (t[1,1] - a1)*(t[2,2] - a2)*(t[3,3] - a3) - (t[1,1] - a1)*(t[2,3] - b23)**2 \
+ 2*(t[1,2] - b12)*(t[2,3] - b23)*(t[1,3] - b13) - (t[1,2] - b12)**2*(t[3,3] - a3) \
- (t[2,2] - a2)*(t[1,3] - b13)**2 < 0:
continue
t1 = matrix(
ZZ,
4, [
a0, b01, b02, b03, b01, a1, b12, b13,
b02, b12, a2, b23, b03, b13, b23, a3
],
check=False)
if t1.det() < 0:
continue
t2 = t - t1
if t2.det() < 0:
continue
t1.set_immutable()
t2.set_immutable()
yield (t1, t2)
raise StopIteration
def iter_indefinite_forms(self):
r"""
Iterate over indices with non-positive discriminant.
TESTS::
sage: from psage.modform.jacobiforms.jacobiformd1nn_fourierexpansion import *
sage: list(JacobiFormD1NNFilter(2, 2, reduced = False, weak_forms = True).iter_indefinite_forms())
[(0, -1), (1, 3), (0, 0), (1, -3), (0, 1), (0, -2), (0, 2)]
sage: list(JacobiFormD1NNFilter(3, 2, reduced = False, weak_forms = True).iter_indefinite_forms())
[(0, -1), (1, 3), (2, -4), (0, 0), (2, 4), (1, -3), (0, 1), (0, -2), (0, 2)]
sage: list(JacobiFormD1NNFilter(10, 2, reduced = True, weak_forms = True).iter_indefinite_forms())
[(0, 0), (0, 1), (0, 2)]
sage: list(JacobiFormD1NNFilter(10, 3, reduced = True, weak_forms = True).iter_indefinite_forms())
[(0, 0), (0, 1), (0, 2), (0, 3)]
sage: list(JacobiFormD1NNFilter(10, 10, reduced = True, weak_forms = True).iter_indefinite_forms())
[(0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0, 6), (0, 7), (0, 8), (0, 9), (0, 10), (1, 7), (1, 8), (1, 9), (1, 10), (2, 9), (2, 10)]
"""
B = self.__bound
m = self.__m
fm = Integer(4 * self.__m)
if self.__reduced:
if self.__weak_forms:
for n in xrange(0, min(self.__m // 4 + 1, self.__bound)):
for r in xrange(
isqrt(fm * n - 1) + 1 if n != 0 else 0,
self.__m + 1):
yield (n, r)
else:
for r in xrange(
0, min(self.__m + 1,
isqrt((self.__bound - 1) * fm) + 1)):
if fm.divides(r**2):
yield (r**2 // fm, r)
else:
if self.__weak_forms:
## We first determine the reduced indices.
for n in xrange(0, min(m // 4 + 1, B)):
if n == 0:
r_iteration = range(-m + 1, m + 1)
else:
r_iteration = range( -m + 1, -isqrt(fm * n - 1) ) \
+ range( isqrt(fm * n - 1) + 1, m + 1 )
for r in r_iteration:
for l in range(
(-r - isqrt(r**2 - 4 * m *
(n - (B - 1))) - 1) // (2 * m) + 1,
(-r + isqrt(r**2 - 4 * m *
(n - (B - 1)))) // (2 * m) + 1):
if n + l * r + m * l**2 >= B:
print l, n, r
yield (n + l * r + m * l**2, r + 2 * m * l)
else:
if self.__bound > 0:
yield (0, 0)
for n in xrange(1, self.__bound):
if (fm * n).is_square():
rt_fmm = isqrt(fm * n)
yield (n, rt_fmm)
yield (n, -rt_fmm)
raise StopIteration
def _wronskian_adjoint(self, weight_parity=0, p=None):
r"""
The matrix `W^\# \pmod{p}`, mentioned on page 142 of Nils Skoruppa's thesis.
This matrix is represented by a list of lists of q-expansions.
The q-expansion is shifted by `-(m + 1) (2*m + 1) / 24` in the case of even weights, and it is
shifted by `-(m - 1) (2*m - 3) / 24` otherwise. This is compensated by the missing q-powers
returned by _wronskian_invdeterminant.
INPUT:
- `p` -- A prime or ``None``.
- ``weight_parity`` -- An integer (default: `0`).
"""
try:
if weight_parity % 2 == 0:
wronskian_adjoint = self.__wronskian_adjoint_even
else:
wronskian_adjoint = self.__wronskian_adjoint_ood
if p is None:
return wronskian_adjoint
else:
P = PowerSeriesRing(GF(p), 'q')
return [map(P, row) for row in wronskian_adjoint]
except AttributeError:
qexp_prec = self._qexp_precision()
if p is None:
PS = self.integral_power_series_ring()
else:
PS = PowerSeriesRing(GF(p), 'q')
m = self.jacobi_index()
twom = 2 * m
frmsq = twom**2
thetas = dict(
((i, j), dict()) for i in xrange(m + 1) for j in xrange(m + 1))
## We want to calculate \hat \theta_{j,l} = sum_r (2 m r + j)**2l q**(m r**2 + j r)
## in the case of even weight, and \hat \theta_{j,l} = sum_r (2 m r + j)**(2l + 1) q**(m r**2 + j r),
## otherwise.
for r in xrange(isqrt((qexp_prec - 1 + m) // m) + 2):
for j in (xrange(m + 1) if weight_parity %
2 == 0 else range(1, m)):
fact_p = (twom * r + j)**2
fact_m = (twom * r - j)**2
if weight_parity % 2 == 0:
coeff_p = 2
coeff_m = 2
else:
coeff_p = 2 * (twom * r + j)
coeff_m = -2 * (twom * r - j)
for l in (xrange(m + 1) if weight_parity %
2 == 0 else range(1, m)):
thetas[(j, l)][m * r**2 + r * j] = coeff_p
thetas[(j, l)][m * r**2 - r * j] = coeff_m
coeff_p = coeff_p * fact_p
coeff_m = coeff_m * fact_m
if weight_parity % 2 == 0:
thetas[(0, 0)][0] = 1
thetas = dict((k, PS(th).add_bigoh(qexp_prec))
for (k, th) in thetas.iteritems())
W = matrix(PS, m + 1 if weight_parity % 2 == 0 else (m - 1), [
thetas[(j, l)] for j in (xrange(m + 1) if weight_parity %
2 == 0 else range(1, m))
for l in (xrange(m + 1) if weight_parity %
2 == 0 else range(1, m))
])
## Since the adjoint of matrices with entries in a general ring
## is extremely slow for matrices of small size, we hard code the
## the cases `m = 2` and `m = 3`. The expressions are obtained by
## computing the adjoint of a matrix with entries `w_{i,j}` in a
## polynomial algebra.
if W.nrows() == 1:
Wadj = matrix(PS, [[1]])
elif W.nrows() == 2:
Wadj = matrix(PS, [[W[1, 1], -W[0, 1]], [-W[1, 0], W[0, 0]]])
elif W.nrows() == 3 and qexp_prec > 10**5:
adj00 = W[1, 1] * W[2, 2] - W[2, 1] * W[1, 2]
adj01 = -W[1, 0] * W[2, 2] + W[2, 0] * W[1, 2]
adj02 = W[1, 0] * W[2, 1] - W[2, 0] * W[1, 1]
adj10 = -W[0, 1] * W[2, 2] + W[2, 1] * W[0, 2]
adj11 = W[0, 0] * W[2, 2] - W[2, 0] * W[0, 2]
adj12 = -W[0, 0] * W[2, 1] + W[2, 0] * W[0, 1]
adj20 = W[0, 1] * W[1, 2] - W[1, 1] * W[0, 2]
adj21 = -W[0, 0] * W[1, 2] + W[1, 0] * W[0, 2]
adj22 = W[0, 0] * W[1, 1] - W[1, 0] * W[0, 1]
Wadj = matrix(PS,
[[adj00, adj01, adj02], [adj10, adj11, adj12],
[adj20, adj21, adj22]])
elif W.nrows() == 4 and qexp_prec > 10**5:
adj00 = -W[0, 2] * W[1, 1] * W[2, 0] + W[0, 1] * W[1, 2] * W[
2, 0] + W[0, 2] * W[1, 0] * W[2, 1] - W[0, 0] * W[
1, 2] * W[2, 1] - W[0, 1] * W[1, 0] * W[2, 2] + W[
0, 0] * W[1, 1] * W[2, 2]
adj01 = -W[0, 3] * W[1, 1] * W[2, 0] + W[0, 1] * W[1, 3] * W[
2, 0] + W[0, 3] * W[1, 0] * W[2, 1] - W[0, 0] * W[
1, 3] * W[2, 1] - W[0, 1] * W[1, 0] * W[2, 3] + W[
0, 0] * W[1, 1] * W[2, 3]
adj02 = -W[0, 3] * W[1, 2] * W[2, 0] + W[0, 2] * W[1, 3] * W[
2, 0] + W[0, 3] * W[1, 0] * W[2, 2] - W[0, 0] * W[
1, 3] * W[2, 2] - W[0, 2] * W[1, 0] * W[2, 3] + W[
0, 0] * W[1, 2] * W[2, 3]
adj03 = -W[0, 3] * W[1, 2] * W[2, 1] + W[0, 2] * W[1, 3] * W[
2, 1] + W[0, 3] * W[1, 1] * W[2, 2] - W[0, 1] * W[
1, 3] * W[2, 2] - W[0, 2] * W[1, 1] * W[2, 3] + W[
0, 1] * W[1, 2] * W[2, 3]
adj10 = -W[0, 2] * W[1, 1] * W[3, 0] + W[0, 1] * W[1, 2] * W[
3, 0] + W[0, 2] * W[1, 0] * W[3, 1] - W[0, 0] * W[
1, 2] * W[3, 1] - W[0, 1] * W[1, 0] * W[3, 2] + W[
0, 0] * W[1, 1] * W[3, 2]
adj11 = -W[0, 3] * W[1, 1] * W[3, 0] + W[0, 1] * W[1, 3] * W[
3, 0] + W[0, 3] * W[1, 0] * W[3, 1] - W[0, 0] * W[
1, 3] * W[3, 1] - W[0, 1] * W[1, 0] * W[3, 3] + W[
0, 0] * W[1, 1] * W[3, 3]
adj12 = -W[0, 3] * W[1, 2] * W[3, 0] + W[0, 2] * W[1, 3] * W[
3, 0] + W[0, 3] * W[1, 0] * W[3, 2] - W[0, 0] * W[
1, 3] * W[3, 2] - W[0, 2] * W[1, 0] * W[3, 3] + W[
0, 0] * W[1, 2] * W[3, 3]
adj13 = -W[0, 3] * W[1, 2] * W[3, 1] + W[0, 2] * W[1, 3] * W[
3, 1] + W[0, 3] * W[1, 1] * W[3, 2] - W[0, 1] * W[
1, 3] * W[3, 2] - W[0, 2] * W[1, 1] * W[3, 3] + W[
0, 1] * W[1, 2] * W[3, 3]
adj20 = -W[0, 2] * W[2, 1] * W[3, 0] + W[0, 1] * W[2, 2] * W[
3, 0] + W[0, 2] * W[2, 0] * W[3, 1] - W[0, 0] * W[
2, 2] * W[3, 1] - W[0, 1] * W[2, 0] * W[3, 2] + W[
0, 0] * W[2, 1] * W[3, 2]
adj21 = -W[0, 3] * W[2, 1] * W[3, 0] + W[0, 1] * W[2, 3] * W[
3, 0] + W[0, 3] * W[2, 0] * W[3, 1] - W[0, 0] * W[
2, 3] * W[3, 1] - W[0, 1] * W[2, 0] * W[3, 3] + W[
0, 0] * W[2, 1] * W[3, 3]
adj22 = -W[0, 3] * W[2, 2] * W[3, 0] + W[0, 2] * W[2, 3] * W[
3, 0] + W[0, 3] * W[2, 0] * W[3, 2] - W[0, 0] * W[
2, 3] * W[3, 2] - W[0, 2] * W[2, 0] * W[3, 3] + W[
0, 0] * W[2, 2] * W[3, 3]
adj23 = -W[0, 3] * W[2, 2] * W[3, 1] + W[0, 2] * W[2, 3] * W[
3, 1] + W[0, 3] * W[2, 1] * W[3, 2] - W[0, 1] * W[
2, 3] * W[3, 2] - W[0, 2] * W[2, 1] * W[3, 3] + W[
0, 1] * W[2, 2] * W[3, 3]
adj30 = -W[1, 2] * W[2, 1] * W[3, 0] + W[1, 1] * W[2, 2] * W[
3, 0] + W[1, 2] * W[2, 0] * W[3, 1] - W[1, 0] * W[
2, 2] * W[3, 1] - W[1, 1] * W[2, 0] * W[3, 2] + W[
1, 0] * W[2, 1] * W[3, 2]
adj31 = -W[1, 3] * W[2, 1] * W[3, 0] + W[1, 1] * W[2, 3] * W[
3, 0] + W[1, 3] * W[2, 0] * W[3, 1] - W[1, 0] * W[
2, 3] * W[3, 1] - W[1, 1] * W[2, 0] * W[3, 3] + W[
1, 0] * W[2, 1] * W[3, 3]
adj32 = -W[1, 3] * W[2, 2] * W[3, 0] + W[1, 2] * W[2, 3] * W[
3, 0] + W[1, 3] * W[2, 0] * W[3, 2] - W[1, 0] * W[
2, 3] * W[3, 2] - W[1, 2] * W[2, 0] * W[3, 3] + W[
1, 0] * W[2, 2] * W[3, 3]
adj33 = -W[1, 3] * W[2, 2] * W[3, 1] + W[1, 2] * W[2, 3] * W[
3, 1] + W[1, 3] * W[2, 1] * W[3, 2] - W[1, 1] * W[
2, 3] * W[3, 2] - W[1, 2] * W[2, 1] * W[3, 3] + W[
1, 1] * W[2, 2] * W[3, 3]
Wadj = matrix(PS, [[adj00, adj01, adj02, adj03],
[adj10, adj11, adj12, adj13],
[adj20, adj21, adj22, adj23],
[adj30, adj31, adj32, adj33]])
else:
Wadj = W.adjoint()
if weight_parity % 2 == 0:
wronskian_adjoint = [[Wadj[i, r] for i in xrange(m + 1)]
for r in xrange(m + 1)]
else:
wronskian_adjoint = [[Wadj[i, r] for i in xrange(m - 1)]
for r in xrange(m - 1)]
if p is None:
if weight_parity % 2 == 0:
self.__wronskian_adjoint_even = wronskian_adjoint
else:
self.__wronskian_adjoint_odd = wronskian_adjoint
return wronskian_adjoint