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 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 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 prime_norm_representative(I, O, D, ell):
"""
Given an order O and a left O-ideal I return another
left O-ideal J in the same class, but with prime norm.
This corresponds to Step 1 in the notes. So given an ideal I it returns
an ideal in the same class but with reduced norm N where N != ell is
a large prime coprime to both D and p, and ell is a quadratic
nonresidue module N.
Args:
I: A left O-ideal.
O: An order in a quaternion algebra.
D: An integer.
ell: A prime.
Returns:
A pair (J, gamma) where J = I * gamma is a left O-ideal in the same
class with prime norm N. N will be coprime to both D and p, and ell
will be a nonquadratic residue module N.
"""
# TODO: Change so O is not an argument.
if not is_minkowski_basis(I.basis()):
print("Warning: The ideal I does not have a minkowski basis"
" precomputed and Sage can not do it for you.")
nrd_I = I.norm()
B = I.quaternion_algebra()
p = B.discriminant()
alpha = B(0)
normalized_norm = Integer(alpha.reduced_norm() / nrd_I)
# Choose random elements in I until one is found with norm N*nrd(I) where N
# is prime.
m_power = 3
m = Integer(2)**m_power # TODO: Change this to a proper bound.
count = 0
while (not is_prime(normalized_norm) or normalized_norm.divides(D)
or normalized_norm == ell or normalized_norm == p
or mod(ell, normalized_norm).is_square()):
# Make a new random element.
alpha = random_combination(I.basis(), bound=m)
normalized_norm = Integer(alpha.reduced_norm() / nrd_I)
# Increase the box we search in if we've been trying for too long. Note
# this was just a random heuristic I came up with, it's not in the
# paper.
count += 1
if count > 4 * m_power:
m_power += 1
m = Integer(2)**m_power
count = 0
# We now have an element alpha with norm N*nrd(I) where N is prime. The
# ideal J = I*gamma has prime norm where gamma = conjugate(alpha) / nrd(I).
gamma = alpha.conjugate() / nrd_I
J = I.scale(gamma)
assert is_prime(Integer(J.norm()))
assert not mod(ell, Integer(J.norm())).is_square()
assert gcd(Integer(J.norm()), D) == 1
return J, gamma