def CO_delta(r, p, N, eps):
r"""
This is used as an intermediate value in computations related to
the paper of Cohen-Oesterle.
INPUT:
- ``r`` - positive integer
- ``p`` - a prime
- ``N`` - positive integer
- ``eps`` - character
OUTPUT: element of the base ring of the character
EXAMPLES::
sage: G.<eps> = DirichletGroup(7)
sage: sage.modular.dims.CO_delta(1,5,7,eps^3)
2
"""
if not is_prime(p):
raise ValueError, "p must be prime"
K = eps.base_ring()
if p % 4 == 3:
return K(0)
if p == 2:
if r == 1:
return K(1)
return K(0)
# interesting case: p=1(mod 4).
# omega is a primitive 4th root of unity mod p.
omega = (IntegerModRing(p).unit_gens()[0])**((p - 1) // 4)
# this n is within a p-power root of a "local" 4th root of 1 modulo p.
n = Mod(int(omega.crt(Mod(1, N // (p**r)))), N)
n = n**(p**(r - 1)) # this is correct now
t = eps(n)
if t == K(1):
return K(2)
if t == K(-1):
return K(-2)
return K(0)
def CO_nu(r, p, N, eps):
r"""
This is used as an intermediate value in computations related to
the paper of Cohen-Oesterle.
INPUT:
- ``r`` - positive integer
- ``p`` - a prime
- ``N`` - positive integer
- ``eps`` - character
OUTPUT: element of the base ring of the character
EXAMPLES::
sage: G.<eps> = DirichletGroup(7)
sage: G.<eps> = DirichletGroup(7)
sage: sage.modular.dims.CO_nu(1,7,7,eps)
-1
"""
K = eps.base_ring()
if p % 3 == 2:
return K(0)
if p == 3:
if r == 1:
return K(1)
return K(0)
# interesting case: p=1(mod 3)
# omega is a cube root of 1 mod p.
omega = (IntegerModRing(p).unit_gens()[0])**((p - 1) // 3)
n = Mod(omega.crt(Mod(1, N // (p**r))),
N) # within a p-power root of a "local" cube root of 1 mod p.
n = n**(p**(r - 1)) # this is right now
t = eps(n)
if t == K(1):
return K(2)
return K(-1)
def additive_order(self):
"""
Return the additive order of this element.
EXAMPLES::
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1, 4*V.2])
sage: Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (4, 12)
sage: Q.0.additive_order()
4
sage: Q.1.additive_order()
12
sage: (Q.0+Q.1).additive_order()
12
sage: V = span([[1/2,1,1],[3/2,2,1],[0,0,1]],ZZ); W = V.span([2*V.0+4*V.1, 9*V.0+12*V.1])
sage: Q = V/W; Q
Finitely generated module V/W over Integer Ring with invariants (12, 0)
sage: Q.0.additive_order()
12
sage: type(Q.0.additive_order())
<type 'sage.rings.integer.Integer'>
sage: Q.1.additive_order()
+Infinity
"""
Q = self.parent()
I = Q.invariants()
v = self.vector()
from sage.rings.all import infinity, Mod, Integer
from sage.arith.all import lcm
n = Integer(1)
for i, a in enumerate(I):
if a == 0:
if v[i] != 0:
return infinity
else:
n = lcm(n, Mod(v[i],a).additive_order())
return n
def __init__(self, disc, level, reduced = True) :
self.__level = level
if isinstance(disc, ParamodularFormD2Filter_discriminant) :
disc = disc.index()
if disc is infinity :
self.__disc = disc
else :
oDmod = (-disc + 1) % (4 * level)
Dmod = oDmod
while Dmod > 0 :
if not Mod(Dmod, 4 * level) :
Dmod = Dmod - 1
else :
break
self.__disc = disc - (oDmod - Dmod)
self.__reduced = reduced
self.__p1list = P1List(level)
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
