def nu3(self):
r"""
Return the number of elliptic points of order 3 for this congruence
subgroup `\Gamma_0(N)`. The number of these is given by a standard formula:
0 if `N` is divisible by 9 or any prime congruent to -1 mod 3, and
otherwise `2^d` where d is the number of primes other than 3 dividing `N`.
EXAMPLE::
sage: Gamma0(2).nu3()
0
sage: Gamma0(3).nu3()
1
sage: Gamma0(9).nu3()
0
sage: Gamma0(7).nu3()
2
sage: Gamma0(21).nu3()
2
sage: Gamma0(1729).nu3()
8
"""
n = self.level()
if (n % 9 == 0):
return ZZ(0)
return prod([1 + kronecker_symbol(-3, p) for p, _ in n.factor()])
def nu3(self):
r"""
Return the number of elliptic points of order 3 for this congruence
subgroup `\Gamma_0(N)`. The number of these is given by a standard formula:
0 if `N` is divisible by 9 or any prime congruent to -1 mod 3, and
otherwise `2^d` where d is the number of primes other than 3 dividing `N`.
EXAMPLE::
sage: Gamma0(2).nu3()
0
sage: Gamma0(3).nu3()
1
sage: Gamma0(9).nu3()
0
sage: Gamma0(7).nu3()
2
sage: Gamma0(21).nu3()
2
sage: Gamma0(1729).nu3()
8
"""
n = self.level()
if (n % 9 == 0):
return ZZ(0)
return prod([ 1 + kronecker_symbol(-3, p) for p, _ in n.factor()])
def residue_ring(N):
fac = N.factor()
if len(fac) != 1:
raise NotImplementedError, "P must be a prime power"
from sage.rings.all import kronecker_symbol
P, e = fac[0]
p = P.smallest_integer()
s = kronecker_symbol(p, 5)
if e == 1:
return P.residue_field()
if s == 1:
return ResidueRing_split(N, P, p, e)
elif s == -1:
return ResidueRing_inert(N, P, p, e)
else:
if e % 2 == 0:
# easy case
return ResidueRing_ramified_even(N, P, p, e)
else:
# hardest case
return ResidueRing_ramified_odd(N, P, p, e)
def residue_ring(N):
fac = N.factor()
if len(fac) != 1:
raise NotImplementedError, "P must be a prime power"
from sage.rings.all import kronecker_symbol
P, e = fac[0]
p = P.smallest_integer()
s = kronecker_symbol(p, 5)
if e == 1:
return P.residue_field()
if s == 1:
return ResidueRing_split(N, P, p, e)
elif s == -1:
return ResidueRing_inert(N, P, p, e)
else:
if e % 2 == 0:
# easy case
return ResidueRing_ramified_even(N, P, p, e)
else:
# hardest case
return ResidueRing_ramified_odd(N, P, p, e)
def nu2(self):
r"""
Return the number of elliptic points of order 2 for this congruence
subgroup `\Gamma_0(N)`. The number of these is given by a standard formula:
0 if `N` is divisible by 4 or any prime congruent to -1 mod 4, and
otherwise `2^d` where d is the number of odd primes dividing `N`.
EXAMPLE::
sage: Gamma0(2).nu2()
1
sage: Gamma0(4).nu2()
0
sage: Gamma0(21).nu2()
0
sage: Gamma0(1105).nu2()
8
sage: [Gamma0(n).nu2() for n in [1..19]]
[1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0]
"""
n = self.level()
if n%4 == 0:
return ZZ(0)
return prod([ 1 + kronecker_symbol(-4, p) for p, _ in n.factor()])
def nu2(self):
r"""
Return the number of elliptic points of order 2 for this congruence
subgroup `\Gamma_0(N)`. The number of these is given by a standard formula:
0 if `N` is divisible by 4 or any prime congruent to -1 mod 4, and
otherwise `2^d` where d is the number of odd primes dividing `N`.
EXAMPLE::
sage: Gamma0(2).nu2()
1
sage: Gamma0(4).nu2()
0
sage: Gamma0(21).nu2()
0
sage: Gamma0(1105).nu2()
8
sage: [Gamma0(n).nu2() for n in [1..19]]
[1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0]
"""
n = self.level()
if n % 4 == 0:
return ZZ(0)
return prod([1 + kronecker_symbol(-4, p) for p, _ in n.factor()])
def dimension__jacobi_scalar_f(k, m, f) :
if moebius(f) != (-1)**k :
return 0
## We use chapter 6 of Skoruppa's thesis
ts = filter(lambda t: gcd(t, m // t) == 1, m.divisors())
## Eisenstein part
eis_dimension = 0
for t in ts :
eis_dimension += moebius(gcd(m // t, f)) \
* (t // t.squarefree_part()).isqrt() \
* (2 if (m // t) % 4 == 0 else 1)
eis_dimension = eis_dimension // len(ts)
if k == 2 and f == 1 :
eis_dimension -= len( (m // m.squarefree_part()).isqrt().divisors() )
## Cuspidal part
cusp_dimension = 0
tmp = ZZ(0)
for t in ts :
tmp += moebius(gcd(m // t, f)) * t
tmp = tmp / len(ts)
cusp_dimension += tmp * (2 * k - 3) / ZZ(12)
print "1: ", cusp_dimension
if m % 2 == 0 :
tmp = ZZ(0)
for t in ts :
tmp += moebius(gcd(m // t, f)) * kronecker_symbol(-4, t)
tmp = tmp / len(ts)
cusp_dimension += 1/ZZ(2) * kronecker_symbol(8, 2 * k - 1) * tmp
print "2: ", 1/ZZ(2) * kronecker_symbol(8, 2 * k - 1) * tmp
tmp = ZZ(0)
for t in ts :
tmp += moebius(gcd(m // t, f)) * kronecker_symbol(t, 3)
tmp = tmp / len(ts)
if m % 3 != 0 :
cusp_dimension += 1 / ZZ(3) * kronecker_symbol(k, 3) * tmp
print ": ", 1 / ZZ(3) * kronecker_symbol(k, 3) * tmp
elif k % 3 == 0 :
cusp_dimension += 2 / ZZ(3) * (-1)**k * tmp
print "3: ", 2 / ZZ(3) * (-1)**k * tmp
else :
cusp_dimension += 1 / ZZ(3) * (kronecker_symbol(k, 3) + (-1)**(k - 1)) * tmp
print "3: ", 1 / ZZ(3) * (kronecker_symbol(k, 3) + (-1)**(k - 1)) * tmp
tmp = ZZ(0)
for t in ts :
tmp += moebius(gcd(m // t, f)) \
* (t // t.squarefree_part()).isqrt() \
* (2 if (m // t) % 4 == 0 else 1)
tmp = tmp / len(ts)
cusp_dimension -= 1 / ZZ(2) * tmp
print "4: ", -1 / ZZ(2) * tmp
tmp = ZZ(0)
for t in ts :
tmp += moebius(gcd(m // t, f)) \
* sum( (( len(BinaryQF_reduced_representatives(-d, True))
if d not in [3, 4] else ( 1 / ZZ(3) if d == 3 else 1 / ZZ(2) ))
if d % 4 == 0 or d % 4 == 3 else 0 )
* kronecker_symbol(-d, m // t)
* ( 1 if (m // t) % 2 != 0 else
( 4 if (m // t) % 4 == 0 else 2 * kronecker_symbol(-d, 2) ))
for d in (4 * m).divisors() )
tmp = tmp / len(ts)
cusp_dimension -= 1 / ZZ(2) * tmp
print "5: ", -1 / ZZ(2) * tmp
if k == 2 :
cusp_dimension += len( (m // f // (m // f).squarefree_part()).isqrt().divisors() )
return eis_dimension + cusp_dimension
def dimension__jacobi_scalar_f(k, m, f):
if moebius(f) != (-1)**k:
return 0
## We use chapter 6 of Skoruppa's thesis
ts = filter(lambda t: gcd(t, m // t) == 1, m.divisors())
## Eisenstein part
eis_dimension = 0
for t in ts:
eis_dimension += moebius(gcd(m // t, f)) \
* (t // t.squarefree_part()).isqrt() \
* (2 if (m // t) % 4 == 0 else 1)
eis_dimension = eis_dimension // len(ts)
if k == 2 and f == 1:
eis_dimension -= len((m // m.squarefree_part()).isqrt().divisors())
## Cuspidal part
cusp_dimension = 0
tmp = ZZ(0)
for t in ts:
tmp += moebius(gcd(m // t, f)) * t
tmp = tmp / len(ts)
cusp_dimension += tmp * (2 * k - 3) / ZZ(12)
print "1: ", cusp_dimension
if m % 2 == 0:
tmp = ZZ(0)
for t in ts:
tmp += moebius(gcd(m // t, f)) * kronecker_symbol(-4, t)
tmp = tmp / len(ts)
cusp_dimension += 1 / ZZ(2) * kronecker_symbol(8, 2 * k - 1) * tmp
print "2: ", 1 / ZZ(2) * kronecker_symbol(8, 2 * k - 1) * tmp
tmp = ZZ(0)
for t in ts:
tmp += moebius(gcd(m // t, f)) * kronecker_symbol(t, 3)
tmp = tmp / len(ts)
if m % 3 != 0:
cusp_dimension += 1 / ZZ(3) * kronecker_symbol(k, 3) * tmp
print ": ", 1 / ZZ(3) * kronecker_symbol(k, 3) * tmp
elif k % 3 == 0:
cusp_dimension += 2 / ZZ(3) * (-1)**k * tmp
print "3: ", 2 / ZZ(3) * (-1)**k * tmp
else:
cusp_dimension += 1 / ZZ(3) * (kronecker_symbol(k, 3) +
(-1)**(k - 1)) * tmp
print "3: ", 1 / ZZ(3) * (kronecker_symbol(k, 3) + (-1)**(k - 1)) * tmp
tmp = ZZ(0)
for t in ts:
tmp += moebius(gcd(m // t, f)) \
* (t // t.squarefree_part()).isqrt() \
* (2 if (m // t) % 4 == 0 else 1)
tmp = tmp / len(ts)
cusp_dimension -= 1 / ZZ(2) * tmp
print "4: ", -1 / ZZ(2) * tmp
tmp = ZZ(0)
for t in ts:
tmp += moebius(gcd(m // t, f)) \
* sum( (( len(BinaryQF_reduced_representatives(-d, True))
if d not in [3, 4] else ( 1 / ZZ(3) if d == 3 else 1 / ZZ(2) ))
if d % 4 == 0 or d % 4 == 3 else 0 )
* kronecker_symbol(-d, m // t)
* ( 1 if (m // t) % 2 != 0 else
( 4 if (m // t) % 4 == 0 else 2 * kronecker_symbol(-d, 2) ))
for d in (4 * m).divisors() )
tmp = tmp / len(ts)
cusp_dimension -= 1 / ZZ(2) * tmp
print "5: ", -1 / ZZ(2) * tmp
if k == 2:
cusp_dimension += len(
(m // f // (m // f).squarefree_part()).isqrt().divisors())
return eis_dimension + cusp_dimension
