def _mul_(self, other):
"""
Multiply this Hecke operator by another element of the same algebra. If
the other element is of the form `T_m` for some m, we check whether the
product is equal to `T_{mn}` and return that; if the product is not
(easily seen to be) of the form `T_{mn}`, then we calculate the product
of the two matrices and return a Hecke algebra element defined by that.
EXAMPLES: We create the space of modular symbols of level
`11` and weight `2`, then compute `T_2`
and `T_3` on it, along with their composition.
::
sage: M = ModularSymbols(11)
sage: t2 = M.hecke_operator(2); t3 = M.hecke_operator(3)
sage: t2*t3 # indirect doctest
Hecke operator T_6 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: t3.matrix() * t2.matrix()
[12 0 -2]
[ 0 2 0]
[ 0 0 2]
sage: (t2*t3).matrix()
[12 0 -2]
[ 0 2 0]
[ 0 0 2]
When we compute `T_2^5` the result is not (easily seen to
be) a Hecke operator of the form `T_n`, so it is returned
as a Hecke module homomorphism defined as a matrix::
sage: t2**5
Hecke operator on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field defined by:
[243 0 -55]
[ 0 -32 0]
[ 0 0 -32]
"""
if isinstance(other,
HeckeOperator) and other.parent() == self.parent():
n = None
if arith.gcd(self.__n, other.__n) == 1:
n = self.__n * other.__n
else:
P = set(arith.prime_divisors(self.domain().level()))
if P.issubset(set(arith.prime_divisors(self.__n))) and \
P.issubset(set(arith.prime_divisors(other.__n))):
n = self.__n * other.__n
if n:
return HeckeOperator(self.parent(), n)
# otherwise
return self.matrix_form() * other
def _mul_(self, other):
"""
Multiply this Hecke operator by another element of the same algebra. If
the other element is of the form `T_m` for some m, we check whether the
product is equal to `T_{mn}` and return that; if the product is not
(easily seen to be) of the form `T_{mn}`, then we calculate the product
of the two matrices and return a Hecke algebra element defined by that.
EXAMPLES: We create the space of modular symbols of level
`11` and weight `2`, then compute `T_2`
and `T_3` on it, along with their composition.
::
sage: M = ModularSymbols(11)
sage: t2 = M.hecke_operator(2); t3 = M.hecke_operator(3)
sage: t2*t3 # indirect doctest
Hecke operator T_6 on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field
sage: t3.matrix() * t2.matrix()
[12 0 -2]
[ 0 2 0]
[ 0 0 2]
sage: (t2*t3).matrix()
[12 0 -2]
[ 0 2 0]
[ 0 0 2]
When we compute `T_2^5` the result is not (easily seen to
be) a Hecke operator of the form `T_n`, so it is returned
as a Hecke module homomorphism defined as a matrix::
sage: t2**5
Hecke operator on Modular Symbols space of dimension 3 for Gamma_0(11) of weight 2 with sign 0 over Rational Field defined by:
[243 0 -55]
[ 0 -32 0]
[ 0 0 -32]
"""
if isinstance(other, HeckeOperator) and other.parent() == self.parent():
n = None
if arith.gcd(self.__n, other.__n) == 1:
n = self.__n * other.__n
else:
P = set(arith.prime_divisors(self.domain().level()))
if P.issubset(set(arith.prime_divisors(self.__n))) and \
P.issubset(set(arith.prime_divisors(other.__n))):
n = self.__n * other.__n
if n:
return HeckeOperator(self.parent(), n)
# otherwise
return self.matrix_form() * other
def __unit_gens_primecase(self, p):
"""
Assuming the modulus is prime, returns the smallest generator
of the group of units.
EXAMPLES::
sage: Zmod(17)._IntegerModRing_generic__unit_gens_primecase(17)
3
"""
if p==2:
return integer_mod.Mod(1,p)
P = prime_divisors(p-1)
ord = integer.Integer(p-1)
one = integer_mod.Mod(1,p)
x = 2
while x < p:
generator = True
z = integer_mod.Mod(x,p)
for q in P:
if z**(ord//q) == one:
generator = False
break
if generator:
return z
x += 1
#end for
assert False, "didn't find primitive root for p=%s"%p
def CS_genus_symbol_list(self, force_recomputation=False):
"""
Returns the list of Conway-Sloane genus symbols in increasing order of primes dividing 2*det.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3,4])
sage: Q.CS_genus_symbol_list()
[Genus symbol at 2 : [[1, 2, 3, 1, 4], [2, 1, 1, 1, 1], [3, 1, 1, 1, 1]],
Genus symbol at 3 : [[0, 3, 1], [1, 1, -1]]]
"""
## Try to use the cached list
if force_recomputation == False:
try:
return self.__CS_genus_symbol_list
except AttributeError:
pass
## Otherwise recompute and cache the list
list_of_CS_genus_symbols = [ ]
for p in prime_divisors(2 * self.det()):
list_of_CS_genus_symbols.append(self.local_genus_symbol(p))
self.__CS_genus_symbol_list = list_of_CS_genus_symbols
return list_of_CS_genus_symbols
def is_locally_equivalent_to(self, other, check_primes_only=False, force_jordan_equivalence_test=False):
"""
Determines if the current quadratic form (defined over ZZ) is
locally equivalent to the given form over the real numbers and the
`p`-adic integers for every prime p.
This works by comparing the local Jordan decompositions at every
prime, and the dimension and signature at the real place.
INPUT:
a QuadraticForm
OUTPUT:
boolean
EXAMPLES::
sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
sage: Q2 = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q1.is_globally_equivalent_to(Q2)
False
sage: Q1.is_locally_equivalent_to(Q2)
True
"""
## TO IMPLEMENT:
if self.det() == 0:
raise NotImplementedError("OOps! We need to think about whether this still works for degenerate forms... especially check the signature.")
## Check that both forms have the same dimension and base ring
if (self.dim() != other.dim()) or (self.base_ring() != other.base_ring()):
return False
## Check that the determinant and level agree
if (self.det() != other.det()) or (self.level() != other.level()):
return False
## -----------------------------------------------------
## Test equivalence over the real numbers
if self.signature() != other.signature():
return False
## Test equivalence over Z_p for all primes
if (self.base_ring() == ZZ) and (force_jordan_equivalence_test == False):
## Test equivalence with Conway-Sloane genus symbols (default over ZZ)
if self.CS_genus_symbol_list() != other.CS_genus_symbol_list():
return False
else:
## Test equivalence via the O'Meara criterion.
for p in prime_divisors(ZZ(2) * self.det()):
#print "checking the prime p = ", p
if not self.has_equivalent_Jordan_decomposition_at_prime(other, p):
return False
## All tests have passed!
return True
def CS_genus_symbol_list(self, force_recomputation=False):
"""
Returns the list of Conway-Sloane genus symbols in increasing order of primes dividing 2*det.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3,4])
sage: Q.CS_genus_symbol_list()
[Genus symbol at 2 : [[1, 2, 3, 1, 4], [2, 1, 1, 1, 1], [3, 1, 1, 1, 1]],
Genus symbol at 3 : [[0, 3, 1], [1, 1, -1]]]
"""
## Try to use the cached list
if force_recomputation == False:
try:
return self.__CS_genus_symbol_list
except AttributeError:
pass
## Otherwise recompute and cache the list
list_of_CS_genus_symbols = []
for p in prime_divisors(2 * self.det()):
list_of_CS_genus_symbols.append(self.local_genus_symbol(p))
self.__CS_genus_symbol_list = list_of_CS_genus_symbols
return list_of_CS_genus_symbols
def __unit_gens_primecase(self, p):
"""
Assuming the modulus is prime, returns the smallest generator
of the group of units.
EXAMPLES::
sage: Zmod(17)._IntegerModRing_generic__unit_gens_primecase(17)
3
"""
if p == 2:
return integer_mod.Mod(1, p)
P = prime_divisors(p - 1)
ord = integer.Integer(p - 1)
one = integer_mod.Mod(1, p)
x = 2
while x < p:
generator = True
z = integer_mod.Mod(x, p)
for q in P:
if z**(ord // q) == one:
generator = False
break
if generator:
return z
x += 1
#end for
assert False, "didn't find primitive root for p=%s" % p
def is_locally_equivalent_to(self, other, check_primes_only=False, force_jordan_equivalence_test=False):
"""
Determines if the current quadratic form (defined over ZZ) is
locally equivalent to the given form over the real numbers and the
`p`-adic integers for every prime p.
This works by comparing the local Jordan decompositions at every
prime, and the dimension and signature at the real place.
INPUT:
a QuadraticForm
OUTPUT:
boolean
EXAMPLES::
sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 2, -1, 5])
sage: Q2 = QuadraticForm(ZZ, 3, [2, 1, 2, 2, 1, 3])
sage: Q1.is_globally_equivalent_to(Q2) # optional -- souvigner
False
sage: Q1.is_locally_equivalent_to(Q2)
True
"""
## TO IMPLEMENT:
if self.det() == 0:
raise NotImplementedError, "OOps! We need to think about whether this still works for degenerate forms... especially check the signature."
## Check that both forms have the same dimension and base ring
if (self.dim() != other.dim()) or (self.base_ring() != other.base_ring()):
return False
## Check that the determinant and level agree
if (self.det() != other.det()) or (self.level() != other.level()):
return False
## -----------------------------------------------------
## Test equivalence over the real numbers
if self.signature() != other.signature():
return False
## Test equivalence over Z_p for all primes
if (self.base_ring() == ZZ) and (force_jordan_equivalence_test == False):
## Test equivalence with Conway-Sloane genus symbols (default over ZZ)
if self.CS_genus_symbol_list() != other.CS_genus_symbol_list():
return False
else:
## Test equivalence via the O'Meara criterion.
for p in prime_divisors(ZZ(2) * self.det()):
#print "checking the prime p = ", p
if not self.has_equivalent_Jordan_decomposition_at_prime(other, p):
return False
## All tests have passed!
return True
def anisotropic_primes(self):
"""
Returns a list with all of the anisotropic primes of the quadratic form.
INPUT:
None
OUTPUT:
Returns a list of prime numbers >0.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.anisotropic_primes()
[2]
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.anisotropic_primes()
[2]
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1,1])
sage: Q.anisotropic_primes()
[]
"""
## Look at all prime divisors of 2 * Det(Q) to find the anisotropic primes...
possible_primes = prime_divisors(2 * self.det())
AnisoPrimes = []
## DIAGNSOTIC
#print " Possible anisotropic primes are: " + str(possible_primes)
for p in possible_primes:
if (self.is_anisotropic(p)):
AnisoPrimes += [p]
## DIAGNSOTIC
#print " leaving anisotropic_primes..."
return AnisoPrimes
def __unit_gens_primecase(self, p):
if p==2:
return integer_mod.Mod(1,p)
P = prime_divisors(p-1)
ord = integer.Integer(p-1)
one = integer_mod.Mod(1,p)
x = 2
while x < p:
generator = True
z = integer_mod.Mod(x,p)
for q in P:
if z**(ord//q) == one:
generator = False
break
if generator:
return z
x += 1
#end for
assert False, "didn't find primitive root for p=%s"%p
def __unit_gens_primecase(self, p):
if p == 2:
return integer_mod.Mod(1, p)
P = prime_divisors(p - 1)
ord = integer.Integer(p - 1)
one = integer_mod.Mod(1, p)
x = 2
while x < p:
generator = True
z = integer_mod.Mod(x, p)
for q in P:
if z**(ord // q) == one:
generator = False
break
if generator:
return z
x += 1
#end for
assert False, "didn't find primitive root for p=%s" % p
def mass__by_Siegel_densities(self,
odd_algorithm="Pall",
even_algorithm="Watson"):
"""
Gives the mass of transformations (det 1 and -1).
WARNING: THIS IS BROKEN RIGHT NOW... =(
Optional Arguments:
- When p > 2 -- odd_algorithm = "Pall" (only one choice for now)
- When p = 2 -- even_algorithm = "Kitaoka" or "Watson"
REFERENCES:
- Nipp's Book "Tables of Quaternary Quadratic Forms".
- Papers of Pall (only for p>2) and Watson (for `p=2` -- tricky!).
- Siegel, Milnor-Hussemoller, Conway-Sloane Paper IV, Kitoaka (all of which
have problems...)
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.mass__by_Siegel_densities()
1/384
sage: Q.mass__by_Siegel_densities() - (2^Q.dim() * factorial(Q.dim()))^(-1)
0
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.mass__by_Siegel_densities()
1/48
sage: Q.mass__by_Siegel_densities() - (2^Q.dim() * factorial(Q.dim()))^(-1)
0
"""
## Setup
n = self.dim()
s = floor((n - 1) / 2)
if n % 2 != 0:
char_d = squarefree_part(2 *
self.det()) ## Accounts for the det as a QF
else:
char_d = squarefree_part(self.det())
## Form the generic zeta product
generic_prod = ZZ(2) * (pi)**(-ZZ(n) * (n + 1) / 4)
##########################################
generic_prod *= (self.det())**(
ZZ(n + 1) / 2) ## ***** This uses the Hessian Determinant ********
##########################################
#print "gp1 = ", generic_prod
generic_prod *= prod([gamma__exact(ZZ(j) / 2) for j in range(1, n + 1)])
#print "\n---", [(ZZ(j)/2, gamma__exact(ZZ(j)/2)) for j in range(1,n+1)]
#print "\n---", prod([gamma__exact(ZZ(j)/2) for j in range(1,n+1)])
#print "gp2 = ", generic_prod
generic_prod *= prod([zeta__exact(ZZ(j)) for j in range(2, 2 * s + 1, 2)])
#print "\n---", [zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)]
#print "\n---", prod([zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)])
#print "gp3 = ", generic_prod
if (n % 2 == 0):
generic_prod *= ZZ(1) * quadratic_L_function__exact(
n / 2, (-1)**(n / 2) * char_d)
#print " NEW = ", ZZ(1) * quadratic_L_function__exact(n/2, (-1)**(n/2) * char_d)
#print
#print "gp4 = ", generic_prod
#print "generic_prod =", generic_prod
## Determine the adjustment factors
adj_prod = 1
for p in prime_divisors(2 * self.det()):
## Cancel out the generic factors
p_adjustment = prod([1 - ZZ(p)**(-j) for j in range(2, 2 * s + 1, 2)])
if (n % 2 == 0):
p_adjustment *= ZZ(1) * (1 - kronecker(
(-1)**(n / 2) * char_d, p) * ZZ(p)**(-n / 2))
#print " EXTRA = ", ZZ(1) * (1 - kronecker((-1)**(n/2) * char_d, p) * ZZ(p)**(-n/2))
#print "Factor to cancel the generic one:", p_adjustment
## Insert the new mass factors
if p == 2:
if even_algorithm == "Kitaoka":
p_adjustment = p_adjustment / self.Kitaoka_mass_at_2()
elif even_algorithm == "Watson":
p_adjustment = p_adjustment / self.Watson_mass_at_2()
else:
raise TypeError(
"There is a problem -- your even_algorithm argument is invalid. Try again. =("
)
else:
if odd_algorithm == "Pall":
p_adjustment = p_adjustment / self.Pall_mass_density_at_odd_prime(
p)
else:
raise TypeError(
"There is a problem -- your optional arguments are invalid. Try again. =("
)
#print "p_adjustment for p =", p, "is", p_adjustment
## Put them together (cumulatively)
adj_prod *= p_adjustment
#print "Cumulative adj_prod =", adj_prod
## Extra adjustment for the case of a 2-dimensional form.
#if (n == 2):
# generic_prod *= 2
## Return the mass
mass = generic_prod * adj_prod
return mass
def basis(self, reduce=True):
r"""
Produce a basis for the free abelian group of eta-products of level
N (under multiplication), attempting to find basis vectors of the
smallest possible degree.
INPUT:
- ``reduce`` - a boolean (default True) indicating
whether or not to apply LLL-reduction to the calculated basis
EXAMPLE::
sage: EtaGroup(5).basis()
[Eta product of level 5 : (eta_1)^6 (eta_5)^-6]
sage: EtaGroup(12).basis()
[Eta product of level 12 : (eta_1)^2 (eta_2)^1 (eta_3)^2 (eta_4)^-1 (eta_6)^-7 (eta_12)^3,
Eta product of level 12 : (eta_1)^-4 (eta_2)^2 (eta_3)^4 (eta_6)^-2,
Eta product of level 12 : (eta_1)^-1 (eta_2)^3 (eta_3)^3 (eta_4)^-2 (eta_6)^-9 (eta_12)^6,
Eta product of level 12 : (eta_1)^1 (eta_2)^-1 (eta_3)^-3 (eta_4)^-2 (eta_6)^7 (eta_12)^-2,
Eta product of level 12 : (eta_1)^-6 (eta_2)^9 (eta_3)^2 (eta_4)^-3 (eta_6)^-3 (eta_12)^1]
sage: EtaGroup(12).basis(reduce=False) # much bigger coefficients
[Eta product of level 12 : (eta_2)^24 (eta_12)^-24,
Eta product of level 12 : (eta_1)^-336 (eta_2)^576 (eta_3)^696 (eta_4)^-216 (eta_6)^-576 (eta_12)^-144,
Eta product of level 12 : (eta_1)^-8 (eta_2)^-2 (eta_6)^2 (eta_12)^8,
Eta product of level 12 : (eta_1)^1 (eta_2)^9 (eta_3)^13 (eta_4)^-4 (eta_6)^-15 (eta_12)^-4,
Eta product of level 12 : (eta_1)^15 (eta_2)^-24 (eta_3)^-29 (eta_4)^9 (eta_6)^24 (eta_12)^5]
ALGORITHM: An eta product of level `N` is uniquely
determined by the integers `r_d` for `d | N` with
`d < N`, since `\sum_{d | N} r_d = 0`. The valid
`r_d` are those that satisfy two congruences modulo 24,
and one congruence modulo 2 for every prime divisor of N. We beef
up the congruences modulo 2 to congruences modulo 24 by multiplying
by 12. To calculate the kernel of the ensuing map
`\ZZ^m \to (\ZZ/24\ZZ)^n`
we lift it arbitrarily to an integer matrix and calculate its Smith
normal form. This gives a basis for the lattice.
This lattice typically contains "large" elements, so by default we
pass it to the reduce_basis() function which performs
LLL-reduction to give a more manageable basis.
"""
N = self.level()
divs = divisors(N)[:-1]
s = len(divs)
primedivs = prime_divisors(N)
rows = []
for i in xrange(s):
# generate a row of relation matrix
row = [ Mod(divs[i], 24) - Mod(N, 24), Mod(N/divs[i], 24) - Mod(1, 24)]
for p in primedivs:
row.append( Mod(12*(N/divs[i]).valuation(p), 24))
rows.append(row)
M = matrix(IntegerModRing(24), rows)
Mlift = M.change_ring(ZZ)
# now we compute elementary factors of Mlift
S,U,V = Mlift.smith_form()
good_vects = []
for i in xrange(U.nrows()):
vect = U.row(i)
nf = (i < S.ncols() and S[i,i]) or 0
good_vects.append((vect * 24/gcd(nf, 24)).list())
for v in good_vects:
v.append(-sum([r for r in v]))
dicts = []
for v in good_vects:
dicts.append({})
for i in xrange(s):
dicts[-1][divs[i]] = v[i]
dicts[-1][N] = v[-1]
if reduce:
return self.reduce_basis([ self(d) for d in dicts])
else:
return [self(d) for d in dicts]
def mass__by_Siegel_densities(self, odd_algorithm="Pall", even_algorithm="Watson"):
"""
Gives the mass of transformations (det 1 and -1).
WARNING: THIS IS BROKEN RIGHT NOW... =(
Optional Arguments:
- When p > 2 -- odd_algorithm = "Pall" (only one choice for now)
- When p = 2 -- even_algorithm = "Kitaoka" or "Watson"
REFERENCES:
- Nipp's Book "Tables of Quaternary Quadratic Forms".
- Papers of Pall (only for p>2) and Watson (for `p=2` -- tricky!).
- Siegel, Milnor-Hussemoller, Conway-Sloane Paper IV, Kitoaka (all of which
have problems...)
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.mass__by_Siegel_densities()
1/384
sage: Q.mass__by_Siegel_densities() - (2^Q.dim() * factorial(Q.dim()))^(-1)
0
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1])
sage: Q.mass__by_Siegel_densities()
1/48
sage: Q.mass__by_Siegel_densities() - (2^Q.dim() * factorial(Q.dim()))^(-1)
0
"""
## Setup
n = self.dim()
s = (n-1) // 2
if n % 2 != 0:
char_d = squarefree_part(2*self.det()) ## Accounts for the det as a QF
else:
char_d = squarefree_part(self.det())
## Form the generic zeta product
generic_prod = ZZ(2) * (pi)**(-ZZ(n) * (n+1) / 4)
##########################################
generic_prod *= (self.det())**(ZZ(n+1)/2) ## ***** This uses the Hessian Determinant ********
##########################################
#print "gp1 = ", generic_prod
generic_prod *= prod([gamma__exact(ZZ(j)/2) for j in range(1,n+1)])
#print "\n---", [(ZZ(j)/2, gamma__exact(ZZ(j)/2)) for j in range(1,n+1)]
#print "\n---", prod([gamma__exact(ZZ(j)/2) for j in range(1,n+1)])
#print "gp2 = ", generic_prod
generic_prod *= prod([zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)])
#print "\n---", [zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)]
#print "\n---", prod([zeta__exact(ZZ(j)) for j in range(2, 2*s+1, 2)])
#print "gp3 = ", generic_prod
if (n % 2 == 0):
generic_prod *= ZZ(1) * quadratic_L_function__exact(n/2, (-1)**(n/2) * char_d)
#print " NEW = ", ZZ(1) * quadratic_L_function__exact(n/2, (-1)**(n/2) * char_d)
#print
#print "gp4 = ", generic_prod
#print "generic_prod =", generic_prod
## Determine the adjustment factors
adj_prod = 1
for p in prime_divisors(2 * self.det()):
## Cancel out the generic factors
p_adjustment = prod([1 - ZZ(p)**(-j) for j in range(2, 2*s+1, 2)])
if (n % 2 == 0):
p_adjustment *= ZZ(1) * (1 - kronecker((-1)**(n/2) * char_d, p) * ZZ(p)**(-n/2))
#print " EXTRA = ", ZZ(1) * (1 - kronecker((-1)**(n/2) * char_d, p) * ZZ(p)**(-n/2))
#print "Factor to cancel the generic one:", p_adjustment
## Insert the new mass factors
if p == 2:
if even_algorithm == "Kitaoka":
p_adjustment = p_adjustment / self.Kitaoka_mass_at_2()
elif even_algorithm == "Watson":
p_adjustment = p_adjustment / self.Watson_mass_at_2()
else:
raise TypeError("There is a problem -- your even_algorithm argument is invalid. Try again. =(")
else:
if odd_algorithm == "Pall":
p_adjustment = p_adjustment / self.Pall_mass_density_at_odd_prime(p)
else:
raise TypeError("There is a problem -- your optional arguments are invalid. Try again. =(")
#print "p_adjustment for p =", p, "is", p_adjustment
## Put them together (cumulatively)
adj_prod *= p_adjustment
#print "Cumulative adj_prod =", adj_prod
## Extra adjustment for the case of a 2-dimensional form.
#if (n == 2):
# generic_prod *= 2
## Return the mass
mass = generic_prod * adj_prod
return mass
def new_submodule(self, p=None):
"""
Returns the new or p-new submodule of self.
INPUT:
- ``p`` - (default: None); if not None, return only
the p-new submodule.
OUTPUT: the new or p-new submodule of self, i.e. the intersection of
the kernel of the degeneracy lowering maps to level `N/p` (for the
given prime `p`, or for all prime divisors of `N` if `p` is not given).
If self is cuspidal this is a Hecke-invariant complement of the
corresponding old submodule, but this may break down on Eisenstein
subspaces (see the amusing example in William Stein's book of a form
which is new and old at the same time).
EXAMPLES::
sage: m = ModularSymbols(33); m.rank()
9
sage: m.new_submodule().rank()
3
sage: m.new_submodule(3).rank()
4
sage: m.new_submodule(11).rank()
8
"""
try:
if self.__is_new[p]:
return self
except AttributeError:
self.__is_new = {}
except KeyError:
pass
if self.rank() == 0:
self.__is_new[p] = True
return self
try:
return self.__new_submodule[p]
except AttributeError:
self.__new_submodule = {}
except KeyError:
pass
# Construct the degeneracy map d.
N = self.level()
d = None
eps = self.character()
if eps == None:
f = 1
else:
f = eps.conductor()
if p == None:
D = arith.prime_divisors(N)
else:
if N % p != 0:
raise ValueError, "p must divide the level."
D = [p]
for q in D:
# Here we are only using degeneracy *lowering* maps, so it is fine
# to be careless and pass an integer for the level. One needs to be
# a bit more careful with degeneracy *raising* maps for the Gamma1
# and GammaH cases.
if ((N//q) % f) == 0:
NN = N//q
d1 = self.degeneracy_map(NN,1).matrix()
if d is None:
d = d1
else:
d = d.augment(d1)
d = d.augment(self.degeneracy_map(NN,q).matrix())
#end if
#end for
if d is None or d == 0:
self.__is_new[p] = True
return self
else:
self.__is_new[p] = False
ns = self.submodule(d.kernel(), check=False)
ns.__is_new = {p:True}
ns._is_full_hecke_module = True
self.__new_submodule[p] = ns
return ns
def old_submodule(self, p=None):
"""
Returns the old or p-old submodule of self, i.e. the sum of the images
of the degeneracy maps from level `N/p` (for the given prime `p`, or
for all primes `p` dividing `N` if `p` is not given).
INPUT:
- ``p`` - (default: None); if not None, return only the p-old
submodule.
OUTPUT: the old or p-old submodule of self
EXAMPLES::
sage: m = ModularSymbols(33); m.rank()
9
sage: m.old_submodule().rank()
7
sage: m.old_submodule(3).rank()
6
sage: m.new_submodule(11).rank()
8
::
sage: e = DirichletGroup(16)([-1, 1])
sage: M = ModularSymbols(e, 3, sign=1); M
Modular Symbols space of dimension 4 and level 16, weight 3, character [-1, 1], sign 1, over Rational Field
sage: M.old_submodule()
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 4 and level 16, weight 3, character [-1, 1], sign 1, over Rational Field
Illustrate that trac 10664 is fixed::
sage: ModularSymbols(DirichletGroup(42)[7], 6, sign=1).old_subspace(3)
Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 40 and level 42, weight 6, character [-1, -1], sign 1, over Rational Field
"""
try:
if self.__is_old[p]:
return self
except AttributeError:
self.__is_old = {}
except KeyError:
pass
if self.rank() == 0:
self.__is_old[p] = True
return self
try:
return self.__old_submodule[p]
except AttributeError:
self.__old_submodule = {}
except KeyError:
pass
# Construct the degeneracy map d.
N = self.level()
d = None
eps = self.character()
if eps is None:
f = 1
else:
f = eps.conductor()
if p is None:
D = arith.prime_divisors(N)
else:
if N % p != 0:
raise ValueError, "p must divide the level."
D = [p]
for q in D:
NN = N//q
if NN % f == 0:
M = self.hecke_module_of_level(NN)
# Here it is vital to pass self as an argument to
# degeneracy_map, because M and the level N don't uniquely
# determine self (e.g. the degeneracy map from level 1 to level
# N could go to Gamma0(N), Gamma1(N) or anything in between)
d1 = M.degeneracy_map(self, 1).matrix()
if d is None:
d = d1
else:
d = d.stack(d1)
d = d.stack(M.degeneracy_map(self, q).matrix())
#end if
#end for
if d is None:
os = self.zero_submodule()
else:
os = self.submodule(d.image(), check=False)
self.__is_old[p] = (os == self)
os.__is_old = {p:True}
os._is_full_hecke_module = True
self.__old_submodule[p] = os
return os
def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
return_BSD=False):
r"""
Attempts to prove the Birch and Swinnerton-Dyer conjectural
formula for `E`, returning a list of primes `p` for which this
function fails to prove BSD(E,p). Here, BSD(E,p) is the
statement: "the Birch and Swinnerton-Dyer formula holds up to a
rational number coprime to `p`."
INPUT:
- ``E`` - an elliptic curve
- ``verbosity`` - int, how much information about the proof to print.
- 0 - print nothing
- 1 - print sketch of proof
- 2 - print information about remaining primes
- ``two_desc`` - string (default ``'mwrank'``), what to use for the
two-descent. Options are ``'mwrank', 'simon', 'sage'``
- ``proof`` - bool or None (default: None, see
proof.elliptic_curve or sage.structure.proof). If False, this
function just immediately returns the empty list.
- ``secs_hi`` - maximum number of seconds to try to compute the
Heegner index before switching over to trying to compute the
Heegner index bound. (Rank 0 only!)
- ``return_BSD`` - bool (default: False) whether to return an object
which contains information to reconstruct a proof
NOTE:
When printing verbose output, phrases such as "by Mazur" are referring
to the following list of papers:
REFERENCES:
.. [Cha] B. Cha. Vanishing of some cohomology goups and bounds for the
Shafarevich-Tate groups of elliptic curves. J. Number Theory, 111:154-
178, 2005.
.. [Jetchev] D. Jetchev. Global divisibility of Heegner points and
Tamagawa numbers. Compos. Math. 144 (2008), no. 4, 811--826.
.. [Kato] K. Kato. p-adic Hodge theory and values of zeta functions of
modular forms. Astérisque, (295):ix, 117-290, 2004.
.. [Kolyvagin] V. A. Kolyvagin. On the structure of Shafarevich-Tate
groups. Algebraic geometry, 94--121, Lecture Notes in Math., 1479,
Springer, Berlin, 1991.
.. [LumStein] A. Lum, W. Stein. Verification of the Birch and
Swinnerton-Dyer Conjecture for Elliptic Curves with Complex
Multiplication (unpublished)
.. [Mazur] B. Mazur. Modular curves and the Eisenstein ideal. Inst.
Hautes Études Sci. Publ. Math. No. 47 (1977), 33--186 (1978).
.. [Rubin] K. Rubin. The "main conjectures" of Iwasawa theory for
imaginary quadratic fields. Invent. Math. 103 (1991), no. 1, 25--68.
.. [SteinWuthrich] W. Stein and C. Wuthrich. Computations about
Tate-Shafarevich groups using Iwasawa theory.
http://wstein.org/papers/shark, February 2008.
.. [SteinEtAl] G. Grigorov, A. Jorza, S. Patrikis, W. Stein,
C. Tarniţǎ. Computational verification of the Birch and
Swinnerton-Dyer conjecture for individual elliptic curves.
Math. Comp. 78 (2009), no. 268, 2397--2425.
EXAMPLES::
sage: EllipticCurve('11a').prove_BSD(verbosity=2)
p = 2: True by 2-descent...
True for p not in {2, 5} by Kolyvagin.
True for p=5 by Mazur
[]
sage: EllipticCurve('14a').prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
Remaining primes:
p = 3: reducible, not surjective, good ordinary, divides a Tamagawa number
(no bounds found)
ord_p(#Sha_an) = 0
[3]
sage: EllipticCurve('14a').prove_BSD(two_desc='simon')
[3]
A rank two curve::
sage: E = EllipticCurve('389a')
We know nothing with proof=True::
sage: E.prove_BSD()
Set of all prime numbers: 2, 3, 5, 7, ...
We (think we) know everything with proof=False::
sage: E.prove_BSD(proof=False)
[]
A curve of rank 0 and prime conductor::
sage: E = EllipticCurve('19a')
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent...
True for p not in {2, 3} by Kolyvagin.
True for p=3 by Mazur
[]
sage: E = EllipticCurve('37a')
sage: E.rank()
1
sage: E._EllipticCurve_rational_field__rank
{True: 1}
sage: E.analytic_rank = lambda : 0
sage: E.prove_BSD()
Traceback (most recent call last):
...
RuntimeError: It seems that the rank conjecture does not hold for this curve (Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field)! This may be a counterexample to BSD, but is more likely a bug.
We test the consistency check for the 2-part of Sha::
sage: E = EllipticCurve('37a')
sage: S = E.sha(); S
Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: def foo(use_database):
... return 4
sage: S.an = foo
sage: E.prove_BSD()
Traceback (most recent call last):
...
RuntimeError: Apparent contradiction: 0 <= rank(sha[2]) <= 0, but ord_2(sha_an) = 2
An example with a Tamagawa number at 5::
sage: E = EllipticCurve('123a1')
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 5} by Kolyvagin.
Remaining primes:
p = 5: reducible, not surjective, good ordinary, divides a Tamagawa number
(no bounds found)
ord_p(#Sha_an) = 0
[5]
A curve for which 3 divides the order of the Tate-Shafarevich group::
sage: E = EllipticCurve('681b')
sage: E.prove_BSD(verbosity=2) # long time
p = 2: True by 2-descent...
True for p not in {2, 3} by Kolyvagin....
Remaining primes:
p = 3: irreducible, surjective, non-split multiplicative
(0 <= ord_p <= 2)
ord_p(#Sha_an) = 2
[3]
A curve for which we need to use ``heegner_index_bound``::
sage: E = EllipticCurve('198b')
sage: E.prove_BSD(verbosity=1, secs_hi=1)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
[3]
The ``return_BSD`` option gives an object with detailed information
about the proof::
sage: E = EllipticCurve('26b')
sage: B = E.prove_BSD(return_BSD=True)
sage: B.two_tor_rk
0
sage: B.N
26
sage: B.gens
[]
sage: B.primes
[7]
sage: B.heegner_indexes
{-23: 2}
TESTS:
This was fixed by trac #8184 and #7575::
sage: EllipticCurve('438e1').prove_BSD(verbosity=1)
p = 2: True by 2-descent...
True for p not in {2} by Kolyvagin.
[]
::
sage: E = EllipticCurve('960d1')
sage: E.prove_BSD(verbosity=1) # long time (4s on sage.math, 2011)
p = 2: True by 2-descent
True for p not in {2} by Kolyvagin.
[]
"""
if proof is None:
from sage.structure.proof.proof import get_flag
proof = get_flag(proof, "elliptic_curve")
else:
proof = bool(proof)
if not proof:
return []
from copy import copy
BSD = BSD_data()
# We replace this curve by the optimal curve, which we can do since
# truth of BSD(E,p) is invariant under isogeny.
BSD.curve = E.optimal_curve()
if BSD.curve.has_cm():
# ensure that CM is by a maximal order
non_max_j_invs = [-12288000, 54000, 287496, 16581375]
if BSD.curve.j_invariant() in non_max_j_invs: # is this possible for optimal curves?
if verbosity > 0:
print 'CM by non maximal order: switching curves'
for E in BSD.curve.isogeny_class(use_tuple=False):
if E.j_invariant() not in non_max_j_invs:
BSD.curve = E
break
BSD.update()
galrep = BSD.curve.galois_representation()
if two_desc=='mwrank':
M = mwrank_two_descent_work(BSD.curve, BSD.two_tor_rk)
elif two_desc=='simon':
M = simon_two_descent_work(BSD.curve, BSD.two_tor_rk)
elif two_desc=='sage':
M = native_two_isogeny_descent_work(BSD.curve, BSD.two_tor_rk)
else:
raise NotImplementedError()
rank_lower_bd, rank_upper_bd, sha2_lower_bd, sha2_upper_bd, gens = M
assert sha2_lower_bd <= sha2_upper_bd
if gens is not None: gens = BSD.curve.saturation(gens)[0]
if rank_lower_bd > rank_upper_bd:
raise RuntimeError("Apparent contradiction: %d <= rank <= %d."%(rank_lower_bd, rank_upper_bd))
BSD.two_selmer_rank = rank_upper_bd + sha2_lower_bd + BSD.two_tor_rk
if sha2_upper_bd == sha2_lower_bd:
BSD.rank = rank_lower_bd
BSD.bounds[2] = (sha2_lower_bd, sha2_upper_bd)
else:
BSD.rank = BSD.curve.rank(use_database=True)
sha2_upper_bd -= (BSD.rank - rank_lower_bd)
BSD.bounds[2] = (sha2_lower_bd, sha2_upper_bd)
if verbosity > 0:
print "Unable to compute the rank exactly -- used database."
if rank_lower_bd > 1:
# We do not know BSD(E,p) for even a single p, since it's
# an open problem to show that L^r(E,1)/(Reg*Omega) is
# rational for any curve with r >= 2.
from sage.sets.all import Primes
BSD.primes = Primes()
if return_BSD:
BSD.rank = rank_lower_bd
return BSD
return BSD.primes
if (BSD.sha_an.ord(2) == 0) != (BSD.bounds[2][1] == 0):
raise RuntimeError("Apparent contradiction: %d <= rank(sha[2]) <= %d, but ord_2(sha_an) = %d"%(sha2_lower_bd, sha2_upper_bd, BSD.sha_an.ord(2)))
if BSD.bounds[2][0] == BSD.sha_an.ord(2) and BSD.sha_an.ord(2) == BSD.bounds[2][1]:
if verbosity > 0:
print 'p = 2: True by 2-descent'
BSD.primes = []
BSD.bounds.pop(2)
BSD.proof[2] = ['2-descent']
else:
BSD.primes = [2]
BSD.proof[2] = [('2-descent',)+BSD.bounds[2]]
if len(gens) > rank_lower_bd or \
rank_lower_bd > rank_upper_bd:
raise RuntimeError("Something went wrong with 2-descent.")
if BSD.rank != len(gens):
if BSD.rank != len(BSD.curve._EllipticCurve_rational_field__gens[True]):
raise RuntimeError("Could not get generators")
gens = BSD.curve._EllipticCurve_rational_field__gens[True]
BSD.gens = [BSD.curve.point(x, check=True) for x in gens]
if BSD.rank != BSD.curve.analytic_rank():
raise RuntimeError("It seems that the rank conjecture does not hold for this curve (%s)! This may be a counterexample to BSD, but is more likely a bug."%(BSD.curve))
# reduce set of remaining primes to a finite set
import signal
kolyvagin_primes = []
heegner_index = None
if BSD.rank == 0:
for D in BSD.curve.heegner_discriminants_list(10):
max_height = max(13,BSD.curve.quadratic_twist(D).CPS_height_bound())
heegner_primes = -1
while heegner_primes == -1:
if max_height > 21: break
heegner_primes, _, exact = BSD.curve.heegner_index_bound(D, max_height=max_height)
max_height += 1
if isinstance(heegner_primes, list):
break
if not isinstance(heegner_primes, list):
raise RuntimeError("Tried 10 Heegner discriminants, and heegner_index_bound failed each time.")
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
else:
BSD.heegner_index_upper_bound[D] = max(heegner_primes+[1])
if 2 in heegner_primes:
heegner_primes.remove(2)
else: # rank 1
for D in BSD.curve.heegner_discriminants_list(10):
I = BSD.curve.heegner_index(D)
J = I.is_int()
if J[0] and J[1]>0:
I = J[1]
else:
J = (2*I).is_int()
if J[0] and J[1]>0:
I = J[1]
else:
continue
heegner_index = I
BSD.heegner_indexes[D] = I
break
heegner_primes = [p for p in arith.prime_divisors(heegner_index) if p!=2]
assert BSD.sha_an in ZZ and BSD.sha_an > 0
if BSD.curve.has_cm():
if BSD.curve.analytic_rank() == 0:
if verbosity > 0:
print ' p >= 5: true by Rubin'
BSD.primes.append(3)
else:
K = rings.QuadraticField(BSD.curve.cm_discriminant(), 'a')
D_K = K.disc()
D_E = BSD.curve.discriminant()
if len(K.factor(3)) == 1: # 3 does not split in K
BSD.primes.append(3)
for p in arith.prime_divisors(D_K):
if p >= 5:
BSD.primes.append(p)
for p in arith.prime_divisors(D_E):
if p >= 5 and D_K%p and len(K.factor(p)) == 1: # p is inert in K
BSD.primes.append(p)
for p in heegner_primes:
if p >= 5 and D_E%p != 0 and D_K%p != 0 and len(K.factor(p)) == 1: # p is good for E and inert in K
kolyvagin_primes.append(p)
for p in arith.prime_divisors(BSD.sha_an):
if p >= 5 and D_K%p != 0 and len(K.factor(p)) == 1:
if BSD.curve.is_good(p):
if verbosity > 2 and p in heegner_primes and heegner_index is None:
print 'ALERT: Prime p (%d) >= 5 dividing sha_an, good for E, inert in K, in heegner_primes, should not divide the actual Heegner index'
# Note that the following check is not entirely
# exhaustive, in case there is a p not dividing
# the Heegner index in heegner_primes,
# for which only an outer bound was computed
if p not in heegner_primes:
raise RuntimeError("p = %d divides sha_an, is of good reduction for E, inert in K, and does not divide the Heegner index. This may be a counterexample to BSD, but is more likely a bug. %s"%(p,BSD.curve))
if verbosity > 0:
print 'True for p not in {%s} by Kolyvagin (via Stein & Lum -- unpublished) and Rubin.'%str(list(set(BSD.primes).union(set(kolyvagin_primes))))[1:-1]
BSD.proof['finite'] = copy(BSD.primes)
else: # no CM
# do some tricks to get to a finite set without calling bound_kolyvagin
BSD.primes += [p for p in galrep.non_surjective() if p != 2]
for p in heegner_primes:
if p not in BSD.primes:
BSD.primes.append(p)
for p in arith.prime_divisors(BSD.sha_an):
if p not in BSD.primes and p != 2:
BSD.primes.append(p)
if verbosity > 0:
s = str(BSD.primes)[1:-1]
if 2 not in BSD.primes:
if len(s) == 0: s = '2'
else: s = '2, '+s
print 'True for p not in {' + s + '} by Kolyvagin.'
BSD.proof['finite'] = copy(BSD.primes)
primes_to_remove = []
for p in BSD.primes:
if p == 2: continue
if galrep.is_surjective(p) and not BSD.curve.has_additive_reduction(p):
if BSD.curve.has_nonsplit_multiplicative_reduction(p):
if BSD.rank > 0:
continue
if p==3:
if (not (BSD.curve.is_ordinary(p) and BSD.curve.is_good(p))) and (not BSD.curve.has_split_multiplicative_reduction(p)):
continue
if BSD.rank > 0:
continue
if verbosity > 1:
print ' p = %d: Trying p_primary_bound'%p
p_bound = BSD.Sha.p_primary_bound(p)
if BSD.proof.has_key(p):
BSD.proof[p].append(('Stein-Wuthrich', p_bound))
else:
BSD.proof[p] = [('Stein-Wuthrich', p_bound)]
if BSD.sha_an.ord(p) == 0 and p_bound == 0:
if verbosity > 0:
print 'True for p=%d by Stein-Wuthrich.'%p
primes_to_remove.append(p)
else:
if BSD.bounds.has_key(p):
BSD.bounds[p][1] = min(BSD.bounds[p][1], p_bound)
else:
BSD.bounds[p] = (0, p_bound)
print 'Analytic %d-rank is '%p + str(BSD.sha_an.ord(p)) + ', actual %d-rank is at most %d.'%(p, p_bound)
print ' by Stein-Wuthrich.\n'
for p in primes_to_remove:
BSD.primes.remove(p)
kolyvagin_primes = []
for p in BSD.primes:
if p == 2: continue
if galrep.is_surjective(p):
kolyvagin_primes.append(p)
for p in kolyvagin_primes:
BSD.primes.remove(p)
# apply other hypotheses which imply Kolyvagin's bound holds
bounded_primes = []
D_K = rings.QuadraticField(D, 'a').disc()
# Cha's hypothesis
for p in BSD.primes:
if p == 2: continue
if D_K%p != 0 and BSD.N%(p**2) != 0 and galrep.is_irreducible(p):
if verbosity > 0:
print 'Kolyvagin\'s bound for p = %d applies by Cha.'%p
if BSD.proof.has_key(p):
BSD.proof[p].append('Cha')
else:
BSD.proof[p] = ['Cha']
kolyvagin_primes.append(p)
# Stein et al.
if not BSD.curve.has_cm():
L = arith.lcm([F.torsion_order() for F in BSD.curve.isogeny_class(use_tuple=False)])
for p in BSD.primes:
if p in kolyvagin_primes or p == 2: continue
if L%p != 0:
if len(arith.prime_divisors(D_K)) == 1:
if D_K%p == 0: continue
if verbosity > 0:
print 'Kolyvagin\'s bound for p = %d applies by Stein et al.'%p
kolyvagin_primes.append(p)
if BSD.proof.has_key(p):
BSD.proof[p].append('Stein et al.')
else:
BSD.proof[p] = ['Stein et al.']
for p in kolyvagin_primes:
if p in BSD.primes:
BSD.primes.remove(p)
# apply Kolyvagin's bound
primes_to_remove = []
for p in kolyvagin_primes:
if p == 2: continue
if p not in heegner_primes:
ord_p_bound = 0
elif heegner_index is not None: # p must divide heegner_index
ord_p_bound = 2*heegner_index.ord(p)
# Here Jetchev's results apply.
m_max = max([BSD.curve.tamagawa_number(q).ord(p) for q in BSD.N.prime_divisors()])
if m_max > 0:
if verbosity > 0:
print 'Jetchev\'s results apply (at p = %d) with m_max ='%p, m_max
if BSD.proof.has_key(p):
BSD.proof[p].append(('Jetchev',m_max))
else:
BSD.proof[p] = [('Jetchev',m_max)]
ord_p_bound -= 2*m_max
else: # Heegner index is None
for D in BSD.heegner_index_upper_bound:
M = BSD.heegner_index_upper_bound[D]
ord_p_bound = 0
while p**(ord_p_bound+1) <= M**2:
ord_p_bound += 1
# now ord_p_bound is one on I_K!!!
ord_p_bound *= 2 # by Kolyvagin, now ord_p_bound is one on #Sha
break
if BSD.proof.has_key(p):
BSD.proof[p].append(('Kolyvagin',ord_p_bound))
else:
BSD.proof[p] = [('Kolyvagin',ord_p_bound)]
if BSD.sha_an.ord(p) == 0 and ord_p_bound == 0:
if verbosity > 0:
print 'True for p = %d by Kolyvagin bound'%p
primes_to_remove.append(p)
elif BSD.sha_an.ord(p) > ord_p_bound:
raise RuntimeError("p = %d: ord_p_bound == %d, but sha_an.ord(p) == %d. This appears to be a counterexample to BSD, but is more likely a bug."%(p,ord_p_bound,BSD.sha_an.ord(p)))
else: # BSD.sha_an.ord(p) <= ord_p_bound != 0:
if BSD.bounds.has_key(p):
low = BSD.bounds[p][0]
BSD.bounds[p] = (low, min(BSD.bounds[p][1], ord_p_bound))
else:
BSD.bounds[p] = (0, ord_p_bound)
for p in primes_to_remove:
kolyvagin_primes.remove(p)
BSD.primes = list( set(BSD.primes).union(set(kolyvagin_primes)) )
# Kato's bound
if BSD.rank == 0 and not BSD.curve.has_cm():
L_over_Omega = BSD.curve.lseries().L_ratio()
kato_primes = BSD.Sha.bound_kato()
primes_to_remove = []
for p in BSD.primes:
if p == 2: continue
if p not in kato_primes:
if verbosity > 0:
print 'Kato further implies that #Sha[%d] is trivial.'%p
primes_to_remove.append(p)
if BSD.proof.has_key(p):
BSD.proof[p].append(('Kato',0))
else:
BSD.proof[p] = [('Kato',0)]
if p not in [2,3] and BSD.N%p != 0:
if galrep.is_surjective(p):
bd = L_over_Omega.valuation(p)
if verbosity > 1:
print 'Kato implies that ord_p(#Sha[%d]) <= %d '%(p,bd)
if BSD.proof.has_key(p):
BSD.proof[p].append(('Kato',bd))
else:
BSD.proof[p] = [('Kato',bd)]
if BSD.bounds.has_key(p):
low = BSD.bounds[p][0]
BSD.bounds[p][1] = (low, min(BSD.bounds[p][1], bd))
else:
BSD.bounds[p] = (0, bd)
for p in primes_to_remove:
BSD.primes.remove(p)
# Mazur
primes_to_remove = []
if BSD.N.is_prime():
for p in BSD.primes:
if p == 2: continue
if galrep.is_reducible(p):
primes_to_remove.append(p)
if verbosity > 0:
print 'True for p=%s by Mazur'%p
for p in primes_to_remove:
BSD.primes.remove(p)
if BSD.proof.has_key(p):
BSD.proof[p].append('Mazur')
else:
BSD.proof[p] = ['Mazur']
BSD.primes.sort()
# Try harder to compute the Heegner index, where it matters
if heegner_index is None:
if max_height < 18:
max_height = 18
for D in BSD.heegner_index_upper_bound:
M = BSD.heegner_index_upper_bound[D]
for p in kolyvagin_primes:
if p not in BSD.primes or p == 3: continue
if verbosity > 0:
print ' p = %d: Trying harder for Heegner index'%p
obt = 0
while p**(BSD.sha_an.ord(p)/2+1) <= M and max_height < 22:
if verbosity > 2:
print ' trying max_height =', max_height
old_bound = M
M, _, exact = BSD.curve.heegner_index_bound(D, max_height=max_height, secs_dc=secs_dc)
if M == -1:
max_height += 1
continue
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
M = exact
if verbosity > 2:
print ' heegner index =', M
else:
M = max(M+[1])
if verbosity > 2:
print ' bound =', M
if old_bound == M:
obt += 1
if obt == 2:
break
max_height += 1
BSD.heegner_index_upper_bound[D] = min(M,BSD.heegner_index_upper_bound[D])
low, upp = BSD.bounds[p]
expn = 0
while p**(expn+1) <= M:
expn += 1
if 2*expn < upp:
upp = 2*expn
BSD.bounds[p] = (low,upp)
if verbosity > 0:
print ' got better bound on ord_p =', upp
if low == upp:
if upp != BSD.sha_an.ord(p):
raise RuntimeError
else:
if verbosity > 0:
print ' proven!'
BSD.primes.remove(p)
break
for p in kolyvagin_primes:
if p not in BSD.primes or p == 3: continue
for D in BSD.curve.heegner_discriminants_list(4):
if D in BSD.heegner_index_upper_bound: continue
print ' discriminant', D
if verbosity > 0:
print 'p = %d: Trying discriminant = %d for Heegner index'%(p,D)
max_height = max(10, BSD.curve.quadratic_twist(D).CPS_height_bound())
obt = 0
while True:
if verbosity > 2:
print ' trying max_height =', max_height
old_bound = M
if p**(BSD.sha_an.ord(p)/2+1) > M or max_height >= 22:
break
M, _, exact = BSD.curve.heegner_index_bound(D, max_height=max_height, secs_dc=secs_dc)
if M == -1:
max_height += 1
continue
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
M = exact
if verbosity > 2:
print ' heegner index =', M
else:
M = max(M+[1])
if verbosity > 2:
print ' bound =', M
if old_bound == M:
obt += 1
if obt == 2:
break
max_height += 1
BSD.heegner_index_upper_bound[D] = M
low, upp = BSD.bounds[p]
expn = 0
while p**(expn+1) <= M:
expn += 1
if 2*expn < upp:
upp = 2*expn
BSD.bounds[p] = (low,upp)
if verbosity > 0:
print ' got better bound =', upp
if low == upp:
if upp != BSD.sha_an.ord(p):
raise RuntimeError
else:
if verbosity > 0:
print ' proven!'
BSD.primes.remove(p)
break
# print some extra information
if verbosity > 1:
if len(BSD.primes) > 0:
print 'Remaining primes:'
for p in BSD.primes:
s = 'p = ' + str(p) + ': '
if galrep.is_irreducible(p):
s += 'ir'
s += 'reducible, '
if not galrep.is_surjective(p):
s += 'not '
s += 'surjective, '
a_p = BSD.curve.an(p)
if BSD.curve.is_good(p):
if a_p%p != 0:
s += 'good ordinary'
else:
s += 'good, non-ordinary'
else:
assert BSD.curve.is_minimal()
if a_p == 0:
s += 'additive'
elif a_p == 1:
s += 'split multiplicative'
elif a_p == -1:
s += 'non-split multiplicative'
if BSD.curve.tamagawa_product()%p==0:
s += ', divides a Tamagawa number'
if BSD.bounds.has_key(p):
s += '\n (%d <= ord_p <= %d)'%BSD.bounds[p]
else:
s += '\n (no bounds found)'
s += '\n ord_p(#Sha_an) = %d'%BSD.sha_an.ord(p)
if heegner_index is None:
may_divide = True
for D in BSD.heegner_index_upper_bound:
if p > BSD.heegner_index_upper_bound[D] or p not in kolyvagin_primes:
may_divide = False
if may_divide:
s += '\n may divide the Heegner index, for which only a bound was computed'
print s
if BSD.curve.has_cm():
if BSD.rank == 1:
BSD.proof['reason_finite'] = 'Rubin&Kolyvagin'
else:
BSD.proof['reason_finite'] = 'Rubin'
else:
BSD.proof['reason_finite'] = 'Kolyvagin'
# reduce memory footprint of BSD object:
BSD.curve = BSD.curve.label()
BSD.Sha = None
return BSD if return_BSD else BSD.primes
def basis(self, reduce=True):
r"""
Produce a basis for the free abelian group of eta-products of level
N (under multiplication), attempting to find basis vectors of the
smallest possible degree.
INPUT:
- ``reduce`` - a boolean (default True) indicating
whether or not to apply LLL-reduction to the calculated basis
EXAMPLE::
sage: EtaGroup(5).basis()
[Eta product of level 5 : (eta_1)^6 (eta_5)^-6]
sage: EtaGroup(12).basis()
[Eta product of level 12 : (eta_1)^2 (eta_2)^1 (eta_3)^2 (eta_4)^-1 (eta_6)^-7 (eta_12)^3,
Eta product of level 12 : (eta_1)^-2 (eta_2)^3 (eta_3)^6 (eta_4)^-1 (eta_6)^-9 (eta_12)^3,
Eta product of level 12 : (eta_1)^-3 (eta_2)^2 (eta_3)^1 (eta_4)^-1 (eta_6)^-2 (eta_12)^3,
Eta product of level 12 : (eta_1)^1 (eta_2)^-1 (eta_3)^-3 (eta_4)^-2 (eta_6)^7 (eta_12)^-2,
Eta product of level 12 : (eta_1)^-6 (eta_2)^9 (eta_3)^2 (eta_4)^-3 (eta_6)^-3 (eta_12)^1]
sage: EtaGroup(12).basis(reduce=False) # much bigger coefficients
[Eta product of level 12 : (eta_2)^24 (eta_12)^-24,
Eta product of level 12 : (eta_1)^-336 (eta_2)^576 (eta_3)^696 (eta_4)^-216 (eta_6)^-576 (eta_12)^-144,
Eta product of level 12 : (eta_1)^-8 (eta_2)^-2 (eta_6)^2 (eta_12)^8,
Eta product of level 12 : (eta_1)^1 (eta_2)^9 (eta_3)^13 (eta_4)^-4 (eta_6)^-15 (eta_12)^-4,
Eta product of level 12 : (eta_1)^15 (eta_2)^-24 (eta_3)^-29 (eta_4)^9 (eta_6)^24 (eta_12)^5]
ALGORITHM: An eta product of level `N` is uniquely
determined by the integers `r_d` for `d | N` with
`d < N`, since `\sum_{d | N} r_d = 0`. The valid
`r_d` are those that satisfy two congruences modulo 24,
and one congruence modulo 2 for every prime divisor of N. We beef
up the congruences modulo 2 to congruences modulo 24 by multiplying
by 12. To calculate the kernel of the ensuing map
`\ZZ^m \to (\ZZ/24\ZZ)^n`
we lift it arbitrarily to an integer matrix and calculate its Smith
normal form. This gives a basis for the lattice.
This lattice typically contains "large" elements, so by default we
pass it to the reduce_basis() function which performs
LLL-reduction to give a more manageable basis.
"""
N = self.level()
divs = divisors(N)[:-1]
s = len(divs)
primedivs = prime_divisors(N)
rows = []
for i in xrange(s):
# generate a row of relation matrix
row = [
Mod(divs[i], 24) - Mod(N, 24),
Mod(N / divs[i], 24) - Mod(1, 24)
]
for p in primedivs:
row.append(Mod(12 * (N / divs[i]).valuation(p), 24))
rows.append(row)
M = matrix(IntegerModRing(24), rows)
Mlift = M.change_ring(ZZ)
# now we compute elementary factors of Mlift
S, U, V = Mlift.smith_form()
good_vects = []
for i in xrange(U.nrows()):
vect = U.row(i)
nf = (i < S.ncols() and S[i, i]) or 0
good_vects.append((vect * 24 / gcd(nf, 24)).list())
for v in good_vects:
v.append(-sum([r for r in v]))
dicts = []
for v in good_vects:
dicts.append({})
for i in xrange(s):
dicts[-1][divs[i]] = v[i]
dicts[-1][N] = v[-1]
if reduce:
return self.reduce_basis([self(d) for d in dicts])
else:
return [self(d) for d in dicts]
def prove_BSD(E, verbosity=0, two_desc='mwrank', proof=None, secs_hi=5,
return_BSD=False):
r"""
Attempts to prove the Birch and Swinnerton-Dyer conjectural
formula for `E`, returning a list of primes `p` for which this
function fails to prove BSD(E,p). Here, BSD(E,p) is the
statement: "the Birch and Swinnerton-Dyer formula holds up to a
rational number coprime to `p`."
INPUT:
- ``E`` - an elliptic curve
- ``verbosity`` - int, how much information about the proof to print.
- 0 - print nothing
- 1 - print sketch of proof
- 2 - print information about remaining primes
- ``two_desc`` - string (default ``'mwrank'``), what to use for the
two-descent. Options are ``'mwrank', 'simon', 'sage'``
- ``proof`` - bool or None (default: None, see
proof.elliptic_curve or sage.structure.proof). If False, this
function just immediately returns the empty list.
- ``secs_hi`` - maximum number of seconds to try to compute the
Heegner index before switching over to trying to compute the
Heegner index bound. (Rank 0 only!)
- ``return_BSD`` - bool (default: False) whether to return an object
which contains information to reconstruct a proof
NOTE:
When printing verbose output, phrases such as "by Mazur" are referring
to the following list of papers:
REFERENCES:
.. [Cha] B. Cha. Vanishing of some cohomology goups and bounds for the
Shafarevich-Tate groups of elliptic curves. J. Number Theory, 111:154-
178, 2005.
.. [Jetchev] D. Jetchev. Global divisibility of Heegner points and
Tamagawa numbers. Compos. Math. 144 (2008), no. 4, 811--826.
.. [Kato] K. Kato. p-adic Hodge theory and values of zeta functions of
modular forms. Astérisque, (295):ix, 117-290, 2004.
.. [Kolyvagin] V. A. Kolyvagin. On the structure of Shafarevich-Tate
groups. Algebraic geometry, 94--121, Lecture Notes in Math., 1479,
Springer, Berlin, 1991.
.. [LumStein] A. Lum, W. Stein. Verification of the Birch and
Swinnerton-Dyer Conjecture for Elliptic Curves with Complex
Multiplication (unpublished)
.. [Mazur] B. Mazur. Modular curves and the Eisenstein ideal. Inst.
Hautes Études Sci. Publ. Math. No. 47 (1977), 33--186 (1978).
.. [Rubin] K. Rubin. The "main conjectures" of Iwasawa theory for
imaginary quadratic fields. Invent. Math. 103 (1991), no. 1, 25--68.
.. [SteinWuthrich] W. Stein and C. Wuthrich, Algorithms
for the Arithmetic of Elliptic Curves using Iwasawa Theory
Mathematics of Computation 82 (2013), 1757-1792.
.. [SteinEtAl] G. Grigorov, A. Jorza, S. Patrikis, W. Stein,
C. Tarniţǎ. Computational verification of the Birch and
Swinnerton-Dyer conjecture for individual elliptic curves.
Math. Comp. 78 (2009), no. 268, 2397--2425.
EXAMPLES::
sage: EllipticCurve('11a').prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 5} by Kolyvagin.
Kato further implies that #Sha[5] is trivial.
[]
sage: EllipticCurve('14a').prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
Kato further implies that #Sha[3] is trivial.
[]
sage: E = EllipticCurve("50b1")
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3, 5} by Kolyvagin.
Kolyvagin's bound for p = 3 applies by Stein et al.
True for p = 3 by Kolyvagin bound
Remaining primes:
p = 5: reducible, not surjective, additive, divides a Tamagawa number
(no bounds found)
ord_p(#Sha_an) = 0
[5]
sage: E.prove_BSD(two_desc='simon')
[5]
A rank two curve::
sage: E = EllipticCurve('389a')
We know nothing with proof=True::
sage: E.prove_BSD()
Set of all prime numbers: 2, 3, 5, 7, ...
We (think we) know everything with proof=False::
sage: E.prove_BSD(proof=False)
[]
A curve of rank 0 and prime conductor::
sage: E = EllipticCurve('19a')
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
Kato further implies that #Sha[3] is trivial.
[]
sage: E = EllipticCurve('37a')
sage: E.rank()
1
sage: E._EllipticCurve_rational_field__rank
{True: 1}
sage: E.analytic_rank = lambda : 0
sage: E.prove_BSD()
Traceback (most recent call last):
...
RuntimeError: It seems that the rank conjecture does not hold for this curve (Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field)! This may be a counterexample to BSD, but is more likely a bug.
We test the consistency check for the 2-part of Sha::
sage: E = EllipticCurve('37a')
sage: S = E.sha(); S
Tate-Shafarevich group for the Elliptic Curve defined by y^2 + y = x^3 - x over Rational Field
sage: def foo(use_database):
... return 4
sage: S.an = foo
sage: E.prove_BSD()
Traceback (most recent call last):
...
RuntimeError: Apparent contradiction: 0 <= rank(sha[2]) <= 0, but ord_2(sha_an) = 2
An example with a Tamagawa number at 5::
sage: E = EllipticCurve('123a1')
sage: E.prove_BSD(verbosity=2)
p = 2: True by 2-descent
True for p not in {2, 5} by Kolyvagin.
Remaining primes:
p = 5: reducible, not surjective, good ordinary, divides a Tamagawa number
(no bounds found)
ord_p(#Sha_an) = 0
[5]
A curve for which 3 divides the order of the Tate-Shafarevich group::
sage: E = EllipticCurve('681b')
sage: E.prove_BSD(verbosity=2) # long time
p = 2: True by 2-descent...
True for p not in {2, 3} by Kolyvagin....
Remaining primes:
p = 3: irreducible, surjective, non-split multiplicative
(0 <= ord_p <= 2)
ord_p(#Sha_an) = 2
[3]
A curve for which we need to use ``heegner_index_bound``::
sage: E = EllipticCurve('198b')
sage: E.prove_BSD(verbosity=1, secs_hi=1)
p = 2: True by 2-descent
True for p not in {2, 3} by Kolyvagin.
[3]
The ``return_BSD`` option gives an object with detailed information
about the proof::
sage: E = EllipticCurve('26b')
sage: B = E.prove_BSD(return_BSD=True)
sage: B.two_tor_rk
0
sage: B.N
26
sage: B.gens
[]
sage: B.primes
[]
sage: B.heegner_indexes
{-23: 2}
TESTS:
This was fixed by trac #8184 and #7575::
sage: EllipticCurve('438e1').prove_BSD(verbosity=1)
p = 2: True by 2-descent...
True for p not in {2} by Kolyvagin.
[]
::
sage: E = EllipticCurve('960d1')
sage: E.prove_BSD(verbosity=1) # long time (4s on sage.math, 2011)
p = 2: True by 2-descent
True for p not in {2} by Kolyvagin.
[]
"""
if proof is None:
from sage.structure.proof.proof import get_flag
proof = get_flag(proof, "elliptic_curve")
else:
proof = bool(proof)
if not proof:
return []
from copy import copy
BSD = BSD_data()
# We replace this curve by the optimal curve, which we can do since
# truth of BSD(E,p) is invariant under isogeny.
BSD.curve = E.optimal_curve()
if BSD.curve.has_cm():
# ensure that CM is by a maximal order
non_max_j_invs = [-12288000, 54000, 287496, 16581375]
if BSD.curve.j_invariant() in non_max_j_invs: # is this possible for optimal curves?
if verbosity > 0:
print 'CM by non maximal order: switching curves'
for E in BSD.curve.isogeny_class():
if E.j_invariant() not in non_max_j_invs:
BSD.curve = E
break
BSD.update()
galrep = BSD.curve.galois_representation()
if two_desc=='mwrank':
M = mwrank_two_descent_work(BSD.curve, BSD.two_tor_rk)
elif two_desc=='simon':
M = simon_two_descent_work(BSD.curve, BSD.two_tor_rk)
elif two_desc=='sage':
M = native_two_isogeny_descent_work(BSD.curve, BSD.two_tor_rk)
else:
raise NotImplementedError()
rank_lower_bd, rank_upper_bd, sha2_lower_bd, sha2_upper_bd, gens = M
assert sha2_lower_bd <= sha2_upper_bd
if gens is not None: gens = BSD.curve.saturation(gens)[0]
if rank_lower_bd > rank_upper_bd:
raise RuntimeError("Apparent contradiction: %d <= rank <= %d."%(rank_lower_bd, rank_upper_bd))
BSD.two_selmer_rank = rank_upper_bd + sha2_lower_bd + BSD.two_tor_rk
if sha2_upper_bd == sha2_lower_bd:
BSD.rank = rank_lower_bd
BSD.bounds[2] = (sha2_lower_bd, sha2_upper_bd)
else:
BSD.rank = BSD.curve.rank(use_database=True)
sha2_upper_bd -= (BSD.rank - rank_lower_bd)
BSD.bounds[2] = (sha2_lower_bd, sha2_upper_bd)
if verbosity > 0:
print "Unable to compute the rank exactly -- used database."
if rank_lower_bd > 1:
# We do not know BSD(E,p) for even a single p, since it's
# an open problem to show that L^r(E,1)/(Reg*Omega) is
# rational for any curve with r >= 2.
from sage.sets.all import Primes
BSD.primes = Primes()
if return_BSD:
BSD.rank = rank_lower_bd
return BSD
return BSD.primes
if (BSD.sha_an.ord(2) == 0) != (BSD.bounds[2][1] == 0):
raise RuntimeError("Apparent contradiction: %d <= rank(sha[2]) <= %d, but ord_2(sha_an) = %d"%(sha2_lower_bd, sha2_upper_bd, BSD.sha_an.ord(2)))
if BSD.bounds[2][0] == BSD.sha_an.ord(2) and BSD.sha_an.ord(2) == BSD.bounds[2][1]:
if verbosity > 0:
print 'p = 2: True by 2-descent'
BSD.primes = []
BSD.bounds.pop(2)
BSD.proof[2] = ['2-descent']
else:
BSD.primes = [2]
BSD.proof[2] = [('2-descent',)+BSD.bounds[2]]
if len(gens) > rank_lower_bd or \
rank_lower_bd > rank_upper_bd:
raise RuntimeError("Something went wrong with 2-descent.")
if BSD.rank != len(gens):
if BSD.rank != len(BSD.curve._EllipticCurve_rational_field__gens[True]):
raise RuntimeError("Could not get generators")
gens = BSD.curve._EllipticCurve_rational_field__gens[True]
BSD.gens = [BSD.curve.point(x, check=True) for x in gens]
if BSD.rank != BSD.curve.analytic_rank():
raise RuntimeError("It seems that the rank conjecture does not hold for this curve (%s)! This may be a counterexample to BSD, but is more likely a bug."%(BSD.curve))
# reduce set of remaining primes to a finite set
import signal
kolyvagin_primes = []
heegner_index = None
if BSD.rank == 0:
for D in BSD.curve.heegner_discriminants_list(10):
max_height = max(13,BSD.curve.quadratic_twist(D).CPS_height_bound())
heegner_primes = -1
while heegner_primes == -1:
if max_height > 21: break
heegner_primes, _, exact = BSD.curve.heegner_index_bound(D, max_height=max_height)
max_height += 1
if isinstance(heegner_primes, list):
break
if not isinstance(heegner_primes, list):
raise RuntimeError("Tried 10 Heegner discriminants, and heegner_index_bound failed each time.")
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
else:
BSD.heegner_index_upper_bound[D] = max(heegner_primes+[1])
if 2 in heegner_primes:
heegner_primes.remove(2)
else: # rank 1
for D in BSD.curve.heegner_discriminants_list(10):
I = BSD.curve.heegner_index(D)
J = I.is_int()
if J[0] and J[1]>0:
I = J[1]
else:
J = (2*I).is_int()
if J[0] and J[1]>0:
I = J[1]
else:
continue
heegner_index = I
BSD.heegner_indexes[D] = I
break
heegner_primes = [p for p in arith.prime_divisors(heegner_index) if p!=2]
assert BSD.sha_an in ZZ and BSD.sha_an > 0
if BSD.curve.has_cm():
if BSD.curve.analytic_rank() == 0:
if verbosity > 0:
print ' p >= 5: true by Rubin'
BSD.primes.append(3)
else:
K = rings.QuadraticField(BSD.curve.cm_discriminant(), 'a')
D_K = K.disc()
D_E = BSD.curve.discriminant()
if len(K.factor(3)) == 1: # 3 does not split in K
BSD.primes.append(3)
for p in arith.prime_divisors(D_K):
if p >= 5:
BSD.primes.append(p)
for p in arith.prime_divisors(D_E):
if p >= 5 and D_K%p and len(K.factor(p)) == 1: # p is inert in K
BSD.primes.append(p)
for p in heegner_primes:
if p >= 5 and D_E%p != 0 and D_K%p != 0 and len(K.factor(p)) == 1: # p is good for E and inert in K
kolyvagin_primes.append(p)
for p in arith.prime_divisors(BSD.sha_an):
if p >= 5 and D_K%p != 0 and len(K.factor(p)) == 1:
if BSD.curve.is_good(p):
if verbosity > 2 and p in heegner_primes and heegner_index is None:
print 'ALERT: Prime p (%d) >= 5 dividing sha_an, good for E, inert in K, in heegner_primes, should not divide the actual Heegner index'
# Note that the following check is not entirely
# exhaustive, in case there is a p not dividing
# the Heegner index in heegner_primes,
# for which only an outer bound was computed
if p not in heegner_primes:
raise RuntimeError("p = %d divides sha_an, is of good reduction for E, inert in K, and does not divide the Heegner index. This may be a counterexample to BSD, but is more likely a bug. %s"%(p,BSD.curve))
if verbosity > 0:
print 'True for p not in {%s} by Kolyvagin (via Stein & Lum -- unpublished) and Rubin.'%str(list(set(BSD.primes).union(set(kolyvagin_primes))))[1:-1]
BSD.proof['finite'] = copy(BSD.primes)
else: # no CM
# do some tricks to get to a finite set without calling bound_kolyvagin
BSD.primes += [p for p in galrep.non_surjective() if p != 2]
for p in heegner_primes:
if p not in BSD.primes:
BSD.primes.append(p)
for p in arith.prime_divisors(BSD.sha_an):
if p not in BSD.primes and p != 2:
BSD.primes.append(p)
if verbosity > 0:
s = str(BSD.primes)[1:-1]
if 2 not in BSD.primes:
if len(s) == 0: s = '2'
else: s = '2, '+s
print 'True for p not in {' + s + '} by Kolyvagin.'
BSD.proof['finite'] = copy(BSD.primes)
primes_to_remove = []
for p in BSD.primes:
if p == 2: continue
if galrep.is_surjective(p) and not BSD.curve.has_additive_reduction(p):
if BSD.curve.has_nonsplit_multiplicative_reduction(p):
if BSD.rank > 0:
continue
if p==3:
if (not (BSD.curve.is_ordinary(p) and BSD.curve.is_good(p))) and (not BSD.curve.has_split_multiplicative_reduction(p)):
continue
if BSD.rank > 0:
continue
if verbosity > 1:
print ' p = %d: Trying p_primary_bound'%p
p_bound = BSD.Sha.p_primary_bound(p)
if p in BSD.proof:
BSD.proof[p].append(('Stein-Wuthrich', p_bound))
else:
BSD.proof[p] = [('Stein-Wuthrich', p_bound)]
if BSD.sha_an.ord(p) == 0 and p_bound == 0:
if verbosity > 0:
print 'True for p=%d by Stein-Wuthrich.'%p
primes_to_remove.append(p)
else:
if p in BSD.bounds:
BSD.bounds[p][1] = min(BSD.bounds[p][1], p_bound)
else:
BSD.bounds[p] = (0, p_bound)
print 'Analytic %d-rank is '%p + str(BSD.sha_an.ord(p)) + ', actual %d-rank is at most %d.'%(p, p_bound)
print ' by Stein-Wuthrich.\n'
for p in primes_to_remove:
BSD.primes.remove(p)
kolyvagin_primes = []
for p in BSD.primes:
if p == 2: continue
if galrep.is_surjective(p):
kolyvagin_primes.append(p)
for p in kolyvagin_primes:
BSD.primes.remove(p)
# apply other hypotheses which imply Kolyvagin's bound holds
bounded_primes = []
D_K = rings.QuadraticField(D, 'a').disc()
# Cha's hypothesis
for p in BSD.primes:
if p == 2: continue
if D_K%p != 0 and BSD.N%(p**2) != 0 and galrep.is_irreducible(p):
if verbosity > 0:
print 'Kolyvagin\'s bound for p = %d applies by Cha.'%p
if p in BSD.proof:
BSD.proof[p].append('Cha')
else:
BSD.proof[p] = ['Cha']
kolyvagin_primes.append(p)
# Stein et al.
if not BSD.curve.has_cm():
L = arith.lcm([F.torsion_order() for F in BSD.curve.isogeny_class()])
for p in BSD.primes:
if p in kolyvagin_primes or p == 2: continue
if L%p != 0:
if len(arith.prime_divisors(D_K)) == 1:
if D_K%p == 0: continue
if verbosity > 0:
print 'Kolyvagin\'s bound for p = %d applies by Stein et al.'%p
kolyvagin_primes.append(p)
if p in BSD.proof:
BSD.proof[p].append('Stein et al.')
else:
BSD.proof[p] = ['Stein et al.']
for p in kolyvagin_primes:
if p in BSD.primes:
BSD.primes.remove(p)
# apply Kolyvagin's bound
primes_to_remove = []
for p in kolyvagin_primes:
if p == 2: continue
if p not in heegner_primes:
ord_p_bound = 0
elif heegner_index is not None: # p must divide heegner_index
ord_p_bound = 2*heegner_index.ord(p)
# Here Jetchev's results apply.
m_max = max([BSD.curve.tamagawa_number(q).ord(p) for q in BSD.N.prime_divisors()])
if m_max > 0:
if verbosity > 0:
print 'Jetchev\'s results apply (at p = %d) with m_max ='%p, m_max
if p in BSD.proof:
BSD.proof[p].append(('Jetchev',m_max))
else:
BSD.proof[p] = [('Jetchev',m_max)]
ord_p_bound -= 2*m_max
else: # Heegner index is None
for D in BSD.heegner_index_upper_bound:
M = BSD.heegner_index_upper_bound[D]
ord_p_bound = 0
while p**(ord_p_bound+1) <= M**2:
ord_p_bound += 1
# now ord_p_bound is one on I_K!!!
ord_p_bound *= 2 # by Kolyvagin, now ord_p_bound is one on #Sha
break
if p in BSD.proof:
BSD.proof[p].append(('Kolyvagin',ord_p_bound))
else:
BSD.proof[p] = [('Kolyvagin',ord_p_bound)]
if BSD.sha_an.ord(p) == 0 and ord_p_bound == 0:
if verbosity > 0:
print 'True for p = %d by Kolyvagin bound'%p
primes_to_remove.append(p)
elif BSD.sha_an.ord(p) > ord_p_bound:
raise RuntimeError("p = %d: ord_p_bound == %d, but sha_an.ord(p) == %d. This appears to be a counterexample to BSD, but is more likely a bug."%(p,ord_p_bound,BSD.sha_an.ord(p)))
else: # BSD.sha_an.ord(p) <= ord_p_bound != 0:
if p in BSD.bounds:
low = BSD.bounds[p][0]
BSD.bounds[p] = (low, min(BSD.bounds[p][1], ord_p_bound))
else:
BSD.bounds[p] = (0, ord_p_bound)
for p in primes_to_remove:
kolyvagin_primes.remove(p)
BSD.primes = list( set(BSD.primes).union(set(kolyvagin_primes)) )
# Kato's bound
if BSD.rank == 0 and not BSD.curve.has_cm():
L_over_Omega = BSD.curve.lseries().L_ratio()
kato_primes = BSD.Sha.bound_kato()
primes_to_remove = []
for p in BSD.primes:
if p == 2: continue
if p not in kato_primes:
if verbosity > 0:
print 'Kato further implies that #Sha[%d] is trivial.'%p
primes_to_remove.append(p)
if p in BSD.proof:
BSD.proof[p].append(('Kato',0))
else:
BSD.proof[p] = [('Kato',0)]
if p not in [2,3] and BSD.N%p != 0:
if galrep.is_surjective(p):
bd = L_over_Omega.valuation(p)
if verbosity > 1:
print 'Kato implies that ord_p(#Sha[%d]) <= %d '%(p,bd)
if p in BSD.proof:
BSD.proof[p].append(('Kato',bd))
else:
BSD.proof[p] = [('Kato',bd)]
if p in BSD.bounds:
low = BSD.bounds[p][0]
BSD.bounds[p][1] = (low, min(BSD.bounds[p][1], bd))
else:
BSD.bounds[p] = (0, bd)
for p in primes_to_remove:
BSD.primes.remove(p)
# Mazur
primes_to_remove = []
if BSD.N.is_prime():
for p in BSD.primes:
if p == 2: continue
if galrep.is_reducible(p):
primes_to_remove.append(p)
if verbosity > 0:
print 'True for p=%s by Mazur'%p
for p in primes_to_remove:
BSD.primes.remove(p)
if p in BSD.proof:
BSD.proof[p].append('Mazur')
else:
BSD.proof[p] = ['Mazur']
BSD.primes.sort()
# Try harder to compute the Heegner index, where it matters
if heegner_index is None:
if max_height < 18:
max_height = 18
for D in BSD.heegner_index_upper_bound:
M = BSD.heegner_index_upper_bound[D]
for p in kolyvagin_primes:
if p not in BSD.primes or p == 3: continue
if verbosity > 0:
print ' p = %d: Trying harder for Heegner index'%p
obt = 0
while p**(BSD.sha_an.ord(p)/2+1) <= M and max_height < 22:
if verbosity > 2:
print ' trying max_height =', max_height
old_bound = M
M, _, exact = BSD.curve.heegner_index_bound(D, max_height=max_height, secs_dc=secs_dc)
if M == -1:
max_height += 1
continue
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
M = exact
if verbosity > 2:
print ' heegner index =', M
else:
M = max(M+[1])
if verbosity > 2:
print ' bound =', M
if old_bound == M:
obt += 1
if obt == 2:
break
max_height += 1
BSD.heegner_index_upper_bound[D] = min(M,BSD.heegner_index_upper_bound[D])
low, upp = BSD.bounds[p]
expn = 0
while p**(expn+1) <= M:
expn += 1
if 2*expn < upp:
upp = 2*expn
BSD.bounds[p] = (low,upp)
if verbosity > 0:
print ' got better bound on ord_p =', upp
if low == upp:
if upp != BSD.sha_an.ord(p):
raise RuntimeError
else:
if verbosity > 0:
print ' proven!'
BSD.primes.remove(p)
break
for p in kolyvagin_primes:
if p not in BSD.primes or p == 3: continue
for D in BSD.curve.heegner_discriminants_list(4):
if D in BSD.heegner_index_upper_bound: continue
print ' discriminant', D
if verbosity > 0:
print 'p = %d: Trying discriminant = %d for Heegner index'%(p,D)
max_height = max(10, BSD.curve.quadratic_twist(D).CPS_height_bound())
obt = 0
while True:
if verbosity > 2:
print ' trying max_height =', max_height
old_bound = M
if p**(BSD.sha_an.ord(p)/2+1) > M or max_height >= 22:
break
M, _, exact = BSD.curve.heegner_index_bound(D, max_height=max_height, secs_dc=secs_dc)
if M == -1:
max_height += 1
continue
if exact is not False:
heegner_index = exact
BSD.heegner_indexes[D] = exact
M = exact
if verbosity > 2:
print ' heegner index =', M
else:
M = max(M+[1])
if verbosity > 2:
print ' bound =', M
if old_bound == M:
obt += 1
if obt == 2:
break
max_height += 1
BSD.heegner_index_upper_bound[D] = M
low, upp = BSD.bounds[p]
expn = 0
while p**(expn+1) <= M:
expn += 1
if 2*expn < upp:
upp = 2*expn
BSD.bounds[p] = (low,upp)
if verbosity > 0:
print ' got better bound =', upp
if low == upp:
if upp != BSD.sha_an.ord(p):
raise RuntimeError
else:
if verbosity > 0:
print ' proven!'
BSD.primes.remove(p)
break
# print some extra information
if verbosity > 1:
if len(BSD.primes) > 0:
print 'Remaining primes:'
for p in BSD.primes:
s = 'p = ' + str(p) + ': '
if galrep.is_irreducible(p):
s += 'ir'
s += 'reducible, '
if not galrep.is_surjective(p):
s += 'not '
s += 'surjective, '
a_p = BSD.curve.an(p)
if BSD.curve.is_good(p):
if a_p%p != 0:
s += 'good ordinary'
else:
s += 'good, non-ordinary'
else:
assert BSD.curve.is_minimal()
if a_p == 0:
s += 'additive'
elif a_p == 1:
s += 'split multiplicative'
elif a_p == -1:
s += 'non-split multiplicative'
if BSD.curve.tamagawa_product()%p==0:
s += ', divides a Tamagawa number'
if p in BSD.bounds:
s += '\n (%d <= ord_p <= %d)'%BSD.bounds[p]
else:
s += '\n (no bounds found)'
s += '\n ord_p(#Sha_an) = %d'%BSD.sha_an.ord(p)
if heegner_index is None:
may_divide = True
for D in BSD.heegner_index_upper_bound:
if p > BSD.heegner_index_upper_bound[D] or p not in kolyvagin_primes:
may_divide = False
if may_divide:
s += '\n may divide the Heegner index, for which only a bound was computed'
print s
if BSD.curve.has_cm():
if BSD.rank == 1:
BSD.proof['reason_finite'] = 'Rubin&Kolyvagin'
else:
BSD.proof['reason_finite'] = 'Rubin'
else:
BSD.proof['reason_finite'] = 'Kolyvagin'
# reduce memory footprint of BSD object:
BSD.curve = BSD.curve.label()
BSD.Sha = None
return BSD if return_BSD else BSD.primes
def siegel_product(self, u):
"""
Computes the infinite product of local densities of the quadratic
form for the number `u`.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.theta_series(11)
1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + 104*q^9 + 144*q^10 + O(q^11)
sage: Q.siegel_product(1)
8
sage: Q.siegel_product(2) ## This one is wrong -- expect 24, and the higher powers of 2 don't work... =(
24
sage: Q.siegel_product(3)
32
sage: Q.siegel_product(5)
48
sage: Q.siegel_product(6)
96
sage: Q.siegel_product(7)
64
sage: Q.siegel_product(9)
104
sage: Q.local_density(2,1)
1
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3
1
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3 # long time (41s on sage.math, 2011)
1
sage: Q.local_density(2,2)
3/2
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3
3/2
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3 # long time (41s on sage.math, 2011)
3/2
TESTS::
sage: [1] + [Q.siegel_product(ZZ(a)) for a in range(1,11)] == Q.theta_series(11).list()
True
"""
## Protect u (since it fails often if it's an just an int!)
u = ZZ(u)
n = self.dim()
d = self.det(
) ## ??? Warning: This is a factor of 2^n larger than it should be!
## DIAGNOSTIC
verbose("n = " + str(n))
verbose("d = " + str(d))
verbose("In siegel_product: d = ", d, "\n")
## Product of "bad" places to omit
S = 2 * d * u
## DIAGNOSTIC
verbose("siegel_product Break 1. \n")
verbose(" u = ", u, "\n")
## Make the odd generic factors
if ((n % 2) == 1):
m = (n - 1) / 2
d1 = fundamental_discriminant(
((-1)**m) * 2 * d *
u) ## Replaced d by 2d here to compensate for the determinant
f = abs(
d1) ## gaining an odd power of 2 by using the matrix of 2Q instead
## of the matrix of Q.
## --> Old d1 = CoreDiscriminant((mpz_class(-1)^m) * d * u);
## Make the ratio of factorials factor: [(2m)! / m!] * prod_{i=1}^m (2*i-1)
factor1 = 1
for i in range(1, m + 1):
factor1 *= 2 * i - 1
for i in range(m + 1, 2 * m + 1):
factor1 *= i
genericfactor = factor1 * ((u / f) ** m) \
* QQ(sqrt((2 ** n) * f) / (u * d)) \
* abs(QuadraticBernoulliNumber(m, d1) / bernoulli(2*m))
## DIAGNOSTIC
verbose("siegel_product Break 2. \n")
## Make the even generic factor
if ((n % 2) == 0):
m = n / 2
d1 = fundamental_discriminant(((-1)**m) * d)
f = abs(d1)
## DIAGNOSTIC
#cout << " mpz_class(-1)^m = " << (mpz_class(-1)^m) << " and d = " << d << endl;
#cout << " f = " << f << " and d1 = " << d1 << endl;
genericfactor = m / QQ(sqrt(f*d)) \
* ((u/2) ** (m-1)) * (f ** m) \
/ abs(QuadraticBernoulliNumber(m, d1)) \
* (2 ** m) ## This last factor compensates for using the matrix of 2*Q
##return genericfactor
## Omit the generic factors in S and compute them separately
omit = 1
include = 1
S_divisors = prime_divisors(S)
## DIAGNOSTIC
#cout << "\n S is " << S << endl;
#cout << " The Prime divisors of S are :";
#PrintV(S_divisors);
for p in S_divisors:
Q_normal = self.local_normal_form(p)
## DIAGNOSTIC
verbose(" p = " + str(p) + " and its Kronecker symbol (d1/p) = (" +
str(d1) + "/" + str(p) + ") is " +
str(kronecker_symbol(d1, p)) + "\n")
omit *= 1 / (1 - (kronecker_symbol(d1, p) / (p**m)))
## DIAGNOSTIC
verbose(" omit = " + str(omit) + "\n")
verbose(" Q_normal is \n" + str(Q_normal) + "\n")
verbose(" Q_normal = \n" + str(Q_normal))
verbose(" p = " + str(p) + "\n")
verbose(" u = " + str(u) + "\n")
verbose(" include = " + str(include) + "\n")
include *= Q_normal.local_density(p, u)
## DIAGNOSTIC
#cout << " Including the p = " << p << " factor: " << local_density(Q_normal, p, u) << endl;
## DIAGNSOTIC
verbose(" --- Exiting loop \n")
#// **************** Important *******************
#// Additional fix (only included for n=4) to deal
#// with the power of 2 introduced at the real place
#// by working with Q instead of 2*Q. This needs to
#// be done for all other n as well...
#/*
#if (n==4)
# genericfactor = 4 * genericfactor;
#*/
## DIAGNSOTIC
#cout << endl;
#cout << " generic factor = " << genericfactor << endl;
#cout << " omit = " << omit << endl;
#cout << " include = " << include << endl;
#cout << endl;
## DIAGNSOTIC
#// cout << "siegel_product Break 3. " << endl;
## Return the final factor (and divide by 2 if n=2)
if (n == 2):
return (genericfactor * omit * include / 2)
else:
return (genericfactor * omit * include)
def new_submodule(self, p=None):
"""
Returns the new or p-new submodule of self.
INPUT:
- ``p`` - (default: None); if not None, return only
the p-new submodule.
OUTPUT: the new or p-new submodule of self, i.e. the intersection of
the kernel of the degeneracy lowering maps to level `N/p` (for the
given prime `p`, or for all prime divisors of `N` if `p` is not given).
If self is cuspidal this is a Hecke-invariant complement of the
corresponding old submodule, but this may break down on Eisenstein
subspaces (see the amusing example in William Stein's book of a form
which is new and old at the same time).
EXAMPLES::
sage: m = ModularSymbols(33); m.rank()
9
sage: m.new_submodule().rank()
3
sage: m.new_submodule(3).rank()
4
sage: m.new_submodule(11).rank()
8
"""
try:
if self.__is_new[p]:
return self
except AttributeError:
self.__is_new = {}
except KeyError:
pass
if self.rank() == 0:
self.__is_new[p] = True
return self
try:
return self.__new_submodule[p]
except AttributeError:
self.__new_submodule = {}
except KeyError:
pass
# Construct the degeneracy map d.
N = self.level()
d = None
eps = self.character()
if eps is None:
f = 1
else:
f = eps.conductor()
if p is None:
D = arith.prime_divisors(N)
else:
if N % p != 0:
raise ValueError("p must divide the level.")
D = [p]
for q in D:
# Here we are only using degeneracy *lowering* maps, so it is fine
# to be careless and pass an integer for the level. One needs to be
# a bit more careful with degeneracy *raising* maps for the Gamma1
# and GammaH cases.
if ((N // q) % f) == 0:
NN = N // q
d1 = self.degeneracy_map(NN, 1).matrix()
if d is None:
d = d1
else:
d = d.augment(d1)
d = d.augment(self.degeneracy_map(NN, q).matrix())
#end if
#end for
if d is None or d == 0:
self.__is_new[p] = True
return self
else:
self.__is_new[p] = False
ns = self.submodule(d.kernel(), check=False)
ns.__is_new = {p: True}
ns._is_full_hecke_module = True
self.__new_submodule[p] = ns
return ns
def siegel_product(self, u):
"""
Computes the infinite product of local densities of the quadratic
form for the number `u`.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.theta_series(11)
1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + 104*q^9 + 144*q^10 + O(q^11)
sage: Q.siegel_product(1)
8
sage: Q.siegel_product(2) ## This one is wrong -- expect 24, and the higher powers of 2 don't work... =(
24
sage: Q.siegel_product(3)
32
sage: Q.siegel_product(5)
48
sage: Q.siegel_product(6)
96
sage: Q.siegel_product(7)
64
sage: Q.siegel_product(9)
104
sage: Q.local_density(2,1)
1
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3
1
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3 # long time (41s on sage.math, 2011)
1
sage: Q.local_density(2,2)
3/2
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3
3/2
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3 # long time (41s on sage.math, 2011)
3/2
TESTS::
sage: [1] + [Q.siegel_product(ZZ(a)) for a in range(1,11)] == Q.theta_series(11).list()
True
"""
## Protect u (since it fails often if it's an just an int!)
u = ZZ(u)
n = self.dim()
d = self.det() ## ??? Warning: This is a factor of 2^n larger than it should be!
## DIAGNOSTIC
verbose("n = " + str(n))
verbose("d = " + str(d))
verbose("In siegel_product: d = ", d, "\n");
## Product of "bad" places to omit
S = 2 * d * u
## DIAGNOSTIC
verbose("siegel_product Break 1. \n")
verbose(" u = ", u, "\n")
## Make the odd generic factors
if ((n % 2) == 1):
m = (n-1) / 2
d1 = fundamental_discriminant(((-1)**m) * 2*d * u) ## Replaced d by 2d here to compensate for the determinant
f = abs(d1) ## gaining an odd power of 2 by using the matrix of 2Q instead
## of the matrix of Q.
## --> Old d1 = CoreDiscriminant((mpz_class(-1)^m) * d * u);
## Make the ratio of factorials factor: [(2m)! / m!] * prod_{i=1}^m (2*i-1)
factor1 = 1
for i in range(1, m+1):
factor1 *= 2*i - 1
for i in range(m+1, 2*m + 1):
factor1 *= i
genericfactor = factor1 * ((u / f) ** m) \
* QQ(sqrt((2 ** n) * f) / (u * d)) \
* abs(QuadraticBernoulliNumber(m, d1) / bernoulli(2*m))
## DIAGNOSTIC
verbose("siegel_product Break 2. \n")
## Make the even generic factor
if ((n % 2) == 0):
m = n / 2
d1 = fundamental_discriminant(((-1)**m) * d)
f = abs(d1)
## DIAGNOSTIC
#cout << " mpz_class(-1)^m = " << (mpz_class(-1)^m) << " and d = " << d << endl;
#cout << " f = " << f << " and d1 = " << d1 << endl;
genericfactor = m / QQ(sqrt(f*d)) \
* ((u/2) ** (m-1)) * (f ** m) \
/ abs(QuadraticBernoulliNumber(m, d1)) \
* (2 ** m) ## This last factor compensates for using the matrix of 2*Q
##return genericfactor
## Omit the generic factors in S and compute them separately
omit = 1
include = 1
S_divisors = prime_divisors(S)
## DIAGNOSTIC
#cout << "\n S is " << S << endl;
#cout << " The Prime divisors of S are :";
#PrintV(S_divisors);
for p in S_divisors:
Q_normal = self.local_normal_form(p)
## DIAGNOSTIC
verbose(" p = " + str(p) + " and its Kronecker symbol (d1/p) = (" + str(d1) + "/" + str(p) + ") is " + str(kronecker_symbol(d1, p)) + "\n")
omit *= 1 / (1 - (kronecker_symbol(d1, p) / (p**m)))
## DIAGNOSTIC
verbose(" omit = " + str(omit) + "\n")
verbose(" Q_normal is \n" + str(Q_normal) + "\n")
verbose(" Q_normal = \n" + str(Q_normal))
verbose(" p = " + str(p) + "\n")
verbose(" u = " +str(u) + "\n")
verbose(" include = " + str(include) + "\n")
include *= Q_normal.local_density(p, u)
## DIAGNOSTIC
#cout << " Including the p = " << p << " factor: " << local_density(Q_normal, p, u) << endl;
## DIAGNSOTIC
verbose(" --- Exiting loop \n")
#// **************** Important *******************
#// Additional fix (only included for n=4) to deal
#// with the power of 2 introduced at the real place
#// by working with Q instead of 2*Q. This needs to
#// be done for all other n as well...
#/*
#if (n==4)
# genericfactor = 4 * genericfactor;
#*/
## DIAGNSOTIC
#cout << endl;
#cout << " generic factor = " << genericfactor << endl;
#cout << " omit = " << omit << endl;
#cout << " include = " << include << endl;
#cout << endl;
## DIAGNSOTIC
#// cout << "siegel_product Break 3. " << endl;
## Return the final factor (and divide by 2 if n=2)
if (n == 2):
return (genericfactor * omit * include / 2)
else:
return (genericfactor * omit * include)
def old_submodule(self, p=None):
"""
Returns the old or p-old submodule of self, i.e. the sum of the images
of the degeneracy maps from level `N/p` (for the given prime `p`, or
for all primes `p` dividing `N` if `p` is not given).
INPUT:
- ``p`` - (default: None); if not None, return only the p-old
submodule.
OUTPUT: the old or p-old submodule of self
EXAMPLES::
sage: m = ModularSymbols(33); m.rank()
9
sage: m.old_submodule().rank()
7
sage: m.old_submodule(3).rank()
6
sage: m.new_submodule(11).rank()
8
::
sage: e = DirichletGroup(16)([-1, 1])
sage: M = ModularSymbols(e, 3, sign=1); M
Modular Symbols space of dimension 4 and level 16, weight 3, character [-1, 1], sign 1, over Rational Field
sage: M.old_submodule()
Modular Symbols subspace of dimension 3 of Modular Symbols space of dimension 4 and level 16, weight 3, character [-1, 1], sign 1, over Rational Field
Illustrate that trac 10664 is fixed::
sage: ModularSymbols(DirichletGroup(42)[7], 6, sign=1).old_subspace(3)
Modular Symbols subspace of dimension 0 of Modular Symbols space of dimension 40 and level 42, weight 6, character [-1, -1], sign 1, over Rational Field
"""
try:
if self.__is_old[p]:
return self
except AttributeError:
self.__is_old = {}
except KeyError:
pass
if self.rank() == 0:
self.__is_old[p] = True
return self
try:
return self.__old_submodule[p]
except AttributeError:
self.__old_submodule = {}
except KeyError:
pass
# Construct the degeneracy map d.
N = self.level()
d = None
eps = self.character()
if eps is None:
f = 1
else:
f = eps.conductor()
if p is None:
D = arith.prime_divisors(N)
else:
if N % p != 0:
raise ValueError("p must divide the level.")
D = [p]
for q in D:
NN = N // q
if NN % f == 0:
M = self.hecke_module_of_level(NN)
# Here it is vital to pass self as an argument to
# degeneracy_map, because M and the level N don't uniquely
# determine self (e.g. the degeneracy map from level 1 to level
# N could go to Gamma0(N), Gamma1(N) or anything in between)
d1 = M.degeneracy_map(self, 1).matrix()
if d is None:
d = d1
else:
d = d.stack(d1)
d = d.stack(M.degeneracy_map(self, q).matrix())
#end if
#end for
if d is None:
os = self.zero_submodule()
else:
os = self.submodule(d.image(), check=False)
self.__is_old[p] = (os == self)
os.__is_old = {p: True}
os._is_full_hecke_module = True
self.__old_submodule[p] = os
return os
def _find_scaling_L_ratio(self):
r"""
This function is use to set ``_scaling``, the factor used to adjust the
scalar multiple of the modular symbol.
If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale
it with respect to the approximation of the L-value. It is known that
the quotient is a rational number with small denominator.
Otherwise we try to scale using quadratic twists.
``_scaling`` will be set to a rational non-zero multiple if we succeed and to 1 otherwise.
Even if we fail we scale at least to make up the difference between the periods
of the `X_0`-optimal curve and our given curve `E` in the isogeny class.
EXAMPLES::
sage : m = EllipticCurve('11a1').modular_symbol(use_eclib=True)
sage : m._scaling
1
sage: m = EllipticCurve('11a2').modular_symbol(use_eclib=True)
sage: m._scaling
5/2
sage: m = EllipticCurve('11a3').modular_symbol(use_eclib=True)
sage: m._scaling
1/10
sage: m = EllipticCurve('11a1').modular_symbol(use_eclib=False)
sage: m._scaling
1/5
sage: m = EllipticCurve('11a2').modular_symbol(use_eclib=False)
sage: m._scaling
1
sage: m = EllipticCurve('11a3').modular_symbol(use_eclib=False)
sage: m._scaling
1/25
sage: m = EllipticCurve('37a1').modular_symbol(use_eclib=False)
sage: m._scaling
1
sage: m = EllipticCurve('37a1').modular_symbol(use_eclib=True)
sage: m._scaling
-1
sage: m = EllipticCurve('389a1').modular_symbol(use_eclib=True)
sage: m._scaling
-1/2
sage: m = EllipticCurve('389a1').modular_symbol(use_eclib=False)
sage: m._scaling
2
sage: m = EllipticCurve('196a1').modular_symbol(use_eclib=False)
sage: m._scaling
1/2
Some harder cases fail::
sage: m = EllipticCurve('121b1').modular_symbol(use_eclib=False)
Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by -1, 2 or -2.
sage: m._scaling
1
TESTS::
sage: rk0 = ['11a1', '11a2', '15a1', '27a1', '37b1']
sage: for la in rk0: # long time (3s on sage.math, 2011)
... E = EllipticCurve(la)
... me = E.modular_symbol(use_eclib = True)
... ms = E.modular_symbol(use_eclib = False)
... print E.lseries().L_ratio()*E.real_components(), me(0), ms(0)
1/5 1/5 1/5
1 1 1
1/4 1/4 1/4
1/3 1/3 1/3
2/3 2/3 2/3
sage: rk1 = ['37a1','43a1','53a1', '91b1','91b2','91b3']
sage: [EllipticCurve(la).modular_symbol(use_eclib=True)(0) for la in rk1] # long time (1s on sage.math, 2011)
[0, 0, 0, 0, 0, 0]
sage: for la in rk1: # long time (8s on sage.math, 2011)
... E = EllipticCurve(la)
... m = E.modular_symbol(use_eclib = True)
... lp = E.padic_lseries(5)
... for D in [5,17,12,8]:
... ED = E.quadratic_twist(D)
... md = sum([kronecker(D,u)*m(ZZ(u)/D) for u in range(D)])
... etaa = lp._quotient_of_periods_to_twist(D)
... assert ED.lseries().L_ratio()*ED.real_components()*etaa == md
"""
E = self._E
self._scaling = 1 # by now.
self._failed_to_scale = False
if self._sign == 1 :
at0 = self(0)
# print 'modular symbol evaluates to ',at0,' at 0'
if at0 != 0 :
l1 = self.__lalg__(1)
if at0 != l1:
verbose('scale modular symbols by %s'%(l1/at0))
self._scaling = l1/at0
else :
# if [0] = 0, we can still hope to scale it correctly by considering twists of E
Dlist = [5,8,12,13,17,21,24,28,29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 76, 77, 85, 88, 89, 92, 93, 97] # a list of positive fundamental discriminants
j = 0
at0 = 0
# computes [0]+ for the twist of E by D until one value is non-zero
while j < 30 and at0 == 0 :
D = Dlist[j]
# the following line checks if the twist of the newform of E by D is a newform
# this is to avoid that we 'twist back'
if all( valuation(E.conductor(),ell)<= valuation(D,ell) for ell in prime_divisors(D) ) :
at0 = sum([kronecker_symbol(D,u) * self(ZZ(u)/D) for u in range(1,abs(D))])
j += 1
if j == 30 and at0 == 0: # curves like "121b1", "225a1", "225e1", "256a1", "256b1", "289a1", "361a1", "400a1", "400c1", "400h1", "441b1", "441c1", "441d1", "441f1 .. will arrive here
self.__scale_by_periods_only__()
else :
l1 = self.__lalg__(D)
if at0 != l1:
verbose('scale modular symbols by %s found at D=%s '%(l1/at0,D), level=2)
self._scaling = l1/at0
else : # that is when sign = -1
Dlist = [-3,-4,-7,-8,-11,-15,-19,-20,-23,-24, -31, -35, -39, -40, -43, -47, -51, -52, -55, -56, -59, -67, -68, -71, -79, -83, -84, -87, -88, -91] # a list of negative fundamental discriminants
j = 0
at0 = 0
while j < 30 and at0 == 0 :
# computes [0]+ for the twist of E by D until one value is non-zero
D = Dlist[j]
if all( valuation(E.conductor(),ell)<= valuation(D,ell) for ell in prime_divisors(D) ) :
at0 = - sum([kronecker_symbol(D,u) * self(ZZ(u)/D) for u in range(1,abs(D))])
j += 1
if j == 30 and at0 == 0: # no more hope for a normalization
# we do at least a scaling with the quotient of the periods
self.__scale_by_periods_only__()
else :
l1 = self.__lalg__(D)
if at0 != l1:
verbose('scale modular symbols by %s'%(l1/at0))
self._scaling = l1/at0
def _find_scaling_L_ratio(self):
r"""
This function is use to set ``_scaling``, the factor used to adjust the
scalar multiple of the modular symbol.
If `[0]`, the modular symbol evaluated at 0, is non-zero, we can just scale
it with respect to the approximation of the L-value. It is known that
the quotient is a rational number with small denominator.
Otherwise we try to scale using quadratic twists.
``_scaling`` will be set to a rational non-zero multiple if we succeed and to 1 otherwise.
Even if we fail we scale at least to make up the difference between the periods
of the `X_0`-optimal curve and our given curve `E` in the isogeny class.
EXAMPLES::
sage : m = EllipticCurve('11a1').modular_symbol(use_eclib=True)
sage : m._scaling
1
sage: m = EllipticCurve('11a2').modular_symbol(use_eclib=True)
sage: m._scaling
5/2
sage: m = EllipticCurve('11a3').modular_symbol(use_eclib=True)
sage: m._scaling
1/10
sage: m = EllipticCurve('11a1').modular_symbol(use_eclib=False)
sage: m._scaling
1/5
sage: m = EllipticCurve('11a2').modular_symbol(use_eclib=False)
sage: m._scaling
1
sage: m = EllipticCurve('11a3').modular_symbol(use_eclib=False)
sage: m._scaling
1/25
sage: m = EllipticCurve('37a1').modular_symbol(use_eclib=False)
sage: m._scaling
1
sage: m = EllipticCurve('37a1').modular_symbol(use_eclib=True)
sage: m._scaling
-1
sage: m = EllipticCurve('389a1').modular_symbol(use_eclib=True)
sage: m._scaling
-1/2
sage: m = EllipticCurve('389a1').modular_symbol(use_eclib=False)
sage: m._scaling
2
sage: m = EllipticCurve('196a1').modular_symbol(use_eclib=False)
sage: m._scaling
1/2
Some harder cases fail::
sage: m = EllipticCurve('121b1').modular_symbol(use_eclib=False)
Warning : Could not normalize the modular symbols, maybe all further results will be multiplied by -1, 2 or -2.
sage: m._scaling
1
TESTS::
sage: rk0 = ['11a1', '11a2', '15a1', '27a1', '37b1']
sage: for la in rk0: # long time (3s on sage.math, 2011)
... E = EllipticCurve(la)
... me = E.modular_symbol(use_eclib = True)
... ms = E.modular_symbol(use_eclib = False)
... print E.lseries().L_ratio()*E.real_components(), me(0), ms(0)
1/5 1/5 1/5
1 1 1
1/4 1/4 1/4
1/3 1/3 1/3
2/3 2/3 2/3
sage: rk1 = ['37a1','43a1','53a1', '91b1','91b2','91b3']
sage: [EllipticCurve(la).modular_symbol(use_eclib=True)(0) for la in rk1] # long time (1s on sage.math, 2011)
[0, 0, 0, 0, 0, 0]
sage: for la in rk1: # long time (8s on sage.math, 2011)
... E = EllipticCurve(la)
... m = E.modular_symbol(use_eclib = True)
... lp = E.padic_lseries(5)
... for D in [5,17,12,8]:
... ED = E.quadratic_twist(D)
... md = sum([kronecker(D,u)*m(ZZ(u)/D) for u in range(D)])
... etaa = lp._quotient_of_periods_to_twist(D)
... assert ED.lseries().L_ratio()*ED.real_components()*etaa == md
"""
E = self._E
self._scaling = 1 # by now.
self._failed_to_scale = False
if self._sign == 1 :
at0 = self(0)
# print 'modular symbol evaluates to ',at0,' at 0'
if at0 != 0 :
l1 = self.__lalg__(1)
if at0 != l1:
verbose('scale modular symbols by %s'%(l1/at0))
self._scaling = l1/at0
else :
# if [0] = 0, we can still hope to scale it correctly by considering twists of E
Dlist = [5,8,12,13,17,21,24,28,29, 33, 37, 40, 41, 44, 53, 56, 57, 60, 61, 65, 69, 73, 76, 77, 85, 88, 89, 92, 93, 97] # a list of positive fundamental discriminants
j = 0
at0 = 0
# computes [0]+ for the twist of E by D until one value is non-zero
while j < 30 and at0 == 0 :
D = Dlist[j]
# the following line checks if the twist of the newform of E by D is a newform
# this is to avoid that we 'twist back'
if all( valuation(E.conductor(),ell)<= valuation(D,ell) for ell in prime_divisors(D) ) :
at0 = sum([kronecker_symbol(D,u) * self(ZZ(u)/D) for u in range(1,abs(D))])
j += 1
if j == 30 and at0 == 0: # curves like "121b1", "225a1", "225e1", "256a1", "256b1", "289a1", "361a1", "400a1", "400c1", "400h1", "441b1", "441c1", "441d1", "441f1 .. will arrive here
self._failed_to_scale = True
self.__scale_by_periods_only__()
else :
l1 = self.__lalg__(D)
if at0 != l1:
verbose('scale modular symbols by %s found at D=%s '%(l1/at0,D), level=2)
self._scaling = l1/at0
else : # that is when sign = -1
Dlist = [-3,-4,-7,-8,-11,-15,-19,-20,-23,-24, -31, -35, -39, -40, -43, -47, -51, -52, -55, -56, -59, -67, -68, -71, -79, -83, -84, -87, -88, -91] # a list of negative fundamental discriminants
j = 0
at0 = 0
while j < 30 and at0 == 0 :
# computes [0]+ for the twist of E by D until one value is non-zero
D = Dlist[j]
if all( valuation(E.conductor(),ell)<= valuation(D,ell) for ell in prime_divisors(D) ) :
at0 = - sum([kronecker_symbol(D,u) * self(ZZ(u)/D) for u in range(1,abs(D))])
j += 1
if j == 30 and at0 == 0: # no more hope for a normalization
# we do at least a scaling with the quotient of the periods
self._failed_to_scale = True
self.__scale_by_periods_only__()
else :
l1 = self.__lalg__(D)
if at0 != l1:
verbose('scale modular symbols by %s'%(l1/at0))
self._scaling = l1/at0
def series(self, n=2, quadratic_twist=+1, prec=5):
r"""
Returns the `n`-th approximation to the `p`-adic L-series as a
power series in `T` (corresponding to `\gamma-1` with
`\gamma=1+p` as a generator of `1+p\ZZ_p`). Each coefficient
is a `p`-adic number whose precision is provably correct.
Here the normalization of the `p`-adic L-series is chosen such
that `L_p(J,1) = (1-1/\alpha)^2 L(J,1)/\Omega_J` where
`\alpha` is the unit root
INPUT:
- ``n`` - (default: 2) a positive integer
- ``quadratic_twist`` - (default: +1) a fundamental
discriminant of a quadratic field, coprime to the
conductor of the curve
- ``prec`` - (default: 5) maximal number of terms of the
series to compute; to compute as many as possible just
give a very large number for ``prec``; the result will
still be correct.
ALIAS: power_series is identical to series.
EXAMPLES:
sage: J = J0(188)[0]
sage: p = 7
sage: L = J.padic_lseries(p)
sage: L.is_ordinary()
True
sage: f = L.series(2)
sage: f[0]
O(7^20)
sage: f[1].norm()
3 + 4*7 + 3*7^2 + 6*7^3 + 5*7^4 + 5*7^5 + 6*7^6 + 4*7^7 + 5*7^8 + 7^10 + 5*7^11 + 4*7^13 + 4*7^14 + 5*7^15 + 2*7^16 + 5*7^17 + 7^18 + 7^19 + O(7^20)
"""
n = ZZ(n)
if n < 1:
raise ValueError, "n (=%s) must be a positive integer"%n
if not self.is_ordinary():
raise ValueError, "p (=%s) must be an ordinary prime"%p
# check if the conditions on quadratic_twist are satisfied
D = ZZ(quadratic_twist)
if D != 1:
if D % 4 == 0:
d = D//4
if not d.is_squarefree() or d % 4 == 1:
raise ValueError, "quadratic_twist (=%s) must be a fundamental discriminant of a quadratic field"%D
else:
if not D.is_squarefree() or D % 4 != 1:
raise ValueError, "quadratic_twist (=%s) must be a fundamental discriminant of a quadratic field"%D
if gcd(D,self._p) != 1:
raise ValueError, "quadratic twist (=%s) must be coprime to p (=%s) "%(D,self._p)
if gcd(D,self._E.conductor())!= 1:
for ell in prime_divisors(D):
if valuation(self._E.conductor(),ell) > valuation(D,ell) :
raise ValueError, "can not twist a curve of conductor (=%s) by the quadratic twist (=%s)."%(self._E.conductor(),D)
p = self._p
if p == 2 and self._normalize :
print 'Warning : For p=2 the normalization might not be correct !'
#verbose("computing L-series for p=%s, n=%s, and prec=%s"%(p,n,prec))
# bounds = self._prec_bounds(n,prec)
# padic_prec = max(bounds[1:]) + 5
padic_prec = 10
# verbose("using p-adic precision of %s"%padic_prec)
res_series_prec = min(p**(n-1), prec)
verbose("using series precision of %s"%res_series_prec)
ans = self._get_series_from_cache(n, res_series_prec,D)
if not ans is None:
verbose("found series in cache")
return ans
K = QQ
gamma = K(1 + p)
R = PowerSeriesRing(K,'T',res_series_prec)
T = R(R.gen(),res_series_prec )
#L = R(0)
one_plus_T_factor = R(1)
gamma_power = K(1)
teich = self.teichmuller(padic_prec)
p_power = p**(n-1)
# F = Qp(p,padic_prec)
verbose("Now iterating over %s summands"%((p-1)*p_power))
verbose_level = get_verbose()
count_verb = 0
alphas = self.alpha()
#print len(alphas)
Lprod = []
self._emb = 0
if len(alphas) == 2:
split = True
else:
split = False
for alpha in alphas:
L = R(0)
self._emb = self._emb + 1
for j in range(p_power):
s = K(0)
if verbose_level >= 2 and j/p_power*100 > count_verb + 3:
verbose("%.2f percent done"%(float(j)/p_power*100))
count_verb += 3
for a in range(1,p):
if split:
# b = ((F.teichmuller(a)).lift() % ZZ(p**n))
b = (teich[a]) % ZZ(p**n)
b = b*gamma_power
else:
b = teich[a] * gamma_power
s += self.measure(b, n, padic_prec,D,alpha)
L += s * one_plus_T_factor
one_plus_T_factor *= 1+T
gamma_power *= gamma
Lprod = Lprod + [L]
if len(Lprod)==1:
return Lprod[0]
else:
return Lprod[0]*Lprod[1]