def local_density(self, p, m):
"""
Gives the local density -- should be called by the user. =)
NOTE: This screens for imprimitive forms, and puts the quadratic
form in local normal form, which is a *requirement* of the
routines performing the computations!
INPUT:
`p` -- a prime number > 0
`m` -- an integer
OUTPUT:
a rational number
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) ## NOTE: This is already in local normal form for *all* primes p!
sage: Q.local_density(p=2, m=1)
1
sage: Q.local_density(p=3, m=1)
8/9
sage: Q.local_density(p=5, m=1)
24/25
sage: Q.local_density(p=7, m=1)
48/49
sage: Q.local_density(p=11, m=1)
120/121
"""
n = self.dim()
if (n == 0):
raise TypeError("Oops! We currently don't handle 0-dim'l forms. =(")
## Find the local normal form and p-scale of Q -- Note: This uses the valuation ordering of local_normal_form.
## TO DO: Write a separate p-scale and p-norm routines!
Q_local = self.local_normal_form(p)
if n == 1:
p_valuation = valuation(Q_local[0, 0], p)
else:
p_valuation = min(valuation(Q_local[0, 0], p),
valuation(Q_local[0, 1], p))
## If m is less p-divisible than the matrix, return zero
if (
(m != 0) and (valuation(m, p) < p_valuation)
): ## Note: The (m != 0) condition protects taking the valuation of zero.
return QQ(0)
## If the form is imprimitive, rescale it and call the local density routine
p_adjustment = QQ(1) / p**p_valuation
m_prim = QQ(m) / p**p_valuation
Q_prim = Q_local.scale_by_factor(p_adjustment)
## Return the densities for the reduced problem
return Q_prim.local_density_congruence(p, m_prim)
def local_density(self, p, m):
"""
Gives the local density -- should be called by the user. =)
NOTE: This screens for imprimitive forms, and puts the quadratic
form in local normal form, which is a *requirement* of the
routines performing the computations!
INPUT:
`p` -- a prime number > 0
`m` -- an integer
OUTPUT:
a rational number
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1]) ## NOTE: This is already in local normal form for *all* primes p!
sage: Q.local_density(p=2, m=1)
1
sage: Q.local_density(p=3, m=1)
8/9
sage: Q.local_density(p=5, m=1)
24/25
sage: Q.local_density(p=7, m=1)
48/49
sage: Q.local_density(p=11, m=1)
120/121
"""
n = self.dim()
if (n == 0):
raise TypeError("Oops! We currently don't handle 0-dim'l forms. =(")
## Find the local normal form and p-scale of Q -- Note: This uses the valuation ordering of local_normal_form.
## TO DO: Write a separate p-scale and p-norm routines!
Q_local = self.local_normal_form(p)
if n == 1:
p_valuation = valuation(Q_local[0,0], p)
else:
p_valuation = min(valuation(Q_local[0,0], p), valuation(Q_local[0,1], p))
## If m is less p-divisible than the matrix, return zero
if ((m != 0) and (valuation(m,p) < p_valuation)): ## Note: The (m != 0) condition protects taking the valuation of zero.
return QQ(0)
## If the form is imprimitive, rescale it and call the local density routine
p_adjustment = QQ(1) / p**p_valuation
m_prim = QQ(m) / p**p_valuation
Q_prim = Q_local.scale_by_factor(p_adjustment)
## Return the densities for the reduced problem
return Q_prim.local_density_congruence(p, m_prim)
def find_entry_with_minimal_scale_at_prime(self, p):
"""
Finds the entry of the quadratic form with minimal scale at the
prime p, preferring diagonal entries in case of a tie. (I.e. If
we write the quadratic form as a symmetric matrix M, then this
entry M[i,j] has the minimal valuation at the prime p.)
Note: This answer is independent of the kind of matrix (Gram or
Hessian) associated to the form.
INPUT:
`p` -- a prime number > 0
OUTPUT:
a pair of integers >= 0
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 2, [6, 2, 20]); Q
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 6 2 ]
[ * 20 ]
sage: Q.find_entry_with_minimal_scale_at_prime(2)
(0, 1)
sage: Q.find_entry_with_minimal_scale_at_prime(3)
(1, 1)
sage: Q.find_entry_with_minimal_scale_at_prime(5)
(0, 0)
"""
n = self.dim()
min_val = Infinity
ij_index = None
val_2 = valuation(2, p)
for d in range(n): ## d = difference j-i
for e in range(n -
d): ## e is the length of the diagonal with value d.
## Compute the valuation of the entry
if d == 0:
tmp_val = valuation(self[e, e + d], p)
else:
tmp_val = valuation(self[e, e + d], p) - val_2
## Check if it's any smaller than what we have
if tmp_val < min_val:
ij_index = (e, e + d)
min_val = tmp_val
## Return the result
return ij_index
def find_entry_with_minimal_scale_at_prime(self, p):
"""
Finds the entry of the quadratic form with minimal scale at the
prime p, preferring diagonal entries in case of a tie. (I.e. If
we write the quadratic form as a symmetric matrix M, then this
entry M[i,j] has the minimal valuation at the prime p.)
Note: This answer is independent of the kind of matrix (Gram or
Hessian) associated to the form.
INPUT:
`p` -- a prime number > 0
OUTPUT:
a pair of integers >= 0
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 2, [6, 2, 20]); Q
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 6 2 ]
[ * 20 ]
sage: Q.find_entry_with_minimal_scale_at_prime(2)
(0, 1)
sage: Q.find_entry_with_minimal_scale_at_prime(3)
(1, 1)
sage: Q.find_entry_with_minimal_scale_at_prime(5)
(0, 0)
"""
n = self.dim()
min_val = Infinity
ij_index = None
val_2 = valuation(2, p)
for d in range(n): ## d = difference j-i
for e in range(n - d): ## e is the length of the diagonal with value d.
## Compute the valuation of the entry
if d == 0:
tmp_val = valuation(self[e, e+d], p)
else:
tmp_val = valuation(self[e, e+d], p) - val_2
## Check if it's any smaller than what we have
if tmp_val < min_val:
ij_index = (e,e+d)
min_val = tmp_val
## Return the result
return ij_index
def _push(self, x):
f = self._t
level = valuation(x.parent().degree(), self._degree)
p = [self[level-1](list(c))
for c in izip_longest(*decompose(x.lift(), f), fillvalue=0)]
return p if p else [self[level-1](0)]
def _e_bounds(self, n, prec):
p = self._p
prec = max(2,prec)
R = PowerSeriesRing(ZZ,'T',prec+1)
T = R(R.gen(),prec +1)
w = (1+T)**(p**n) - 1
return [infinity] + [valuation(w[j],p) for j in range(1,min(w.degree()+1,prec))]
def _push(self, x):
level = valuation(x.parent().degree(), self._degree)
f, g = self._rel_polys[-level % len(self._rel_polys)]
deg = self._degree**(level - 1) - 1
x *= x.parent(g**deg)
p = [self[level-1](list(c))
for c in izip_longest(*decompose(x.lift(), f, g, deg), fillvalue=0)]
return p if p else [self[level-1](0)]
def _lift(self, xs):
if not xs:
raise RuntimeError("Don't know where to lift to.")
f = self._t
Ps = map(self._P.__call__, izip_longest(*xs))
level = valuation(xs[0].parent().degree(), self._degree)
return self[level+1](compose(Ps, f))
def _lift(self, xs):
if not xs:
raise RuntimeError("Don't know where to lift to.")
level = valuation(xs[0].parent().degree(), self._degree)
f, g = self._rel_polys[(-level-1) % len(self._rel_polys)]
Ps = map(self._P.__call__, izip_longest(*xs))
return (self[level+1](compose(Ps, f, g)) /
self[level+1](g**(len(Ps)-1)))
def valuation(self,p): """returns the exponent of the highest power of p which divides all coefficients of self""" #assert self.base_ring==QQ, "need to be working over Q in valuation" k=self.weight v=self.vars() X=v[0] Y=v[1] v=[] for j in range(k+1): v=v+[valuation(QQ(self.poly.coefficient((X**j)*(Y**(k-j)))),p)] return min(v)
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 local_primitive_density(self, p, m):
"""
Gives the local primitive density -- should be called by the user. =)
NOTE: This screens for imprimitive forms, and puts the
quadratic form in local normal form, which is a *requirement* of
the routines performing the computations!
INPUT:
`p` -- a prime number > 0
`m` -- an integer
OUTPUT:
a rational number
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 4, range(10))
sage: Q[0,0] = 5
sage: Q[1,1] = 10
sage: Q[2,2] = 15
sage: Q[3,3] = 20
sage: Q
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 5 1 2 3 ]
[ * 10 5 6 ]
[ * * 15 8 ]
[ * * * 20 ]
sage: Q.theta_series(20)
1 + 2*q^5 + 2*q^10 + 2*q^14 + 2*q^15 + 2*q^16 + 2*q^18 + O(q^20)
sage: Q.local_normal_form(2)
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 0 1 0 0 ]
[ * 0 0 0 ]
[ * * 0 1 ]
[ * * * 0 ]
sage: Q.local_primitive_density(2, 1)
3/4
sage: Q.local_primitive_density(5, 1)
24/25
sage: Q.local_primitive_density(2, 5)
3/4
sage: Q.local_density(2, 5)
3/4
"""
n = self.dim()
if (n == 0):
raise TypeError("Oops! We currently don't handle 0-dim'l forms. =(")
## Find the local normal form and p-scale of Q -- Note: This uses the valuation ordering of local_normal_form.
## TO DO: Write a separate p-scale and p-norm routines!
Q_local = self.local_normal_form(p)
if n == 1:
p_valuation = valuation(Q_local[0, 0], p)
else:
p_valuation = min(valuation(Q_local[0, 0], p),
valuation(Q_local[0, 1], p))
## If m is less p-divisible than the matrix, return zero
if (
(m != 0) and (valuation(m, p) < p_valuation)
): ## Note: The (m != 0) condition protects taking the valuation of zero.
return QQ(0)
## If the form is imprimitive, rescale it and call the local density routine
p_adjustment = QQ(1) / p**p_valuation
m_prim = QQ(m) / p**p_valuation
Q_prim = Q_local.scale_by_factor(p_adjustment)
## Return the densities for the reduced problem
return Q_prim.local_primitive_density_congruence(p, m_prim)
def local_badII_density_congruence(self, p, m, Zvec=None, NZvec=None):
"""
Finds the Bad-type II local density of Q representing `m` at `p`.
(Assuming that `p` > 2 and Q is given in local diagonal form.)
INPUT:
Q -- quadratic form assumed to be block diagonal and p-integral
`p` -- a prime number
`m` -- an integer
Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None
OUTPUT:
a rational number
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3])
sage: Q.local_badII_density_congruence(2, 1, None, None)
0
sage: Q.local_badII_density_congruence(2, 2, None, None)
0
sage: Q.local_badII_density_congruence(2, 4, None, None)
0
sage: Q.local_badII_density_congruence(3, 1, None, None)
0
sage: Q.local_badII_density_congruence(3, 6, None, None)
0
sage: Q.local_badII_density_congruence(3, 9, None, None)
0
sage: Q.local_badII_density_congruence(3, 27, None, None)
0
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,3,9,9])
sage: Q.local_badII_density_congruence(3, 1, None, None)
0
sage: Q.local_badII_density_congruence(3, 3, None, None)
0
sage: Q.local_badII_density_congruence(3, 6, None, None)
0
sage: Q.local_badII_density_congruence(3, 9, None, None)
4/27
sage: Q.local_badII_density_congruence(3, 18, None, None)
4/9
"""
## DIAGNOSTIC
verbose(" In local_badII_density_congruence with ")
verbose(" Q is: \n" + str(self))
verbose(" p = " + str(p))
verbose(" m = " + str(m))
verbose(" Zvec = " + str(Zvec))
verbose(" NZvec = " + str(NZvec))
## Put the Zvec congruence condition in a standard form
if Zvec is None:
Zvec = []
n = self.dim()
## Sanity Check on Zvec and NZvec:
## -------------------------------
Sn = Set(range(n))
if (Zvec is not None) and (len(Set(Zvec) + Sn) > n):
raise RuntimeError("Zvec must be a subset of {0, ..., n-1}.")
if (NZvec is not None) and (len(Set(NZvec) + Sn) > n):
raise RuntimeError("NZvec must be a subset of {0, ..., n-1}.")
## Define the indexing sets S_i:
## -----------------------------
S0 = []
S1 = []
S2plus = []
for i in range(n):
## Compute the valuation of each index, allowing for off-diagonal terms
if (self[i,i] == 0):
if (i == 0):
val = valuation(self[i,i+1], p) ## Look at the term to the right
elif (i == n-1):
val = valuation(self[i-1,i], p) ## Look at the term above
else:
val = valuation(self[i,i+1] + self[i-1,i], p) ## Finds the valuation of the off-diagonal term since only one isn't zero
else:
val = valuation(self[i,i], p)
## Sort the indices into disjoint sets by their valuation
if (val == 0):
S0 += [i]
elif (val == 1):
S1 += [i]
elif (val >= 2):
S2plus += [i]
## Check that S2 is non-empty and p^2 divides m to proceed, otherwise return no solutions.
p2 = p * p
if (S2plus == []) or (m % p2 != 0):
return 0
## Check some conditions for no bad-type II solutions to exist
if (NZvec is not None) and (len(Set(S2plus).intersection(Set(NZvec))) == 0):
return 0
## Check that the form is primitive... WHY IS THIS NECESSARY?
if (S0 == []):
print " Using Q = " + str(self)
print " and p = " + str(p)
raise RuntimeError("Oops! The form is not primitive!")
## DIAGNOSTIC
verbose("\n Entering BII routine ")
verbose(" S0 is " + str(S0))
verbose(" S1 is " + str(S1))
verbose(" S2plus is " + str(S2plus))
verbose(" m = " + str(m) + " p = " + str(p))
## Make the form Qnew for the reduction procedure:
## -----------------------------------------------
Qnew = deepcopy(self) ## TO DO: DO THIS WITHOUT A copy(). =)
for i in range(n):
if i in S2plus:
Qnew[i,i] = Qnew[i,i] / p2
if (p == 2) and (i < n-1):
Qnew[i,i+1] = Qnew[i,i+1] / p2
## DIAGNOSTIC
verbose("\n\n Check of Bad-type II reduction: \n")
verbose(" Q is " + str(self))
verbose(" Qnew is " + str(Qnew))
## Perform the reduction formula
Zvec_geq_2 = list(Set([i for i in Zvec if i in S2plus]))
if NZvec is None:
NZvec_geq_2 = NZvec
else:
NZvec_geq_2 = list(Set([i for i in NZvec if i in S2plus]))
return QQ(p**(len(S2plus) + 2 - n)) \
* (Qnew.local_density_congruence(p, m / p2, Zvec_geq_2, NZvec_geq_2) \
- Qnew.local_density_congruence(p, m / p2, S2plus , NZvec_geq_2))
def has_equivalent_Jordan_decomposition_at_prime(self, other, p):
"""
Determines if the given quadratic form has a Jordan decomposition
equivalent to that of self.
INPUT:
a QuadraticForm
OUTPUT:
boolean
EXAMPLES::
sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 1, 0, 3])
sage: Q2 = QuadraticForm(ZZ, 3, [1, 0, 0, 2, -2, 6])
sage: Q3 = QuadraticForm(ZZ, 3, [1, 0, 0, 1, 0, 11])
sage: [Q1.level(), Q2.level(), Q3.level()]
[44, 44, 44]
sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q2,2)
False
sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q2,11)
False
sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q3,2)
False
sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q3,11)
True
sage: Q2.has_equivalent_Jordan_decomposition_at_prime(Q3,2)
True
sage: Q2.has_equivalent_Jordan_decomposition_at_prime(Q3,11)
False
"""
## Sanity Checks
#if not isinstance(other, QuadraticForm):
if not isinstance(other, type(self)):
raise TypeError(
"Oops! The first argument must be of type QuadraticForm.")
if not is_prime(p):
raise TypeError("Oops! The second argument must be a prime number.")
## Get the relevant local normal forms quickly
self_jordan = self.jordan_blocks_by_scale_and_unimodular(p,
safe_flag=False)
other_jordan = other.jordan_blocks_by_scale_and_unimodular(p,
safe_flag=False)
## DIAGNOSTIC
#print "self_jordan = ", self_jordan
#print "other_jordan = ", other_jordan
## Check for the same number of Jordan components
if len(self_jordan) != len(other_jordan):
return False
## Deal with odd primes: Check that the Jordan component scales, dimensions, and discriminants are the same
if p != 2:
for i in range(len(self_jordan)):
if (self_jordan[i][0] != other_jordan[i][0]) \
or (self_jordan[i][1].dim() != other_jordan[i][1].dim()) \
or (legendre_symbol(self_jordan[i][1].det() * other_jordan[i][1].det(), p) != 1):
return False
## All tests passed for an odd prime.
return True
## For p = 2: Check that all Jordan Invariants are the same.
elif p == 2:
## Useful definition
t = len(self_jordan) ## Define t = Number of Jordan components
## Check that all Jordan Invariants are the same (scale, dim, and norm)
for i in range(t):
if (self_jordan[i][0] != other_jordan[i][0]) \
or (self_jordan[i][1].dim() != other_jordan[i][1].dim()) \
or (valuation(GCD(self_jordan[i][1].coefficients()), p) != valuation(GCD(other_jordan[i][1].coefficients()), p)):
return False
## DIAGNOSTIC
#print "Passed the Jordan invariant test."
## Use O'Meara's isometry test 93:29 on p277.
## ------------------------------------------
## List of norms, scales, and dimensions for each i
scale_list = [ZZ(2)**self_jordan[i][0] for i in range(t)]
norm_list = [
ZZ(2)**(self_jordan[i][0] +
valuation(GCD(self_jordan[i][1].coefficients()), 2))
for i in range(t)
]
dim_list = [(self_jordan[i][1].dim()) for i in range(t)]
## List of Hessian determinants and Hasse invariants for each Jordan (sub)chain
## (Note: This is not the same as O'Meara's Gram determinants, but ratios are the same!) -- NOT SO GOOD...
## But it matters in condition (ii), so we multiply all by 2 (instead of dividing by 2 since only square-factors matter, and it's easier.)
j = 0
self_chain_det_list = [
self_jordan[j][1].Gram_det() * (scale_list[j]**dim_list[j])
]
other_chain_det_list = [
other_jordan[j][1].Gram_det() * (scale_list[j]**dim_list[j])
]
self_hasse_chain_list = [
self_jordan[j][1].scale_by_factor(
ZZ(2)**self_jordan[j][0]).hasse_invariant__OMeara(2)
]
other_hasse_chain_list = [
other_jordan[j][1].scale_by_factor(
ZZ(2)**other_jordan[j][0]).hasse_invariant__OMeara(2)
]
for j in range(1, t):
self_chain_det_list.append(self_chain_det_list[j - 1] *
self_jordan[j][1].Gram_det() *
(scale_list[j]**dim_list[j]))
other_chain_det_list.append(other_chain_det_list[j - 1] *
other_jordan[j][1].Gram_det() *
(scale_list[j]**dim_list[j]))
self_hasse_chain_list.append(self_hasse_chain_list[j-1] \
* hilbert_symbol(self_chain_det_list[j-1], self_jordan[j][1].Gram_det(), 2) \
* self_jordan[j][1].hasse_invariant__OMeara(2))
other_hasse_chain_list.append(other_hasse_chain_list[j-1] \
* hilbert_symbol(other_chain_det_list[j-1], other_jordan[j][1].Gram_det(), 2) \
* other_jordan[j][1].hasse_invariant__OMeara(2))
## SANITY CHECK -- check that the scale powers are strictly increasing
for i in range(1, len(scale_list)):
if scale_list[i - 1] >= scale_list[i]:
raise RuntimeError(
"Oops! There is something wrong with the Jordan Decomposition -- the given scales are not strictly increasing!"
)
## DIAGNOSTIC
#print "scale_list = ", scale_list
#print "norm_list = ", norm_list
#print "dim_list = ", dim_list
#print
#print "self_chain_det_list = ", self_chain_det_list
#print "other_chain_det_list = ", other_chain_det_list
#print "self_hasse_chain_list = ", self_hasse_chain_list
#print "other_hasse_chain_det_list = ", other_hasse_chain_list
## Test O'Meara's two conditions
for i in range(t - 1):
## Condition (i): Check that their (unit) ratio is a square (but it suffices to check at most mod 8).
modulus = norm_list[i] * norm_list[i + 1] / (scale_list[i]**2)
if modulus > 8:
modulus = 8
if (modulus > 1) and ((
(self_chain_det_list[i] / other_chain_det_list[i]) % modulus)
!= 1):
#print "Failed when i =", i, " in condition 1."
return False
## Check O'Meara's condition (ii) when appropriate
if norm_list[i + 1] % (4 * norm_list[i]) == 0:
if self_hasse_chain_list[i] * hilbert_symbol(norm_list[i] * other_chain_det_list[i], -self_chain_det_list[i], 2) \
!= other_hasse_chain_list[i] * hilbert_symbol(norm_list[i], -other_chain_det_list[i], 2): ## Nipp conditions
#print "Failed when i =", i, " in condition 2."
return False
## All tests passed for the prime 2.
return True
else:
raise TypeError("Oops! This should not have happened.")
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 local_badI_density_congruence(self, p, m, Zvec=None, NZvec=None):
"""
Finds the Bad-type I local density of Q representing `m` at `p`.
(Assuming that p > 2 and Q is given in local diagonal form.)
INPUT:
Q -- quadratic form assumed to be block diagonal and `p`-integral
`p` -- a prime number
`m` -- an integer
Zvec, NZvec -- non-repeating lists of integers in range(self.dim()) or None
OUTPUT:
a rational number
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,2,3])
sage: Q.local_badI_density_congruence(2, 1, None, None)
0
sage: Q.local_badI_density_congruence(2, 2, None, None)
1
sage: Q.local_badI_density_congruence(2, 4, None, None)
0
sage: Q.local_badI_density_congruence(3, 1, None, None)
0
sage: Q.local_badI_density_congruence(3, 6, None, None)
0
sage: Q.local_badI_density_congruence(3, 9, None, None)
0
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.local_badI_density_congruence(2, 1, None, None)
0
sage: Q.local_badI_density_congruence(2, 2, None, None)
0
sage: Q.local_badI_density_congruence(2, 4, None, None)
0
sage: Q.local_badI_density_congruence(3, 2, None, None)
0
sage: Q.local_badI_density_congruence(3, 6, None, None)
0
sage: Q.local_badI_density_congruence(3, 9, None, None)
0
::
sage: Q = DiagonalQuadraticForm(ZZ, [1,3,3,9])
sage: Q.local_badI_density_congruence(3, 1, None, None)
0
sage: Q.local_badI_density_congruence(3, 3, None, None)
4/3
sage: Q.local_badI_density_congruence(3, 6, None, None)
4/3
sage: Q.local_badI_density_congruence(3, 9, None, None)
0
sage: Q.local_badI_density_congruence(3, 18, None, None)
0
"""
## DIAGNOSTIC
verbose(" In local_badI_density_congruence with ")
verbose(" Q is: \n" + str(self))
verbose(" p = " + str(p))
verbose(" m = " + str(m))
verbose(" Zvec = " + str(Zvec))
verbose(" NZvec = " + str(NZvec))
## Put the Zvec congruence condition in a standard form
if Zvec is None:
Zvec = []
n = self.dim()
## Sanity Check on Zvec and NZvec:
## -------------------------------
Sn = Set(range(n))
if (Zvec is not None) and (len(Set(Zvec) + Sn) > n):
raise RuntimeError("Zvec must be a subset of {0, ..., n-1}.")
if (NZvec is not None) and (len(Set(NZvec) + Sn) > n):
raise RuntimeError("NZvec must be a subset of {0, ..., n-1}.")
## Define the indexing set S_0, and determine if S_1 is empty:
## -----------------------------------------------------------
S0 = []
S1_empty_flag = True ## This is used to check if we should be computing BI solutions at all!
## (We should really to this earlier, but S1 must be non-zero to proceed.)
## Find the valuation of each variable (which will be the same over 2x2 blocks),
## remembering those of valuation 0 and if an entry of valuation 1 exists.
for i in range(n):
## Compute the valuation of each index, allowing for off-diagonal terms
if (self[i,i] == 0):
if (i == 0):
val = valuation(self[i,i+1], p) ## Look at the term to the right
else:
if (i == n-1):
val = valuation(self[i-1,i], p) ## Look at the term above
else:
val = valuation(self[i,i+1] + self[i-1,i], p) ## Finds the valuation of the off-diagonal term since only one isn't zero
else:
val = valuation(self[i,i], p)
if (val == 0):
S0 += [i]
elif (val == 1):
S1_empty_flag = False ## Need to have a non-empty S1 set to proceed with Bad-type I reduction...
## Check that S1 is non-empty and p|m to proceed, otherwise return no solutions.
if (S1_empty_flag == True) or (m % p != 0):
return 0
## Check some conditions for no bad-type I solutions to exist
if (NZvec is not None) and (len(Set(S0).intersection(Set(NZvec))) != 0):
return 0
## Check that the form is primitive... WHY DO WE NEED TO DO THIS?!?
if (S0 == []):
print " Using Q = " + str(self)
print " and p = " + str(p)
raise RuntimeError("Oops! The form is not primitive!")
## DIAGNOSTIC
verbose(" m = " + str(m) + " p = " + str(p))
verbose(" S0 = " + str(S0))
verbose(" len(S0) = " + str(len(S0)))
## Make the form Qnew for the reduction procedure:
## -----------------------------------------------
Qnew = deepcopy(self) ## TO DO: DO THIS WITHOUT A copy(). =)
for i in range(n):
if i in S0:
Qnew[i,i] = p * Qnew[i,i]
if ((p == 2) and (i < n-1)):
Qnew[i,i+1] = p * Qnew[i,i+1]
else:
Qnew[i,i] = Qnew[i,i] / p
if ((p == 2) and (i < n-1)):
Qnew[i,i+1] = Qnew[i,i+1] / p
## DIAGNOSTIC
verbose("\n\n Check of Bad-type I reduction: \n")
verbose(" Q is " + str(self))
verbose(" Qnew is " + str(Qnew))
verbose(" p = " + str(p))
verbose(" m / p = " + str(m/p))
verbose(" NZvec " + str(NZvec))
## Do the reduction
Zvec_geq_1 = list(Set([i for i in Zvec if i not in S0]))
if NZvec is None:
NZvec_geq_1 = NZvec
else:
NZvec_geq_1 = list(Set([i for i in NZvec if i not in S0]))
return QQ(p**(1 - len(S0))) * Qnew.local_good_density_congruence(p, m / p, Zvec_geq_1, NZvec_geq_1)
def local_primitive_density(self, p, m):
"""
Gives the local primitive density -- should be called by the user. =)
NOTE: This screens for imprimitive forms, and puts the
quadratic form in local normal form, which is a *requirement* of
the routines performing the computations!
INPUT:
`p` -- a prime number > 0
`m` -- an integer
OUTPUT:
a rational number
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 4, range(10))
sage: Q[0,0] = 5
sage: Q[1,1] = 10
sage: Q[2,2] = 15
sage: Q[3,3] = 20
sage: Q
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 5 1 2 3 ]
[ * 10 5 6 ]
[ * * 15 8 ]
[ * * * 20 ]
sage: Q.theta_series(20)
1 + 2*q^5 + 2*q^10 + 2*q^14 + 2*q^15 + 2*q^16 + 2*q^18 + O(q^20)
sage: Q.local_normal_form(2)
Quadratic form in 4 variables over Integer Ring with coefficients:
[ 0 1 0 0 ]
[ * 0 0 0 ]
[ * * 0 1 ]
[ * * * 0 ]
sage: Q.local_primitive_density(2, 1)
3/4
sage: Q.local_primitive_density(5, 1)
24/25
sage: Q.local_primitive_density(2, 5)
3/4
sage: Q.local_density(2, 5)
3/4
"""
n = self.dim()
if (n == 0):
raise TypeError("Oops! We currently don't handle 0-dim'l forms. =(")
## Find the local normal form and p-scale of Q -- Note: This uses the valuation ordering of local_normal_form.
## TO DO: Write a separate p-scale and p-norm routines!
Q_local = self.local_normal_form(p)
if n == 1:
p_valuation = valuation(Q_local[0,0], p)
else:
p_valuation = min(valuation(Q_local[0,0], p), valuation(Q_local[0,1], p))
## If m is less p-divisible than the matrix, return zero
if ((m != 0) and (valuation(m,p) < p_valuation)): ## Note: The (m != 0) condition protects taking the valuation of zero.
return QQ(0)
## If the form is imprimitive, rescale it and call the local density routine
p_adjustment = QQ(1) / p**p_valuation
m_prim = QQ(m) / p**p_valuation
Q_prim = Q_local.scale_by_factor(p_adjustment)
## Return the densities for the reduced problem
return Q_prim.local_primitive_density_congruence(p, m_prim)
def CohenOesterle(eps, k):
r"""
Compute the Cohen-Oesterle function associate to eps, `k`.
This is a summand in the formula for the dimension of the space of
cusp forms of weight `2` with character
`\varepsilon`.
INPUT:
- ``eps`` - Dirichlet character
- ``k`` - integer
OUTPUT: element of the base ring of eps.
EXAMPLES::
sage: G.<eps> = DirichletGroup(7)
sage: sage.modular.dims.CohenOesterle(eps, 2)
-2/3
sage: sage.modular.dims.CohenOesterle(eps, 4)
-1
"""
N = eps.modulus()
facN = factor(N)
f = eps.conductor()
gamma_k = 0
if k % 4 == 2:
gamma_k = frac(-1, 4)
elif k % 4 == 0:
gamma_k = frac(1, 4)
mu_k = 0
if k % 3 == 2:
mu_k = frac(-1, 3)
elif k % 3 == 0:
mu_k = frac(1, 3)
def _lambda(r, s, p):
"""
Used internally by the CohenOesterle function.
INPUT:
- ``r, s, p`` - integers
OUTPUT: Integer
EXAMPLES: (indirect doctest)
::
sage: K = CyclotomicField(3)
sage: eps = DirichletGroup(7*43,K).0^2
sage: sage.modular.dims.CohenOesterle(eps,2)
-4/3
"""
if 2 * s <= r:
if r % 2 == 0:
return p**(r // 2) + p**((r // 2) - 1)
return 2 * p**((r - 1) // 2)
return 2 * (p**(r - s))
#end def of lambda
K = eps.base_ring()
return K(frac(-1,2) * mul([_lambda(r,valuation(f,p),p) for p, r in facN]) + \
gamma_k * mul([CO_delta(r,p,N,eps) for p, r in facN]) + \
mu_k * mul([CO_nu(r,p,N,eps) for p, r in facN]))
def local_normal_form(self, p):
"""
Returns the a locally integrally equivalent quadratic form over
the p-adic integers Z_p which gives the Jordan decomposition. The
Jordan components are written as sums of blocks of size <= 2 and
are arranged by increasing scale, and then by increasing norm.
(This is equivalent to saying that we put the 1x1 blocks before
the 2x2 blocks in each Jordan component.)
INPUT:
`p` -- a positive prime number.
OUTPUT:
a quadratic form over ZZ
WARNING: Currently this only works for quadratic forms defined over ZZ.
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 2, [10,4,1])
sage: Q.local_normal_form(5)
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 1 0 ]
[ * 6 ]
::
sage: Q.local_normal_form(3)
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 10 0 ]
[ * 15 ]
sage: Q.local_normal_form(2)
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 1 0 ]
[ * 6 ]
"""
## Sanity Checks
if (self.base_ring() != IntegerRing()):
raise NotImplementedError(
"Oops! This currently only works for quadratic forms defined over IntegerRing(). =("
)
if not ((p >= 2) and is_prime(p)):
raise TypeError("Oops! p is not a positive prime number. =(")
## Some useful local variables
Q = copy.deepcopy(self)
Q.__init__(self.base_ring(), self.dim(), self.coefficients())
## Prepare the final form to return
Q_Jordan = copy.deepcopy(self)
Q_Jordan.__init__(self.base_ring(), 0)
while Q.dim() > 0:
n = Q.dim()
## Step 1: Find the minimally p-divisible matrix entry, preferring diagonals
## -------------------------------------------------------------------------
(min_i, min_j) = Q.find_entry_with_minimal_scale_at_prime(p)
if min_i == min_j:
min_val = valuation(2 * Q[min_i, min_j], p)
else:
min_val = valuation(Q[min_i, min_j], p)
## Error if we still haven't seen non-zero coefficients!
if (min_val == Infinity):
raise RuntimeError("Oops! The original matrix is degenerate. =(")
## Step 2: Arrange for the upper leftmost entry to have minimal valuation
## ----------------------------------------------------------------------
if (min_i == min_j):
block_size = 1
Q.swap_variables(0, min_i, in_place=True)
else:
## Work in the upper-left 2x2 block, and replace it by its 2-adic equivalent form
Q.swap_variables(0, min_i, in_place=True)
Q.swap_variables(1, min_j, in_place=True)
## 1x1 => make upper left the smallest
if (p != 2):
block_size = 1
Q.add_symmetric(1, 0, 1, in_place=True)
## 2x2 => replace it with the appropriate 2x2 matrix
else:
block_size = 2
## DIAGNOSTIC
#print "\n Finished Step 2 \n";
#print "\n Q is: \n" + str(Q) + "\n";
#print " p is: " + str(p)
#print " min_val is: " + str( min_val)
#print " block_size is: " + str(block_size)
#print "\n Starting Step 3 \n"
## Step 3: Clear out the remaining entries
## ---------------------------------------
min_scale = p**min_val ## This is the minimal valuation of the Hessian matrix entries.
##DIAGNOSTIC
#print "Starting Step 3:"
#print "----------------"
#print " min_scale is: " + str(min_scale)
## Perform cancellation over Z by ensuring divisibility
if (block_size == 1):
a = 2 * Q[0, 0]
for j in range(block_size, n):
b = Q[0, j]
g = GCD(a, b)
## DIAGNSOTIC
#print "Cancelling from a 1x1 block:"
#print "----------------------------"
#print " Cancelling entry with index (" + str(upper_left) + ", " + str(j) + ")"
#print " entry = " + str(b)
#print " gcd = " + str(g)
#print " a = " + str(a)
#print " b = " + str(b)
#print " a/g = " + str(a/g) + " (used for stretching)"
#print " -b/g = " + str(-b/g) + " (used for cancelling)"
## Sanity Check: a/g is a p-unit
if valuation(g, p) != valuation(a, p):
raise RuntimeError(
"Oops! We have a problem with our rescaling not preserving p-integrality!"
)
Q.multiply_variable(
ZZ(a / g), j, in_place=True
) ## Ensures that the new b entry is divisible by a
Q.add_symmetric(ZZ(-b / g), j, 0,
in_place=True) ## Performs the cancellation
elif (block_size == 2):
a1 = 2 * Q[0, 0]
a2 = Q[0, 1]
b1 = Q[1, 0] ## This is the same as a2
b2 = 2 * Q[1, 1]
big_det = (a1 * b2 - a2 * b1)
small_det = big_det / (min_scale * min_scale)
## Cancels out the rows/columns of the 2x2 block
for j in range(block_size, n):
a = Q[0, j]
b = Q[1, j]
## Ensures an integral result (scale jth row/column by big_det)
Q.multiply_variable(big_det, j, in_place=True)
## Performs the cancellation (by producing -big_det * jth row/column)
Q.add_symmetric(ZZ(-(a * b2 - b * a2)), j, 0, in_place=True)
Q.add_symmetric(ZZ(-(-a * b1 + b * a1)), j, 1, in_place=True)
## Now remove the extra factor (non p-unit factor) in big_det we introduced above
Q.divide_variable(ZZ(min_scale * min_scale), j, in_place=True)
## DIAGNOSTIC
#print "Cancelling out a 2x2 block:"
#print "---------------------------"
#print " a1 = " + str(a1)
#print " a2 = " + str(a2)
#print " b1 = " + str(b1)
#print " b2 = " + str(b2)
#print " big_det = " + str(big_det)
#print " min_scale = " + str(min_scale)
#print " small_det = " + str(small_det)
#print " Q = \n", Q
## Uses Cassels's proof to replace the remaining 2 x 2 block
if (((1 + small_det) % 8) == 0):
Q[0, 0] = 0
Q[1, 1] = 0
Q[0, 1] = min_scale
elif (((5 + small_det) % 8) == 0):
Q[0, 0] = min_scale
Q[1, 1] = min_scale
Q[0, 1] = min_scale
else:
raise RuntimeError(
"Error in LocalNormal: Impossible behavior for a 2x2 block! \n"
)
## Check that the cancellation worked, extract the upper-left block, and trim Q to handle the next block.
for i in range(block_size):
for j in range(block_size, n):
if Q[i, j] != 0:
raise RuntimeError(
"Oops! The cancellation didn't work properly at entry ("
+ str(i) + ", " + str(j) + ").")
Q_Jordan = Q_Jordan + Q.extract_variables(range(block_size))
Q = Q.extract_variables(range(block_size, n))
return Q_Jordan
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]
def jordan_blocks_by_scale_and_unimodular(self, p, safe_flag=True):
"""
Returns a list of pairs `(s_i, L_i)` where `L_i` is a maximal
`p^{s_i}`-unimodular Jordan component which is further decomposed into
block diagonals of block size `\le 2`. For each `L_i` the 2x2 blocks are
listed after the 1x1 blocks (which follows from the convention of the
:meth:`local_normal_form` method).
..note ::
The decomposition of each `L_i` into smaller block is not unique!
The ``safe_flag`` argument allows us to select whether we want a copy of
the output, or the original output. By default ``safe_flag = True``, so we
return a copy of the cached information. If this is set to ``False``, then
the routine is much faster but the return values are vulnerable to being
corrupted by the user.
INPUT:
- `p` -- a prime number > 0.
OUTPUT:
A list of pairs `(s_i, L_i)` where:
- `s_i` is an integer,
- `L_i` is a block-diagonal unimodular quadratic form over `\ZZ_p`.
.. note::
These forms `L_i` are defined over the `p`-adic integers, but by a
matrix over `\ZZ` (or `\QQ`?).
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,9,5,7])
sage: Q.jordan_blocks_by_scale_and_unimodular(3)
[(0, Quadratic form in 3 variables over Integer Ring with coefficients:
[ 1 0 0 ]
[ * 5 0 ]
[ * * 7 ]), (2, Quadratic form in 1 variables over Integer Ring with coefficients:
[ 1 ])]
::
sage: Q2 = QuadraticForm(ZZ, 2, [1,1,1])
sage: Q2.jordan_blocks_by_scale_and_unimodular(2)
[(-1, Quadratic form in 2 variables over Integer Ring with coefficients:
[ 2 2 ]
[ * 2 ])]
sage: Q = Q2 + Q2.scale_by_factor(2)
sage: Q.jordan_blocks_by_scale_and_unimodular(2)
[(-1, Quadratic form in 2 variables over Integer Ring with coefficients:
[ 2 2 ]
[ * 2 ]), (0, Quadratic form in 2 variables over Integer Ring with coefficients:
[ 2 2 ]
[ * 2 ])]
"""
## Try to use the cached result
try:
if safe_flag:
return copy.deepcopy(self.__jordan_blocks_by_scale_and_unimodular_dict[p])
else:
return self.__jordan_blocks_by_scale_and_unimodular_dict[p]
except StandardError:
## Initialize the global dictionary if it doesn't exist
if not hasattr(self, '__jordan_blocks_by_scale_and_unimodular_dict'):
self.__jordan_blocks_by_scale_and_unimodular_dict = {}
## Deal with zero dim'l forms
if self.dim() == 0:
return []
## Find the Local Normal form of Q at p
Q1 = self.local_normal_form(p)
## Parse this into Jordan Blocks
n = Q1.dim()
tmp_Jordan_list = []
i = 0
start_ind = 0
if (n >= 2) and (Q1[0,1] != 0):
start_scale = valuation(Q1[0,1], p) - 1
else:
start_scale = valuation(Q1[0,0], p)
while (i < n):
## Determine the size of the current block
if (i == n-1) or (Q1[i,i+1] == 0):
block_size = 1
else:
block_size = 2
## Determine the valuation of the current block
if block_size == 1:
block_scale = valuation(Q1[i,i], p)
else:
block_scale = valuation(Q1[i,i+1], p) - 1
## Process the previous block if the valuation increased
if block_scale > start_scale:
tmp_Jordan_list += [(start_scale, Q1.extract_variables(range(start_ind, i)).scale_by_factor(ZZ(1) / (QQ(p)**(start_scale))))]
start_ind = i
start_scale = block_scale
## Increment the index
i += block_size
## Add the last block
tmp_Jordan_list += [(start_scale, Q1.extract_variables(range(start_ind, n)).scale_by_factor(ZZ(1) / QQ(p)**(start_scale)))]
## Cache the result
self.__jordan_blocks_by_scale_and_unimodular_dict[p] = tmp_Jordan_list
## Return the result
return tmp_Jordan_list
def has_equivalent_Jordan_decomposition_at_prime(self, other, p):
"""
Determines if the given quadratic form has a Jordan decomposition
equivalent to that of self.
INPUT:
a QuadraticForm
OUTPUT:
boolean
EXAMPLES::
sage: Q1 = QuadraticForm(ZZ, 3, [1, 0, -1, 1, 0, 3])
sage: Q2 = QuadraticForm(ZZ, 3, [1, 0, 0, 2, -2, 6])
sage: Q3 = QuadraticForm(ZZ, 3, [1, 0, 0, 1, 0, 11])
sage: [Q1.level(), Q2.level(), Q3.level()]
[44, 44, 44]
sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q2,2)
False
sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q2,11)
False
sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q3,2)
False
sage: Q1.has_equivalent_Jordan_decomposition_at_prime(Q3,11)
True
sage: Q2.has_equivalent_Jordan_decomposition_at_prime(Q3,2)
True
sage: Q2.has_equivalent_Jordan_decomposition_at_prime(Q3,11)
False
"""
## Sanity Checks
#if not isinstance(other, QuadraticForm):
if type(other) != type(self):
raise TypeError, "Oops! The first argument must be of type QuadraticForm."
if not is_prime(p):
raise TypeError, "Oops! The second argument must be a prime number."
## Get the relevant local normal forms quickly
self_jordan = self.jordan_blocks_by_scale_and_unimodular(p, safe_flag= False)
other_jordan = other.jordan_blocks_by_scale_and_unimodular(p, safe_flag=False)
## DIAGNOSTIC
#print "self_jordan = ", self_jordan
#print "other_jordan = ", other_jordan
## Check for the same number of Jordan components
if len(self_jordan) != len(other_jordan):
return False
## Deal with odd primes: Check that the Jordan component scales, dimensions, and discriminants are the same
if p != 2:
for i in range(len(self_jordan)):
if (self_jordan[i][0] != other_jordan[i][0]) \
or (self_jordan[i][1].dim() != other_jordan[i][1].dim()) \
or (legendre_symbol(self_jordan[i][1].det() * other_jordan[i][1].det(), p) != 1):
return False
## All tests passed for an odd prime.
return True
## For p = 2: Check that all Jordan Invariants are the same.
elif p == 2:
## Useful definition
t = len(self_jordan) ## Define t = Number of Jordan components
## Check that all Jordan Invariants are the same (scale, dim, and norm)
for i in range(t):
if (self_jordan[i][0] != other_jordan[i][0]) \
or (self_jordan[i][1].dim() != other_jordan[i][1].dim()) \
or (valuation(GCD(self_jordan[i][1].coefficients()), p) != valuation(GCD(other_jordan[i][1].coefficients()), p)):
return False
## DIAGNOSTIC
#print "Passed the Jordan invariant test."
## Use O'Meara's isometry test 93:29 on p277.
## ------------------------------------------
## List of norms, scales, and dimensions for each i
scale_list = [ZZ(2)**self_jordan[i][0] for i in range(t)]
norm_list = [ZZ(2)**(self_jordan[i][0] + valuation(GCD(self_jordan[i][1].coefficients()), 2)) for i in range(t)]
dim_list = [(self_jordan[i][1].dim()) for i in range(t)]
## List of Hessian determinants and Hasse invariants for each Jordan (sub)chain
## (Note: This is not the same as O'Meara's Gram determinants, but ratios are the same!) -- NOT SO GOOD...
## But it matters in condition (ii), so we multiply all by 2 (instead of dividing by 2 since only square-factors matter, and it's easier.)
j = 0
self_chain_det_list = [ self_jordan[j][1].Gram_det() * (scale_list[j]**dim_list[j])]
other_chain_det_list = [ other_jordan[j][1].Gram_det() * (scale_list[j]**dim_list[j])]
self_hasse_chain_list = [ self_jordan[j][1].scale_by_factor(ZZ(2)**self_jordan[j][0]).hasse_invariant__OMeara(2) ]
other_hasse_chain_list = [ other_jordan[j][1].scale_by_factor(ZZ(2)**other_jordan[j][0]).hasse_invariant__OMeara(2) ]
for j in range(1, t):
self_chain_det_list.append(self_chain_det_list[j-1] * self_jordan[j][1].Gram_det() * (scale_list[j]**dim_list[j]))
other_chain_det_list.append(other_chain_det_list[j-1] * other_jordan[j][1].Gram_det() * (scale_list[j]**dim_list[j]))
self_hasse_chain_list.append(self_hasse_chain_list[j-1] \
* hilbert_symbol(self_chain_det_list[j-1], self_jordan[j][1].Gram_det(), 2) \
* self_jordan[j][1].hasse_invariant__OMeara(2))
other_hasse_chain_list.append(other_hasse_chain_list[j-1] \
* hilbert_symbol(other_chain_det_list[j-1], other_jordan[j][1].Gram_det(), 2) \
* other_jordan[j][1].hasse_invariant__OMeara(2))
## SANITY CHECK -- check that the scale powers are strictly increasing
for i in range(1, len(scale_list)):
if scale_list[i-1] >= scale_list[i]:
raise RuntimeError, "Oops! There is something wrong with the Jordan Decomposition -- the given scales are not strictly increasing!"
## DIAGNOSTIC
#print "scale_list = ", scale_list
#print "norm_list = ", norm_list
#print "dim_list = ", dim_list
#print
#print "self_chain_det_list = ", self_chain_det_list
#print "other_chain_det_list = ", other_chain_det_list
#print "self_hasse_chain_list = ", self_hasse_chain_list
#print "other_hasse_chain_det_list = ", other_hasse_chain_list
## Test O'Meara's two conditions
for i in range(t-1):
## Condition (i): Check that their (unit) ratio is a square (but it suffices to check at most mod 8).
modulus = norm_list[i] * norm_list[i+1] / (scale_list[i] ** 2)
if modulus > 8:
modulus = 8
if (modulus > 1) and (((self_chain_det_list[i] / other_chain_det_list[i]) % modulus) != 1):
#print "Failed when i =", i, " in condition 1."
return False
## Check O'Meara's condition (ii) when appropriate
if norm_list[i+1] % (4 * norm_list[i]) == 0:
if self_hasse_chain_list[i] * hilbert_symbol(norm_list[i] * other_chain_det_list[i], -self_chain_det_list[i], 2) \
!= other_hasse_chain_list[i] * hilbert_symbol(norm_list[i], -other_chain_det_list[i], 2): ## Nipp conditions
#print "Failed when i =", i, " in condition 2."
return False
## All tests passed for the prime 2.
return True
else:
raise TypeError, "Oops! This should not have happened."
def CohenOesterle(eps, k):
r"""
Compute the Cohen-Oesterle function associate to eps, `k`.
This is a summand in the formula for the dimension of the space of
cusp forms of weight `2` with character
`\varepsilon`.
INPUT:
- ``eps`` - Dirichlet character
- ``k`` - integer
OUTPUT: element of the base ring of eps.
EXAMPLES::
sage: G.<eps> = DirichletGroup(7)
sage: sage.modular.dims.CohenOesterle(eps, 2)
-2/3
sage: sage.modular.dims.CohenOesterle(eps, 4)
-1
"""
N = eps.modulus()
facN = factor(N)
f = eps.conductor()
gamma_k = 0
if k%4==2:
gamma_k = frac(-1,4)
elif k%4==0:
gamma_k = frac(1,4)
mu_k = 0
if k%3==2:
mu_k = frac(-1,3)
elif k%3==0:
mu_k = frac(1,3)
def _lambda(r,s,p):
"""
Used internally by the CohenOesterle function.
INPUT:
- ``r, s, p`` - integers
OUTPUT: Integer
EXAMPLES: (indirect doctest)
::
sage: K = CyclotomicField(3)
sage: eps = DirichletGroup(7*43,K).0^2
sage: sage.modular.dims.CohenOesterle(eps,2)
-4/3
"""
if 2*s<=r:
if r%2==0:
return p**(r//2) + p**((r//2)-1)
return 2*p**((r-1)//2)
return 2*(p**(r-s))
#end def of lambda
K = eps.base_ring()
return K(frac(-1,2) * mul([_lambda(r,valuation(f,p),p) for p, r in facN]) + \
gamma_k * mul([CO_delta(r,p,N,eps) for p, r in facN]) + \
mu_k * mul([CO_nu(r,p,N,eps) for p, r in facN]))
def local_normal_form(self, p):
"""
Returns the a locally integrally equivalent quadratic form over
the p-adic integers Z_p which gives the Jordan decomposition. The
Jordan components are written as sums of blocks of size <= 2 and
are arranged by increasing scale, and then by increasing norm.
(This is equivalent to saying that we put the 1x1 blocks before
the 2x2 blocks in each Jordan component.)
INPUT:
`p` -- a positive prime number.
OUTPUT:
a quadratic form over ZZ
WARNING: Currently this only works for quadratic forms defined over ZZ.
EXAMPLES::
sage: Q = QuadraticForm(ZZ, 2, [10,4,1])
sage: Q.local_normal_form(5)
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 1 0 ]
[ * 6 ]
::
sage: Q.local_normal_form(3)
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 10 0 ]
[ * 15 ]
sage: Q.local_normal_form(2)
Quadratic form in 2 variables over Integer Ring with coefficients:
[ 1 0 ]
[ * 6 ]
"""
## Sanity Checks
if (self.base_ring() != IntegerRing()):
raise NotImplementedError, "Oops! This currently only works for quadratic forms defined over IntegerRing(). =("
if not ((p>=2) and is_prime(p)):
raise TypeError, "Oops! p is not a positive prime number. =("
## Some useful local variables
Q = copy.deepcopy(self)
Q.__init__(self.base_ring(), self.dim(), self.coefficients())
## Prepare the final form to return
Q_Jordan = copy.deepcopy(self)
Q_Jordan.__init__(self.base_ring(), 0)
while Q.dim() > 0:
n = Q.dim()
## Step 1: Find the minimally p-divisible matrix entry, preferring diagonals
## -------------------------------------------------------------------------
(min_i, min_j) = Q.find_entry_with_minimal_scale_at_prime(p)
if min_i == min_j:
min_val = valuation(2 * Q[min_i, min_j], p)
else:
min_val = valuation(Q[min_i, min_j], p)
## Error if we still haven't seen non-zero coefficients!
if (min_val == Infinity):
raise RuntimeError, "Oops! The original matrix is degenerate. =("
## Step 2: Arrange for the upper leftmost entry to have minimal valuation
## ----------------------------------------------------------------------
if (min_i == min_j):
block_size = 1
Q.swap_variables(0, min_i, in_place = True)
else:
## Work in the upper-left 2x2 block, and replace it by its 2-adic equivalent form
Q.swap_variables(0, min_i, in_place = True)
Q.swap_variables(1, min_j, in_place = True)
## 1x1 => make upper left the smallest
if (p != 2):
block_size = 1;
Q.add_symmetric(1, 0, 1, in_place = True)
## 2x2 => replace it with the appropriate 2x2 matrix
else:
block_size = 2
## DIAGNOSTIC
#print "\n Finished Step 2 \n";
#print "\n Q is: \n" + str(Q) + "\n";
#print " p is: " + str(p)
#print " min_val is: " + str( min_val)
#print " block_size is: " + str(block_size)
#print "\n Starting Step 3 \n"
## Step 3: Clear out the remaining entries
## ---------------------------------------
min_scale = p ** min_val ## This is the minimal valuation of the Hessian matrix entries.
##DIAGNOSTIC
#print "Starting Step 3:"
#print "----------------"
#print " min_scale is: " + str(min_scale)
## Perform cancellation over Z by ensuring divisibility
if (block_size == 1):
a = 2 * Q[0,0]
for j in range(block_size, n):
b = Q[0, j]
g = GCD(a, b)
## DIAGNSOTIC
#print "Cancelling from a 1x1 block:"
#print "----------------------------"
#print " Cancelling entry with index (" + str(upper_left) + ", " + str(j) + ")"
#print " entry = " + str(b)
#print " gcd = " + str(g)
#print " a = " + str(a)
#print " b = " + str(b)
#print " a/g = " + str(a/g) + " (used for stretching)"
#print " -b/g = " + str(-b/g) + " (used for cancelling)"
## Sanity Check: a/g is a p-unit
if valuation (g, p) != valuation(a, p):
raise RuntimeError, "Oops! We have a problem with our rescaling not preserving p-integrality!"
Q.multiply_variable(ZZ(a/g), j, in_place = True) ## Ensures that the new b entry is divisible by a
Q.add_symmetric(ZZ(-b/g), j, 0, in_place = True) ## Performs the cancellation
elif (block_size == 2):
a1 = 2 * Q[0,0]
a2 = Q[0, 1]
b1 = Q[1, 0] ## This is the same as a2
b2 = 2 * Q[1, 1]
big_det = (a1*b2 - a2*b1)
small_det = big_det / (min_scale * min_scale)
## Cancels out the rows/columns of the 2x2 block
for j in range(block_size, n):
a = Q[0, j]
b = Q[1, j]
## Ensures an integral result (scale jth row/column by big_det)
Q.multiply_variable(big_det, j, in_place = True)
## Performs the cancellation (by producing -big_det * jth row/column)
Q.add_symmetric(ZZ(-(a*b2 - b*a2)), j, 0, in_place = True)
Q.add_symmetric(ZZ(-(-a*b1 + b*a1)), j, 1, in_place = True)
## Now remove the extra factor (non p-unit factor) in big_det we introduced above
Q.divide_variable(ZZ(min_scale * min_scale), j, in_place = True)
## DIAGNOSTIC
#print "Cancelling out a 2x2 block:"
#print "---------------------------"
#print " a1 = " + str(a1)
#print " a2 = " + str(a2)
#print " b1 = " + str(b1)
#print " b2 = " + str(b2)
#print " big_det = " + str(big_det)
#print " min_scale = " + str(min_scale)
#print " small_det = " + str(small_det)
#print " Q = \n", Q
## Uses Cassels's proof to replace the remaining 2 x 2 block
if (((1 + small_det) % 8) == 0):
Q[0, 0] = 0
Q[1, 1] = 0
Q[0, 1] = min_scale
elif (((5 + small_det) % 8) == 0):
Q[0, 0] = min_scale
Q[1, 1] = min_scale
Q[0, 1] = min_scale
else:
raise RuntimeError, "Error in LocalNormal: Impossible behavior for a 2x2 block! \n"
## Check that the cancellation worked, extract the upper-left block, and trim Q to handle the next block.
for i in range(block_size):
for j in range(block_size, n):
if Q[i,j] != 0:
raise RuntimeError, "Oops! The cancellation didn't work properly at entry (" + str(i) + ", " + str(j) + ")."
Q_Jordan = Q_Jordan + Q.extract_variables(range(block_size))
Q = Q.extract_variables(range(block_size, n))
return Q_Jordan
def jordan_blocks_by_scale_and_unimodular(self, p, safe_flag=True):
"""
Returns a list of pairs `(s_i, L_i)` where `L_i` is a maximal
`p^{s_i}`-unimodular Jordan component which is further decomposed into
block diagonals of block size `\le 2`. For each `L_i` the 2x2 blocks are
listed after the 1x1 blocks (which follows from the convention of the
:meth:`local_normal_form` method).
..note ::
The decomposition of each `L_i` into smaller block is not unique!
The ``safe_flag`` argument allows us to select whether we want a copy of
the output, or the original output. By default ``safe_flag = True``, so we
return a copy of the cached information. If this is set to ``False``, then
the routine is much faster but the return values are vulnerable to being
corrupted by the user.
INPUT:
- `p` -- a prime number > 0.
OUTPUT:
A list of pairs `(s_i, L_i)` where:
- `s_i` is an integer,
- `L_i` is a block-diagonal unimodular quadratic form over `\ZZ_p`.
.. note::
These forms `L_i` are defined over the `p`-adic integers, but by a
matrix over `\ZZ` (or `\QQ`?).
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,9,5,7])
sage: Q.jordan_blocks_by_scale_and_unimodular(3)
[(0, Quadratic form in 3 variables over Integer Ring with coefficients:
[ 1 0 0 ]
[ * 5 0 ]
[ * * 7 ]), (2, Quadratic form in 1 variables over Integer Ring with coefficients:
[ 1 ])]
::
sage: Q2 = QuadraticForm(ZZ, 2, [1,1,1])
sage: Q2.jordan_blocks_by_scale_and_unimodular(2)
[(-1, Quadratic form in 2 variables over Integer Ring with coefficients:
[ 2 2 ]
[ * 2 ])]
sage: Q = Q2 + Q2.scale_by_factor(2)
sage: Q.jordan_blocks_by_scale_and_unimodular(2)
[(-1, Quadratic form in 2 variables over Integer Ring with coefficients:
[ 2 2 ]
[ * 2 ]), (0, Quadratic form in 2 variables over Integer Ring with coefficients:
[ 2 2 ]
[ * 2 ])]
"""
## Try to use the cached result
try:
if safe_flag:
return copy.deepcopy(
self.__jordan_blocks_by_scale_and_unimodular_dict[p])
else:
return self.__jordan_blocks_by_scale_and_unimodular_dict[p]
except Exception:
## Initialize the global dictionary if it doesn't exist
if not hasattr(self, '__jordan_blocks_by_scale_and_unimodular_dict'):
self.__jordan_blocks_by_scale_and_unimodular_dict = {}
## Deal with zero dim'l forms
if self.dim() == 0:
return []
## Find the Local Normal form of Q at p
Q1 = self.local_normal_form(p)
## Parse this into Jordan Blocks
n = Q1.dim()
tmp_Jordan_list = []
i = 0
start_ind = 0
if (n >= 2) and (Q1[0, 1] != 0):
start_scale = valuation(Q1[0, 1], p) - 1
else:
start_scale = valuation(Q1[0, 0], p)
while (i < n):
## Determine the size of the current block
if (i == n - 1) or (Q1[i, i + 1] == 0):
block_size = 1
else:
block_size = 2
## Determine the valuation of the current block
if block_size == 1:
block_scale = valuation(Q1[i, i], p)
else:
block_scale = valuation(Q1[i, i + 1], p) - 1
## Process the previous block if the valuation increased
if block_scale > start_scale:
tmp_Jordan_list += [
(start_scale, Q1.extract_variables(
range(start_ind,
i)).scale_by_factor(ZZ(1) / (QQ(p)**(start_scale))))
]
start_ind = i
start_scale = block_scale
## Increment the index
i += block_size
## Add the last block
tmp_Jordan_list += [(start_scale, Q1.extract_variables(range(
start_ind, n)).scale_by_factor(ZZ(1) / QQ(p)**(start_scale)))]
## Cache the result
self.__jordan_blocks_by_scale_and_unimodular_dict[p] = tmp_Jordan_list
## Return the result
return tmp_Jordan_list