def _find_cusps(self):
r"""
Calculate the reduced representatives of the equivalence classes of
cusps for this group. Adapted from code by Ron Evans.
EXAMPLE::
sage: Gamma(8).cusps() # indirect doctest
[0, 1/4, 1/3, 3/8, 1/2, 2/3, 3/4, 1, 4/3, 3/2, 5/3, 2, 7/3, 5/2, 8/3, 3, 7/2, 11/3, 4, 14/3, 5, 6, 7, Infinity]
"""
n = self.level()
C = [QQ(x) for x in xrange(n)]
n0 = n // 2
n1 = (n + 1) // 2
for r in xrange(1, n1):
if r > 1 and gcd(r, n) == 1:
C.append(ZZ(r) / ZZ(n))
if n0 == n / 2 and gcd(r, n0) == 1:
C.append(ZZ(r) / ZZ(n0))
for s in xrange(2, n1):
for r in xrange(1, 1 + n):
if GCD_list([s, r, n]) == 1:
# GCD_list is ~40x faster than gcd, since gcd wastes loads
# of time initialising a Sequence type.
u, v = _lift_pair(r, s, n)
C.append(ZZ(u) / ZZ(v))
return [Cusp(x) for x in sorted(C)] + [Cusp(1, 0)]
def _find_cusps(self):
r"""
Calculate the reduced representatives of the equivalence classes of
cusps for this group. Adapted from code by Ron Evans.
EXAMPLE::
sage: Gamma(8).cusps() # indirect doctest
[0, 1/4, 1/3, 3/8, 1/2, 2/3, 3/4, 1, 4/3, 3/2, 5/3, 2, 7/3, 5/2, 8/3, 3, 7/2, 11/3, 4, 14/3, 5, 6, 7, Infinity]
"""
n = self.level()
C=[QQ(x) for x in xrange(n)]
n0=n//2
n1=(n+1)//2
for r in xrange(1, n1):
if r > 1 and gcd(r,n)==1:
C.append(ZZ(r)/ZZ(n))
if n0==n/2 and gcd(r,n0)==1:
C.append(ZZ(r)/ZZ(n0))
for s in xrange(2,n1):
for r in xrange(1, 1+n):
if GCD_list([s,r,n])==1:
# GCD_list is ~40x faster than gcd, since gcd wastes loads
# of time initialising a Sequence type.
u,v = _lift_pair(r,s,n)
C.append(ZZ(u)/ZZ(v))
return [Cusp(x) for x in sorted(C)] + [Cusp(1,0)]
def _lift_pair(U,V,N):
r"""
Utility function. Given integers ``U, V, N``, with `N \ge 1` and `{\rm
gcd}(U, V, N) = 1`, return a pair `(u, v)` congruent to `(U, V) \bmod N`,
such that `{\rm gcd}(u,v) = 1`, `u, v \ge 0`, `v` is as small as possible,
and `u` is as small as possible for that `v`.
*Warning*: As this function is for internal use, it does not do a
preliminary sanity check on its input, for efficiency. It will recover
reasonably gracefully if ``(U, V, N)`` are not coprime, but only after
wasting quite a lot of cycles!
EXAMPLE::
sage: from sage.modular.arithgroup.congroup_gamma import _lift_pair
sage: _lift_pair(2,4,7)
(9, 4)
sage: _lift_pair(2,4,8) # don't do this
Traceback (most recent call last):
...
ValueError: (U, V, N) must be coprime
"""
u = U % N
v = V % N
if v == 0:
if u == 1:
return (1,0)
else:
v = N
while gcd(u, v) > 1:
u = u+N
if u > N*v: raise ValueError, "(U, V, N) must be coprime"
return (u, v)
def _lift_pair(U, V, N):
r"""
Utility function. Given integers ``U, V, N``, with `N \ge 1` and `{\rm
gcd}(U, V, N) = 1`, return a pair `(u, v)` congruent to `(U, V) \bmod N`,
such that `{\rm gcd}(u,v) = 1`, `u, v \ge 0`, `v` is as small as possible,
and `u` is as small as possible for that `v`.
*Warning*: As this function is for internal use, it does not do a
preliminary sanity check on its input, for efficiency. It will recover
reasonably gracefully if ``(U, V, N)`` are not coprime, but only after
wasting quite a lot of cycles!
EXAMPLE::
sage: from sage.modular.arithgroup.congroup_gamma import _lift_pair
sage: _lift_pair(2,4,7)
(9, 4)
sage: _lift_pair(2,4,8) # don't do this
Traceback (most recent call last):
...
ValueError: (U, V, N) must be coprime
"""
u = U % N
v = V % N
if v == 0:
if u == 1:
return (1, 0)
else:
v = N
while gcd(u, v) > 1:
u = u + N
if u > N * v: raise ValueError, "(U, V, N) must be coprime"
return (u, v)
def cardinality(self):
"""
Returns the number of Lyndon words with the evaluation e.
EXAMPLES::
sage: LyndonWords([]).cardinality()
0
sage: LyndonWords([2,2]).cardinality()
1
sage: LyndonWords([2,3,2]).cardinality()
30
Check to make sure that the count matches up with the number of
Lyndon words generated.
::
sage: comps = [[],[2,2],[3,2,7],[4,2]]+Compositions(4).list()
sage: lws = [ LyndonWords(comp) for comp in comps]
sage: all( [ lw.cardinality() == len(lw.list()) for lw in lws] )
True
"""
evaluation = self.e
le = __builtin__.list(evaluation)
if len(evaluation) == 0:
return 0
n = sum(evaluation)
return sum([
moebius(j) * factorial(n / j) /
prod([factorial(ni / j) for ni in evaluation])
for j in divisors(gcd(le))
]) / n
def primitive(self, signed=True):
"""
Return hyperplane defined by primitive equation.
INPUT:
- ``signed`` -- boolean (optional, default: ``True``); whether
to preserve the overall sign
OUTPUT:
Hyperplane whose linear expression has common factors and
denominators cleared. That is, the same hyperplane (with the
same sign) but defined by a rescaled equation. Note that
different linear expressions must define different hyperplanes
as comparison is used in caching.
If ``signed``, the overall rescaling is by a positive constant
only.
EXAMPLES::
sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: h = -1/3*x + 1/2*y - 1; h
Hyperplane -1/3*x + 1/2*y - 1
sage: h.primitive()
Hyperplane -2*x + 3*y - 6
sage: h == h.primitive()
False
sage: (4*x + 8).primitive()
Hyperplane x + 0*y + 2
sage: (4*x - y - 8).primitive(signed=True) # default
Hyperplane 4*x - y - 8
sage: (4*x - y - 8).primitive(signed=False)
Hyperplane -4*x + y + 8
"""
from sage.rings.all import lcm, gcd
coeffs = self.coefficients()
try:
d = lcm([x.denom() for x in coeffs])
n = gcd([x.numer() for x in coeffs])
except AttributeError:
return self
if not signed:
for x in coeffs:
if x > 0:
break
if x < 0:
d = -d
break
parent = self.parent()
d = parent.base_ring()(d)
n = parent.base_ring()(n)
if n == 0:
n = parent.base_ring().one()
return parent(self * d / n)
def primitive(self, signed=True):
"""
Return hyperplane defined by primitive equation.
INPUT:
- ``signed`` -- boolean (optional, default: ``True``); whether
to preserve the overall sign
OUTPUT:
Hyperplane whose linear expression has common factors and
denominators cleared. That is, the same hyperplane (with the
same sign) but defined by a rescaled equation. Note that
different linear expressions must define different hyperplanes
as comparison is used in caching.
If ``signed``, the overall rescaling is by a positive constant
only.
EXAMPLES::
sage: H.<x,y> = HyperplaneArrangements(QQ)
sage: h = -1/3*x + 1/2*y - 1; h
Hyperplane -1/3*x + 1/2*y - 1
sage: h.primitive()
Hyperplane -2*x + 3*y - 6
sage: h == h.primitive()
False
sage: (4*x + 8).primitive()
Hyperplane x + 0*y + 2
sage: (4*x - y - 8).primitive(signed=True) # default
Hyperplane 4*x - y - 8
sage: (4*x - y - 8).primitive(signed=False)
Hyperplane -4*x + y + 8
"""
from sage.rings.all import lcm, gcd
coeffs = self.coefficients()
try:
d = lcm([x.denom() for x in coeffs])
n = gcd([x.numer() for x in coeffs])
except AttributeError:
return self
if not signed:
for x in coeffs:
if x > 0:
break
if x < 0:
d = -d
break
parent = self.parent()
d = parent.base_ring()(d)
n = parent.base_ring()(n)
if n == 0:
n = parent.base_ring().one()
return parent(self * d / n)
def _normalize_coefficients(self, c):
"""
If our coefficient ring is the field of fractions over a univariate
polynomial ring over the rationals, then we should clear both the
numerator and denominator of the denominators of their
coefficients.
EXAMPLES::
sage: P = JackPolynomialsP(QQ)
sage: t = P.base_ring().gen()
sage: a = 2/(1/2*t+1/2)
sage: P._normalize_coefficients(a)
4/(t + 1)
sage: a = 1/(1/3+1/6*t)
sage: P._normalize_coefficients(a)
6/(t + 2)
sage: a = 24/(4*t^2 + 12*t + 8)
sage: P._normalize_coefficients(a)
6/(t^2 + 3*t + 2)
"""
BR = self.base_ring()
if is_FractionField(BR) and BR.base_ring() == QQ:
denom = c.denominator()
numer = c.numerator()
#Clear the denominators
a = lcm([i.denominator() for i in denom.coeffs()])
b = lcm([i.denominator() for i in numer.coeffs()])
l = Integer(a).lcm(Integer(b))
denom *= l
numer *= l
#Divide through by the gcd of the numerators
a = gcd([i.numerator() for i in denom.coeffs()])
b = gcd([i.numerator() for i in numer.coeffs()])
l = Integer(a).gcd(Integer(b))
denom = denom / l
numer = numer / l
return c.parent()(numer, denom)
else:
return c
def _normalize_coefficients(self, c):
"""
If our coefficient ring is the field of fractions over a univariate
polynomial ring over the rationals, then we should clear both the
numerator and denominator of the denominators of their
coefficients.
EXAMPLES::
sage: P = JackPolynomialsP(QQ)
sage: t = P.base_ring().gen()
sage: a = 2/(1/2*t+1/2)
sage: P._normalize_coefficients(a)
4/(t + 1)
sage: a = 1/(1/3+1/6*t)
sage: P._normalize_coefficients(a)
6/(t + 2)
sage: a = 24/(4*t^2 + 12*t + 8)
sage: P._normalize_coefficients(a)
6/(t^2 + 3*t + 2)
"""
BR = self.base_ring()
if is_FractionField(BR) and BR.base_ring() == QQ:
denom = c.denominator()
numer = c.numerator()
#Clear the denominators
a = lcm([i.denominator() for i in denom.coeffs()])
b = lcm([i.denominator() for i in numer.coeffs()])
l = Integer(a).lcm(Integer(b))
denom *= l
numer *= l
#Divide through by the gcd of the numerators
a = gcd([i.numerator() for i in denom.coeffs()])
b = gcd([i.numerator() for i in numer.coeffs()])
l = Integer(a).gcd(Integer(b))
denom = denom / l
numer = numer / l
return c.parent()(numer, denom)
else:
return c
def cardinality(self):
"""
Returns the number of Lyndon words with the evaluation e.
EXAMPLES::
sage: LyndonWords([]).cardinality()
0
sage: LyndonWords([2,2]).cardinality()
1
sage: LyndonWords([2,3,2]).cardinality()
30
Check to make sure that the count matches up with the number of
Lyndon words generated.
::
sage: comps = [[],[2,2],[3,2,7],[4,2]]+Compositions(4).list()
sage: lws = [ LyndonWords(comp) for comp in comps]
sage: all( [ lw.cardinality() == len(lw.list()) for lw in lws] )
True
"""
evaluation = self.e
le = __builtin__.list(evaluation)
if len(evaluation) == 0:
return 0
n = sum(evaluation)
return (
sum(
[
moebius(j) * factorial(n / j) / prod([factorial(ni / j) for ni in evaluation])
for j in divisors(gcd(le))
]
)
/ n
)
def WP(self, *q, **kw):
# Specific keyword arguments instead of **kw would be preferable,
# later versions of Python might support specific (optional) keyword
# arguments after *q.
r"""
Construct weighted projective `n`-space over a field.
INPUT:
- ``q`` -- a sequence of positive integers relatively prime to
one another. The weights ``q`` can be given either as a list
or tuple, or as positional arguments.
Two keyword arguments:
- ``K`` -- a field (default: `\QQ`).
- ``names`` -- string or list (tuple) of strings (default 'z+'). See
:func:`~sage.schemes.toric.variety.normalize_names` for
acceptable formats.
OUTPUT:
- A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
If `q=(q_0,\dots,q_n)`, then the output is the weighted projective
space `\mathbb{P}(q_0,\dots,q_n)` over `K`. ``names`` are the names
of the generators of the homogeneous coordinate ring.
EXAMPLES:
A hyperelliptic curve `C` of genus 2 as a subscheme of the weighted
projective plane `\mathbb{P}(1,3,1)`::
sage: X = toric_varieties.WP([1,3,1], names='x y z')
sage: X.inject_variables()
Defining x, y, z
sage: g = y^2-(x^6-z^6)
sage: C = X.subscheme([g]); C
Closed subscheme of 2-d toric variety covered by 3 affine patches defined by:
-x^6 + z^6 + y^2
"""
if len(q) == 1:
# tuples and lists of weights are acceptable input
if isinstance(q[0], (list, tuple)):
q = q[0]
q = list(q)
m = len(q)
# allow case q=[1]? (not allowed presently)
if m < 2:
raise ValueError("more than one weight must be provided (got %s)" % q)
for i in range(m):
try:
q[i] = ZZ(q[i])
except (TypeError):
raise TypeError("the weights (=%s) must be integers" % q)
if q[i] <= 0:
raise ValueError("the weights (=%s) must be positive integers" % q)
if not gcd(q) == 1:
raise ValueError("the weights (=%s) must be relatively prime" % q)
# set default values for K and names
K = QQ
names = "z+"
for key in kw:
if key == "K":
K = kw["K"]
if K not in _Fields:
raise TypeError("K (=%r) must be a field" % K)
elif key == "names":
names = kw["names"]
names = normalize_names(names, m, DEFAULT_PREFIX)
else:
raise TypeError("got an unexpected keyword argument %r" % key)
L = ToricLattice(m)
L_sub = L.submodule([L(q)])
Q = L / L_sub
rays = []
cones = []
w = range(m)
L_basis = L.basis()
for i in w:
b = L_basis[i]
v = Q.coordinate_vector(Q(b))
rays = rays + [v]
w_c = w[:i] + w[i + 1 :]
cones = cones + [tuple(w_c)]
fan = Fan(cones, rays)
return ToricVariety(fan, coordinate_names=names, base_field=K)
def _compute_lattice(self, rational_only=False, rational_subgroup=False):
r"""
Return a list of vectors that define elements of the rational
homology that generate this finite subgroup.
INPUT:
- ``rational_only`` - bool (default: False); if
``True``, only use rational cusps.
OUTPUT:
- ``list`` - list of vectors
EXAMPLES::
sage: J = J0(37)
sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
sage: C._compute_lattice()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[ 1 0 0 0]
[ 0 1 0 0]
[ 0 0 1 0]
[ 0 0 0 1/3]
sage: J = J0(43)
sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
sage: C._compute_lattice()
Free module of degree 6 and rank 6 over Integer Ring
Echelon basis matrix:
[ 1 0 0 0 0 0]
[ 0 1/7 0 6/7 0 5/7]
[ 0 0 1 0 0 0]
[ 0 0 0 1 0 0]
[ 0 0 0 0 1 0]
[ 0 0 0 0 0 1]
sage: J = J0(22)
sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
sage: C._compute_lattice()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[1/5 1/5 4/5 0]
[ 0 1 0 0]
[ 0 0 1 0]
[ 0 0 0 1/5]
sage: J = J1(13)
sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
sage: C._compute_lattice()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[ 1/19 0 0 9/19]
[ 0 1/19 1/19 18/19]
[ 0 0 1 0]
[ 0 0 0 1]
We compute with and without the optional
``rational_only`` option.
::
sage: J = J0(27); G = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
sage: G._compute_lattice()
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1/3 0]
[ 0 1/3]
sage: G._compute_lattice(rational_only=True)
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1/3 0]
[ 0 1]
"""
A = self.abelian_variety()
Cusp = A.modular_symbols()
Amb = Cusp.ambient_module()
Eis = Amb.eisenstein_submodule()
C = Amb.cusps()
N = Amb.level()
if rational_subgroup:
# QQ-rational subgroup of cuspidal subgroup
assert A.is_ambient()
Q = Cusp.abvarquo_rational_cuspidal_subgroup()
return Q.V()
if rational_only:
# subgroup generated by differences of rational cusps
if not is_Gamma0(A.group()):
raise NotImplementedError, 'computation of rational cusps only implemented in Gamma0 case.'
if not N.is_squarefree():
data = [n for n in range(2,N) if gcd(n,N) == 1]
C = [c for c in C if is_rational_cusp_gamma0(c, N, data)]
v = [Amb([infinity, alpha]).element() for alpha in C]
cusp_matrix = matrix(QQ, len(v), Amb.dimension(), v)
# TODO -- refactor something out here
# Now we project onto the cuspidal part.
B = Cusp.free_module().basis_matrix().stack(Eis.free_module().basis_matrix())
X = B.solve_left(cusp_matrix)
X = X.matrix_from_columns(range(Cusp.dimension()))
lattice = X.row_module(ZZ) + A.lattice()
return lattice
def multiple_of_order(self, maxp=None):
"""
Return a multiple of the order of this torsion group.
The multiple is computed using characteristic polynomials of Hecke
operators of odd index not dividing the level.
INPUT:
- ``maxp`` - (default: None) If maxp is None (the
default), return gcd of best bound computed so far with bound
obtained by computing GCD's of orders modulo p until this gcd
stabilizes for 3 successive primes. If maxp is given, just use all
primes up to and including maxp.
EXAMPLES::
sage: J = J0(11)
sage: G = J.rational_torsion_subgroup()
sage: G.multiple_of_order(11)
5
sage: J = J0(389)
sage: G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(389) of dimension 32
sage: G.multiple_of_order()
97
sage: [G.multiple_of_order(p) for p in prime_range(3,11)]
[92645296242160800, 7275, 291]
sage: [G.multiple_of_order(p) for p in prime_range(3,13)]
[92645296242160800, 7275, 291, 97]
sage: [G.multiple_of_order(p) for p in prime_range(3,19)]
[92645296242160800, 7275, 291, 97, 97, 97]
::
sage: J = J0(33) * J0(11) ; J.rational_torsion_subgroup().order()
Traceback (most recent call last):
...
NotImplementedError: torsion multiple only implemented for Gamma0
The next example illustrates calling this function with a larger
input and how the result may be cached when maxp is None::
sage: T = J0(43)[1].rational_torsion_subgroup()
sage: T.multiple_of_order()
14
sage: T.multiple_of_order(50)
7
sage: T.multiple_of_order()
7
"""
if maxp is None:
try:
return self.__multiple_of_order
except AttributeError:
pass
bnd = ZZ(0)
A = self.abelian_variety()
if A.dimension() == 0:
T = ZZ(1)
self.__multiple_of_order = T
return T
N = A.level()
if not (len(A.groups()) == 1 and is_Gamma0(A.groups()[0])):
# to generalize to this case, you'll need to
# (1) define a charpoly_of_frob function:
# this is tricky because I don't know a simple
# way to do this for Gamma1 and GammaH. Will
# probably have to compute explicit matrix for
# <p> operator (add to modular symbols code),
# then compute some charpoly involving
# that directly...
# (2) use (1) -- see my MAGMA code.
raise NotImplementedError("torsion multiple only implemented for Gamma0")
cnt = 0
if maxp is None:
X = Primes()
else:
X = prime_range(maxp+1)
for p in X:
if (2*N) % p == 0:
continue
f = A.hecke_polynomial(p)
b = ZZ(f(p+1))
if bnd == 0:
bnd = b
else:
bnd_last = bnd
bnd = ZZ(gcd(bnd, b))
if bnd == bnd_last:
cnt += 1
else:
cnt = 0
if maxp is None and cnt >= 2:
break
# The code below caches the computed bound and
# will be used if this function is called
# again with maxp equal to None (the default).
if maxp is None:
# maxp is None but self.__multiple_of_order is
# not set, since otherwise we would have immediately
# returned at the top of this function
self.__multiple_of_order = bnd
else:
# maxp is given -- record new info we get as
# a gcd...
try:
self.__multiple_of_order = gcd(self.__multiple_of_order, bnd)
except AttributeError:
# ... except in the case when self.__multiple_of_order
# was never set. In this case, we just set
# it as long as the gcd stabilized for 3 in a row.
if cnt >= 2:
self.__multiple_of_order = bnd
return bnd
def saturation(A, proof=True, p=0, max_dets=5):
"""
Compute a saturation matrix of A.
INPUT:
- A -- a matrix over ZZ
- proof -- bool (default: True)
- p -- int (default: 0); if not 0 only guarantees that output is
p-saturated
- max_dets -- int (default: 4) max number of dets of submatrices to
compute.
OUTPUT:
matrix -- saturation of the matrix A.
EXAMPLES::
sage: from sage.matrix.matrix_integer_dense_saturation import saturation
sage: A = matrix(ZZ, 2, 2, [3,2,3,4]); B = matrix(ZZ, 2,3,[1,2,3,4,5,6]); C = A*B
sage: C
[11 16 21]
[19 26 33]
sage: C.index_in_saturation()
18
sage: S = saturation(C); S
[11 16 21]
[-2 -3 -4]
sage: S.index_in_saturation()
1
sage: saturation(C, proof=False)
[11 16 21]
[-2 -3 -4]
sage: saturation(C, p=2)
[11 16 21]
[-2 -3 -4]
sage: saturation(C, p=2, max_dets=1)
[11 16 21]
[-2 -3 -4]
"""
# Find a submatrix of full rank and instead saturate that matrix.
r = A.rank()
if A.is_square() and r == A.nrows():
return identity_matrix(ZZ, r)
if A.nrows() > r:
P = []
while len(P) < r:
P = matrix_integer_dense_hnf.probable_pivot_rows(A)
A = A.matrix_from_rows(P)
# Factor out all common factors from all rows, just in case.
A = copy(A)
A._factor_out_common_factors_from_each_row()
if A.nrows() <= 1:
return A
A, zero_cols = A._delete_zero_columns()
if max_dets > 0:
# Take the GCD of at most num_dets randomly chosen determinants.
nr = A.nrows()
nc = A.ncols()
d = 0
trials = min(binomial(nc, nr), max_dets)
already_tried = []
while len(already_tried) < trials:
v = random_sublist_of_size(nc, nr)
tm = verbose("saturation -- checking det condition on submatrix")
d = gcd(d, A.matrix_from_columns(v).determinant(proof=proof))
verbose("saturation -- got det down to %s" % d, tm)
if gcd(d, p) == 1:
return A._insert_zero_columns(zero_cols)
already_tried.append(v)
if gcd(d, p) == 1:
# already p-saturated
return A._insert_zero_columns(zero_cols)
# Factor and p-saturate at each p.
# This is not a good algorithm, because all the HNF's in it are really slow!
#
# tm = verbose('factoring gcd %s of determinants'%d)
# limit = 2**31-1
# F = d.factor(limit = limit)
# D = [p for p, e in F if p <= limit]
# B = [n for n, e in F if n > limit] # all big factors -- there will only be at most one
# assert len(B) <= 1
# C = B[0]
# for p in D:
# A = p_saturation(A, p=p, proof=proof)
# This is a really simple but powerful algorithm.
# FACT: If A is a matrix of full rank, then hnf(transpose(A))^(-1)*A is a saturation of A.
# To make this practical we use solve_system_with_difficult_last_row, since the
# last column of HNF's are typically the only really big ones.
B = A.transpose().hermite_form(include_zero_rows=False, proof=proof)
B = B.transpose()
# Now compute B^(-1) * A
C = solve_system_with_difficult_last_row(B, A)
return C.change_ring(ZZ)._insert_zero_columns(zero_cols)
def charpolys(v, B, filename=None):
"""
Compute characteristic polynomials of T_P for primes P with norm
<= B coprime to the level, for all spaces of Hilbert modular forms
for all the levels in v.
INPUT:
- `v` -- list of positive integers
- `B` -- positive integer
- ``filename`` -- optional string; if given, output is also written
to that file (in addition to stdout).
OUTPUT:
- outputs a table with rows corresponding to the ideals
of Q(sqrt(5)) with norm in v, and optionally creates a file
EXAMPLES::
sage: from sage.modular.hilbert.sqrt5_tables import charpolys
sage: out = charpolys([1..20], 10)
4 2 ... [(5,x-6),(3,x-10)]
5 -2*a+1 ... [(2,x-5),(3,x-10)]
9 3 ... [(2,x-5),(5,x-6)]
11 -3*a+1 ... [(2,x-5),(5,x-6),(3,x-10)]
11 -3*a+2 ... [(2,x-5),(5,x-6),(3,x-10)]
16 4 ... [(5,x-6),(3,x-10)]
19 -4*a+1 ... [(2,x-5),(5,x-6),(3,x-10)]
19 -4*a+3 ... [(2,x-5),(5,x-6),(3,x-10)]
20 -4*a+2 ... [(3,x-10)]
sage: out = charpolys([20, 11], 10)
20 -4*a+2 ... [(3,x-10)]
11 -3*a+1 ... [(2,x-5),(5,x-6),(3,x-10)]
11 -3*a+2 ... [(2,x-5),(5,x-6),(3,x-10)]
Test writing to a file::
sage: if os.path.exists('tmp_table.txt'): os.unlink('tmp_table.txt')
sage: out = charpolys([20, 11], 10, 'tmp_table.txt')
20 -4*a+2 ... [(3,x-10)]
11 -3*a+1 ... [(2,x-5),(5,x-6),(3,x-10)]
11 -3*a+2 ... [(2,x-5),(5,x-6),(3,x-10)]
sage: r = open('tmp_table.txt').read()
sage: 'x-10' in r
True
sage: r.count('\n')
3
sage: os.unlink('tmp_table.txt')
"""
if len(v) == 0:
return ""
out = ""
F = open(filename, "a") if filename else None
P = [p for p in ideals_of_bounded_norm(B) if p.is_prime()]
for N in ideals_of_norm(v):
t = cputime()
H = IcosiansModP1ModN(N)
T = [
(p.smallest_integer(), H.hecke_matrix(p).fcp()) for p in P if gcd(Integer(p.norm()), Integer(N.norm())) == 1
]
tm = "%.2f" % cputime(t)
s = "%s %s %s %s" % (N.norm(), no_space(reduced_gen(N)), tm, no_space(T))
print s
out += s + "\n"
if F:
F.write(s + "\n")
F.flush()
return out
def _subpoly_parallel_facets(self):
"""
Generator for all lattice sub-polyhedra with parallel facets.
In a sub-polyhedron `Y\subset X` not all edges of `Y` need to
be parallel to `X`. This method iterates over all
sub-polyhedra where they are parallel, up to an overall
translation of the sub-polyhedron. Degenerate sub-polyhedra of
dimension strictly smaller are included.
OUTPUT:
A generator yielding `\ZZ`-polyhedra. By construction, each
facet of the returned polyhedron is parallel to one of the
facets of ``self``.
EXAMPLES::
sage: X = Polyhedron(vertices=[(0,0), (0,1), (1,0), (1,1)])
sage: X._subpoly_parallel_facets()
<generator object _subpoly_parallel_facets at 0x...>
sage: for p in X._subpoly_parallel_facets():
... print p.Vrepresentation()
(A vertex at (0, 0),)
(A vertex at (0, -1), A vertex at (0, 0))
(A vertex at (-1, 0), A vertex at (0, 0))
(A vertex at (-1, -1), A vertex at (-1, 0), A vertex at (0, -1), A vertex at (0, 0))
TESTS::
sage: X = Polyhedron(vertices=[(0,), (3,)])
sage: [ p.vertices() for p in X._subpoly_parallel_facets() ]
[(A vertex at (0),),
(A vertex at (-1), A vertex at (0)),
(A vertex at (-2), A vertex at (0)),
(A vertex at (-3), A vertex at (0))]
sage: list( Polyhedron(vertices=[[0,0]])._subpoly_parallel_facets() )
[A 0-dimensional polyhedron in ZZ^2 defined as the convex hull of 1 vertex]
sage: list( Polyhedron()._subpoly_parallel_facets() )
[The empty polyhedron in ZZ^0]
"""
if self.dim()>2 or not self.is_compact():
raise NotImplementedError('only implemented for bounded polygons')
from sage.geometry.polyhedron.plot import cyclic_sort_vertices_2d
vertices = cyclic_sort_vertices_2d(self.vertices())
n = len(vertices)
if n==1: # single point
yield self
return
edge_vectors = []
for i in range(0,n):
v = vertices[(i+1) % n].vector() - vertices[i].vector()
d = gcd(list(v))
v_prim = (v/d).change_ring(ZZ)
edge_vectors.append([ v_prim*i for i in range(d+1) ])
origin = self.ambient_space().zero()
parent = self.parent()
from sage.combinat.cartesian_product import CartesianProduct
for edges in CartesianProduct(*edge_vectors):
v = []
point = origin
for e in edges:
point += e
v.append(point)
if point!=origin: # does not close up, not a subpolygon
continue
yield parent([v, [], []], None)
def hecke_operator_on_qexp(f,
n,
k,
eps=None,
prec=None,
check=True,
_return_list=False):
r"""
Given the `q`-expansion `f` of a modular form with character
`\varepsilon`, this function computes the image of `f` under the
Hecke operator `T_{n,k}` of weight `k`.
EXAMPLES::
sage: M = ModularForms(1,12)
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + O(q^5)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12, prec=7)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + O(q^14)
sage: M.prec(20)
20
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 + 987136*q^16 - 6905934*q^17 + 2727432*q^18 + 10661420*q^19 - 7109760*q^20 + O(q^21)
sage: (hecke_operator_on_qexp(M.basis()[0], 1, 12)*252).add_bigoh(7)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 6, 12)
-6048*q + 145152*q^2 - 1524096*q^3 + O(q^4)
An example on a formal power series::
sage: R.<q> = QQ[[]]
sage: f = q + q^2 + q^3 + q^7 + O(q^8)
sage: hecke_operator_on_qexp(f, 3, 12)
q + O(q^3)
sage: hecke_operator_on_qexp(delta_qexp(24), 3, 12).prec()
8
sage: hecke_operator_on_qexp(delta_qexp(25), 3, 12).prec()
9
An example of computing `T_{p,k}` in characteristic `p`::
sage: p = 199
sage: fp = delta_qexp(prec=p^2+1, K=GF(p))
sage: tfp = hecke_operator_on_qexp(fp, p, 12)
sage: tfp == fp[p] * fp
True
sage: tf = hecke_operator_on_qexp(delta_qexp(prec=p^2+1), p, 12).change_ring(GF(p))
sage: tfp == tf
True
"""
if eps is None:
# Need to have base_ring=ZZ to work over finite fields, since
# ZZ can coerce to GF(p), but QQ can't.
eps = DirichletGroup(1, base_ring=ZZ).gen(0)
if check:
if not (is_PowerSeries(f) or is_ModularFormElement(f)):
raise TypeError, "f (=%s) must be a power series or modular form" % f
if not is_DirichletCharacter(eps):
raise TypeError, "eps (=%s) must be a Dirichlet character" % eps
k = Integer(k)
n = Integer(n)
v = []
if prec is None:
if is_ModularFormElement(f):
# always want at least three coefficients, but not too many, unless
# requested
pr = max(f.prec(), f.parent().prec(), (n + 1) * 3)
pr = min(pr, 100 * (n + 1))
prec = pr // n + 1
else:
prec = (f.prec() / ZZ(n)).ceil()
if prec == Infinity: prec = f.parent().default_prec() // n + 1
if f.prec() < prec:
f._compute_q_expansion(prec)
p = Integer(f.base_ring().characteristic())
if k != 1 and p.is_prime() and n.is_power_of(p):
# if computing T_{p^a} in characteristic p, use the simpler (and faster)
# formula
v = [f[m * n] for m in range(prec)]
else:
l = k - 1
for m in range(prec):
am = sum([eps(d) * d**l * f[m*n//(d*d)] for \
d in divisors(gcd(n, m)) if (m*n) % (d*d) == 0])
v.append(am)
if _return_list:
return v
if is_ModularFormElement(f):
R = f.parent()._q_expansion_ring()
else:
R = f.parent()
return R(v, prec)
def multiple_of_order(self, maxp=None):
"""
Return a multiple of the order of this torsion group.
The multiple is computed using characteristic polynomials of Hecke
operators of odd index not dividing the level.
INPUT:
- ``maxp`` - (default: None) If maxp is None (the
default), return gcd of best bound computed so far with bound
obtained by computing GCD's of orders modulo p until this gcd
stabilizes for 3 successive primes. If maxp is given, just use all
primes up to and including maxp.
EXAMPLES::
sage: J = J0(11)
sage: G = J.rational_torsion_subgroup()
sage: G.multiple_of_order(11)
5
sage: J = J0(389)
sage: G = J.rational_torsion_subgroup(); G
Torsion subgroup of Abelian variety J0(389) of dimension 32
sage: G.multiple_of_order()
97
sage: [G.multiple_of_order(p) for p in prime_range(3,11)]
[92645296242160800, 7275, 291]
sage: [G.multiple_of_order(p) for p in prime_range(3,13)]
[92645296242160800, 7275, 291, 97]
sage: [G.multiple_of_order(p) for p in prime_range(3,19)]
[92645296242160800, 7275, 291, 97, 97, 97]
::
sage: J = J0(33) * J0(11) ; J.rational_torsion_subgroup().order()
Traceback (most recent call last):
...
NotImplementedError: torsion multiple only implemented for Gamma0
The next example illustrates calling this function with a larger
input and how the result may be cached when maxp is None::
sage: T = J0(43)[1].rational_torsion_subgroup()
sage: T.multiple_of_order()
14
sage: T.multiple_of_order(50)
7
sage: T.multiple_of_order()
7
"""
if maxp is None:
try:
return self.__multiple_of_order
except AttributeError:
pass
bnd = ZZ(0)
A = self.abelian_variety()
if A.dimension() == 0:
T = ZZ(1)
self.__multiple_of_order = T
return T
N = A.level()
if not (len(A.groups()) == 1 and is_Gamma0(A.groups()[0])):
# to generalize to this case, you'll need to
# (1) define a charpoly_of_frob function:
# this is tricky because I don't know a simple
# way to do this for Gamma1 and GammaH. Will
# probably have to compute explicit matrix for
# <p> operator (add to modular symbols code),
# then compute some charpoly involving
# that directly...
# (2) use (1) -- see my MAGMA code.
raise NotImplementedError(
"torsion multiple only implemented for Gamma0")
cnt = 0
if maxp is None:
X = Primes()
else:
X = prime_range(maxp + 1)
for p in X:
if (2 * N) % p == 0:
continue
f = A.hecke_polynomial(p)
b = ZZ(f(p + 1))
if bnd == 0:
bnd = b
else:
bnd_last = bnd
bnd = ZZ(gcd(bnd, b))
if bnd == bnd_last:
cnt += 1
else:
cnt = 0
if maxp is None and cnt >= 2:
break
# The code below caches the computed bound and
# will be used if this function is called
# again with maxp equal to None (the default).
if maxp is None:
# maxp is None but self.__multiple_of_order is
# not set, since otherwise we would have immediately
# returned at the top of this function
self.__multiple_of_order = bnd
else:
# maxp is given -- record new info we get as
# a gcd...
try:
self.__multiple_of_order = gcd(self.__multiple_of_order, bnd)
except AttributeError:
# ... except in the case when self.__multiple_of_order
# was never set. In this case, we just set
# it as long as the gcd stabilized for 3 in a row.
if cnt >= 2:
self.__multiple_of_order = bnd
return bnd
def hecke_operator_on_qexp(f, n, k, eps=None, prec=None, check=True, _return_list=False):
r"""
Given the `q`-expansion `f` of a modular form with character
`\varepsilon`, this function computes the image of `f` under the
Hecke operator `T_{n,k}` of weight `k`.
EXAMPLES::
sage: M = ModularForms(1,12)
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + O(q^5)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12, prec=7)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + O(q^14)
sage: M.prec(20)
20
sage: hecke_operator_on_qexp(M.basis()[0], 3, 12)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 1, 12)
q - 24*q^2 + 252*q^3 - 1472*q^4 + 4830*q^5 - 6048*q^6 - 16744*q^7 + 84480*q^8 - 113643*q^9 - 115920*q^10 + 534612*q^11 - 370944*q^12 - 577738*q^13 + 401856*q^14 + 1217160*q^15 + 987136*q^16 - 6905934*q^17 + 2727432*q^18 + 10661420*q^19 - 7109760*q^20 + O(q^21)
sage: (hecke_operator_on_qexp(M.basis()[0], 1, 12)*252).add_bigoh(7)
252*q - 6048*q^2 + 63504*q^3 - 370944*q^4 + 1217160*q^5 - 1524096*q^6 + O(q^7)
sage: hecke_operator_on_qexp(M.basis()[0], 6, 12)
-6048*q + 145152*q^2 - 1524096*q^3 + O(q^4)
An example on a formal power series::
sage: R.<q> = QQ[[]]
sage: f = q + q^2 + q^3 + q^7 + O(q^8)
sage: hecke_operator_on_qexp(f, 3, 12)
q + O(q^3)
sage: hecke_operator_on_qexp(delta_qexp(24), 3, 12).prec()
8
sage: hecke_operator_on_qexp(delta_qexp(25), 3, 12).prec()
9
An example of computing `T_{p,k}` in characteristic `p`::
sage: p = 199
sage: fp = delta_qexp(prec=p^2+1, K=GF(p))
sage: tfp = hecke_operator_on_qexp(fp, p, 12)
sage: tfp == fp[p] * fp
True
sage: tf = hecke_operator_on_qexp(delta_qexp(prec=p^2+1), p, 12).change_ring(GF(p))
sage: tfp == tf
True
"""
if eps is None:
# Need to have base_ring=ZZ to work over finite fields, since
# ZZ can coerce to GF(p), but QQ can't.
eps = DirichletGroup(1, base_ring=ZZ)[0]
if check:
if not (is_PowerSeries(f) or is_ModularFormElement(f)):
raise TypeError("f (=%s) must be a power series or modular form" % f)
if not is_DirichletCharacter(eps):
raise TypeError("eps (=%s) must be a Dirichlet character" % eps)
k = Integer(k)
n = Integer(n)
v = []
if prec is None:
if is_ModularFormElement(f):
# always want at least three coefficients, but not too many, unless
# requested
pr = max(f.prec(), f.parent().prec(), (n + 1) * 3)
pr = min(pr, 100 * (n + 1))
prec = pr // n + 1
else:
prec = (f.prec() / ZZ(n)).ceil()
if prec == Infinity:
prec = f.parent().default_prec() // n + 1
if f.prec() < prec:
f._compute_q_expansion(prec)
p = Integer(f.base_ring().characteristic())
if k != 1 and p.is_prime() and n.is_power_of(p):
# if computing T_{p^a} in characteristic p, use the simpler (and faster)
# formula
v = [f[m * n] for m in range(prec)]
else:
l = k - 1
for m in range(prec):
am = sum([eps(d) * d ** l * f[m * n // (d * d)] for d in divisors(gcd(n, m)) if (m * n) % (d * d) == 0])
v.append(am)
if _return_list:
return v
if is_ModularFormElement(f):
R = f.parent()._q_expansion_ring()
else:
R = f.parent()
return R(v, prec)
def saturation(A, proof=True, p=0, max_dets=5):
"""
Compute a saturation matrix of A.
INPUT:
A -- a matrix over ZZ
proof -- bool (default: True)
p -- int (default: 0); if not 0
only guarantees that output is p-saturated
max_dets -- int (default: 4) max number of dets of
submatrices to compute.
OUTPUT:
matrix -- saturation of the matrix A.
EXAMPLES:
sage: from sage.matrix.matrix_integer_dense_saturation import saturation
sage: A = matrix(ZZ, 2, 2, [3,2,3,4]); B = matrix(ZZ, 2,3,[1,2,3,4,5,6]); C = A*B
sage: C
[11 16 21]
[19 26 33]
sage: C.index_in_saturation()
18
sage: S = saturation(C); S
[11 16 21]
[-2 -3 -4]
sage: S.index_in_saturation()
1
sage: saturation(C, proof=False)
[11 16 21]
[-2 -3 -4]
sage: saturation(C, p=2)
[11 16 21]
[-2 -3 -4]
sage: saturation(C, p=2, max_dets=1)
[11 16 21]
[-2 -3 -4]
"""
# Find a submatrix of full rank and instead saturate that matrix.
r = A.rank()
if A.is_square() and r == A.nrows():
return identity_matrix(ZZ, r)
if A.nrows() > r:
P = []
while len(P) < r:
P = matrix_integer_dense_hnf.probable_pivot_rows(A)
A = A.matrix_from_rows(P)
# Factor out all common factors from all rows, just in case.
A = copy(A)
A._factor_out_common_factors_from_each_row()
if A.nrows() <= 1:
return A
A, zero_cols = A._delete_zero_columns()
if max_dets > 0:
# Take the GCD of at most num_dets randomly chosen determinants.
nr = A.nrows()
nc = A.ncols()
d = 0
trials = min(binomial(nc, nr), max_dets)
already_tried = []
while len(already_tried) < trials:
v = random_sublist_of_size(nc, nr)
tm = verbose('saturation -- checking det condition on submatrix')
d = gcd(d, A.matrix_from_columns(v).determinant(proof=proof))
verbose('saturation -- got det down to %s' % d, tm)
if gcd(d, p) == 1:
return A._insert_zero_columns(zero_cols)
already_tried.append(v)
if gcd(d, p) == 1:
# already p-saturated
return A._insert_zero_columns(zero_cols)
# Factor and p-saturate at each p.
# This is not a good algorithm, because all the HNF's in it are really slow!
#
#tm = verbose('factoring gcd %s of determinants'%d)
#limit = 2**31-1
#F = d.factor(limit = limit)
#D = [p for p, e in F if p <= limit]
#B = [n for n, e in F if n > limit] # all big factors -- there will only be at most one
#assert len(B) <= 1
#C = B[0]
#for p in D:
# A = p_saturation(A, p=p, proof=proof)
# This is a really simple but powerful algorithm.
# FACT: If A is a matrix of full rank, then hnf(transpose(A))^(-1)*A is a saturation of A.
# To make this practical we use solve_system_with_difficult_last_row, since the
# last column of HNF's are typically the only really big ones.
B = A.transpose().hermite_form(include_zero_rows=False, proof=proof)
B = B.transpose()
# Now compute B^(-1) * A
C = solve_system_with_difficult_last_row(B, A)
return C.change_ring(ZZ)._insert_zero_columns(zero_cols)
def WP(self, *q, **kw):
# Specific keyword arguments instead of **kw would be preferable,
# later versions of Python might support specific (optional) keyword
# arguments after *q.
r"""
Construct weighted projective `n`-space over a field.
INPUT:
- ``q`` -- a sequence of positive integers relatively prime to
one another. The weights ``q`` can be given either as a list
or tuple, or as positional arguments.
Two keyword arguments:
- ``K`` -- a field (default: `\QQ`).
- ``names`` -- string or list (tuple) of strings (default 'z+'). See
:func:`~sage.schemes.toric.variety.normalize_names` for
acceptable formats.
OUTPUT:
- A :class:`toric variety
<sage.schemes.toric.variety.ToricVariety_field>`.
If `q=(q_0,\dots,q_n)`, then the output is the weighted projective
space `\mathbb{P}(q_0,\dots,q_n)` over `K`. ``names`` are the names
of the generators of the homogeneous coordinate ring.
EXAMPLES:
A hyperelliptic curve `C` of genus 2 as a subscheme of the weighted
projective plane `\mathbb{P}(1,3,1)`::
sage: X = toric_varieties.WP([1,3,1], names='x y z')
sage: X.inject_variables()
Defining x, y, z
sage: g = y^2-(x^6-z^6)
sage: C = X.subscheme([g]); C
Closed subscheme of 2-d toric variety covered by 3 affine patches defined by:
-x^6 + z^6 + y^2
"""
if len(q)==1:
# tuples and lists of weights are acceptable input
if isinstance(q[0], (list, tuple)):
q = q[0]
q = list(q)
m = len(q)
# allow case q=[1]? (not allowed presently)
if m < 2:
raise ValueError("more than one weight must be provided (got %s)" % q)
for i in range(m):
try:
q[i] = ZZ(q[i])
except(TypeError):
raise TypeError("the weights (=%s) must be integers" % q)
if q[i] <= 0:
raise ValueError("the weights (=%s) must be positive integers" % q)
if not gcd(q) == 1:
raise ValueError("the weights (=%s) must be relatively prime" % q)
# set default values for K and names
K = QQ
names = 'z+'
for key in kw:
if key == 'K':
K = kw['K']
if K not in _Fields:
raise TypeError("K (=%r) must be a field" % K)
elif key == 'names':
names = kw['names']
names = normalize_names(names, m, DEFAULT_PREFIX)
else:
raise TypeError("got an unexpected keyword argument %r" % key)
L = ToricLattice(m)
L_sub = L.submodule([L(q)])
Q = L/L_sub
rays = []
cones = []
w = range(m)
L_basis = L.basis()
for i in w:
b = L_basis[i]
v = Q.coordinate_vector(Q(b))
rays = rays + [v]
w_c = w[:i] + w[i+1:]
cones = cones + [tuple(w_c)]
fan = Fan(cones,rays)
return ToricVariety(fan, coordinate_names=names, base_field=K)
def _compute_lattice(self, rational_only=False, rational_subgroup=False):
r"""
Return a list of vectors that define elements of the rational
homology that generate this finite subgroup.
INPUT:
- ``rational_only`` - bool (default: False); if
``True``, only use rational cusps.
OUTPUT:
- ``list`` - list of vectors
EXAMPLES::
sage: J = J0(37)
sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
sage: C._compute_lattice()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[ 1 0 0 0]
[ 0 1 0 0]
[ 0 0 1 0]
[ 0 0 0 1/3]
sage: J = J0(43)
sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
sage: C._compute_lattice()
Free module of degree 6 and rank 6 over Integer Ring
Echelon basis matrix:
[ 1 0 0 0 0 0]
[ 0 1/7 0 6/7 0 5/7]
[ 0 0 1 0 0 0]
[ 0 0 0 1 0 0]
[ 0 0 0 0 1 0]
[ 0 0 0 0 0 1]
sage: J = J0(22)
sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
sage: C._compute_lattice()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[1/5 1/5 4/5 0]
[ 0 1 0 0]
[ 0 0 1 0]
[ 0 0 0 1/5]
sage: J = J1(13)
sage: C = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
sage: C._compute_lattice()
Free module of degree 4 and rank 4 over Integer Ring
Echelon basis matrix:
[ 1/19 0 0 9/19]
[ 0 1/19 1/19 18/19]
[ 0 0 1 0]
[ 0 0 0 1]
We compute with and without the optional
``rational_only`` option.
::
sage: J = J0(27); G = sage.modular.abvar.cuspidal_subgroup.CuspidalSubgroup(J)
sage: G._compute_lattice()
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1/3 0]
[ 0 1/3]
sage: G._compute_lattice(rational_only=True)
Free module of degree 2 and rank 2 over Integer Ring
Echelon basis matrix:
[1/3 0]
[ 0 1]
"""
A = self.abelian_variety()
Cusp = A.modular_symbols()
Amb = Cusp.ambient_module()
Eis = Amb.eisenstein_submodule()
C = Amb.cusps()
N = Amb.level()
if rational_subgroup:
# QQ-rational subgroup of cuspidal subgroup
assert A.is_ambient()
Q = Cusp.abvarquo_rational_cuspidal_subgroup()
return Q.V()
if rational_only:
# subgroup generated by differences of rational cusps
if not is_Gamma0(A.group()):
raise NotImplementedError, 'computation of rational cusps only implemented in Gamma0 case.'
if not N.is_squarefree():
data = [n for n in range(2, N) if gcd(n, N) == 1]
C = [c for c in C if is_rational_cusp_gamma0(c, N, data)]
v = [Amb([infinity, alpha]).element() for alpha in C]
cusp_matrix = matrix(QQ, len(v), Amb.dimension(), v)
# TODO -- refactor something out here
# Now we project onto the cuspidal part.
B = Cusp.free_module().basis_matrix().stack(
Eis.free_module().basis_matrix())
X = B.solve_left(cusp_matrix)
X = X.matrix_from_columns(range(Cusp.dimension()))
lattice = X.row_module(ZZ) + A.lattice()
return lattice
def dimension__jacobi_scalar_f(k, m, f):
if moebius(f) != (-1)**k:
return 0
## We use chapter 6 of Skoruppa's thesis
ts = filter(lambda t: gcd(t, m // t) == 1, m.divisors())
## Eisenstein part
eis_dimension = 0
for t in ts:
eis_dimension += moebius(gcd(m // t, f)) \
* (t // t.squarefree_part()).isqrt() \
* (2 if (m // t) % 4 == 0 else 1)
eis_dimension = eis_dimension // len(ts)
if k == 2 and f == 1:
eis_dimension -= len((m // m.squarefree_part()).isqrt().divisors())
## Cuspidal part
cusp_dimension = 0
tmp = ZZ(0)
for t in ts:
tmp += moebius(gcd(m // t, f)) * t
tmp = tmp / len(ts)
cusp_dimension += tmp * (2 * k - 3) / ZZ(12)
print "1: ", cusp_dimension
if m % 2 == 0:
tmp = ZZ(0)
for t in ts:
tmp += moebius(gcd(m // t, f)) * kronecker_symbol(-4, t)
tmp = tmp / len(ts)
cusp_dimension += 1 / ZZ(2) * kronecker_symbol(8, 2 * k - 1) * tmp
print "2: ", 1 / ZZ(2) * kronecker_symbol(8, 2 * k - 1) * tmp
tmp = ZZ(0)
for t in ts:
tmp += moebius(gcd(m // t, f)) * kronecker_symbol(t, 3)
tmp = tmp / len(ts)
if m % 3 != 0:
cusp_dimension += 1 / ZZ(3) * kronecker_symbol(k, 3) * tmp
print ": ", 1 / ZZ(3) * kronecker_symbol(k, 3) * tmp
elif k % 3 == 0:
cusp_dimension += 2 / ZZ(3) * (-1)**k * tmp
print "3: ", 2 / ZZ(3) * (-1)**k * tmp
else:
cusp_dimension += 1 / ZZ(3) * (kronecker_symbol(k, 3) +
(-1)**(k - 1)) * tmp
print "3: ", 1 / ZZ(3) * (kronecker_symbol(k, 3) + (-1)**(k - 1)) * tmp
tmp = ZZ(0)
for t in ts:
tmp += moebius(gcd(m // t, f)) \
* (t // t.squarefree_part()).isqrt() \
* (2 if (m // t) % 4 == 0 else 1)
tmp = tmp / len(ts)
cusp_dimension -= 1 / ZZ(2) * tmp
print "4: ", -1 / ZZ(2) * tmp
tmp = ZZ(0)
for t in ts:
tmp += moebius(gcd(m // t, f)) \
* sum( (( len(BinaryQF_reduced_representatives(-d, True))
if d not in [3, 4] else ( 1 / ZZ(3) if d == 3 else 1 / ZZ(2) ))
if d % 4 == 0 or d % 4 == 3 else 0 )
* kronecker_symbol(-d, m // t)
* ( 1 if (m // t) % 2 != 0 else
( 4 if (m // t) % 4 == 0 else 2 * kronecker_symbol(-d, 2) ))
for d in (4 * m).divisors() )
tmp = tmp / len(ts)
cusp_dimension -= 1 / ZZ(2) * tmp
print "5: ", -1 / ZZ(2) * tmp
if k == 2:
cusp_dimension += len(
(m // f // (m // f).squarefree_part()).isqrt().divisors())
return eis_dimension + cusp_dimension
def dimension__jacobi_scalar_f(k, m, f) :
if moebius(f) != (-1)**k :
return 0
## We use chapter 6 of Skoruppa's thesis
ts = filter(lambda t: gcd(t, m // t) == 1, m.divisors())
## Eisenstein part
eis_dimension = 0
for t in ts :
eis_dimension += moebius(gcd(m // t, f)) \
* (t // t.squarefree_part()).isqrt() \
* (2 if (m // t) % 4 == 0 else 1)
eis_dimension = eis_dimension // len(ts)
if k == 2 and f == 1 :
eis_dimension -= len( (m // m.squarefree_part()).isqrt().divisors() )
## Cuspidal part
cusp_dimension = 0
tmp = ZZ(0)
for t in ts :
tmp += moebius(gcd(m // t, f)) * t
tmp = tmp / len(ts)
cusp_dimension += tmp * (2 * k - 3) / ZZ(12)
print "1: ", cusp_dimension
if m % 2 == 0 :
tmp = ZZ(0)
for t in ts :
tmp += moebius(gcd(m // t, f)) * kronecker_symbol(-4, t)
tmp = tmp / len(ts)
cusp_dimension += 1/ZZ(2) * kronecker_symbol(8, 2 * k - 1) * tmp
print "2: ", 1/ZZ(2) * kronecker_symbol(8, 2 * k - 1) * tmp
tmp = ZZ(0)
for t in ts :
tmp += moebius(gcd(m // t, f)) * kronecker_symbol(t, 3)
tmp = tmp / len(ts)
if m % 3 != 0 :
cusp_dimension += 1 / ZZ(3) * kronecker_symbol(k, 3) * tmp
print ": ", 1 / ZZ(3) * kronecker_symbol(k, 3) * tmp
elif k % 3 == 0 :
cusp_dimension += 2 / ZZ(3) * (-1)**k * tmp
print "3: ", 2 / ZZ(3) * (-1)**k * tmp
else :
cusp_dimension += 1 / ZZ(3) * (kronecker_symbol(k, 3) + (-1)**(k - 1)) * tmp
print "3: ", 1 / ZZ(3) * (kronecker_symbol(k, 3) + (-1)**(k - 1)) * tmp
tmp = ZZ(0)
for t in ts :
tmp += moebius(gcd(m // t, f)) \
* (t // t.squarefree_part()).isqrt() \
* (2 if (m // t) % 4 == 0 else 1)
tmp = tmp / len(ts)
cusp_dimension -= 1 / ZZ(2) * tmp
print "4: ", -1 / ZZ(2) * tmp
tmp = ZZ(0)
for t in ts :
tmp += moebius(gcd(m // t, f)) \
* sum( (( len(BinaryQF_reduced_representatives(-d, True))
if d not in [3, 4] else ( 1 / ZZ(3) if d == 3 else 1 / ZZ(2) ))
if d % 4 == 0 or d % 4 == 3 else 0 )
* kronecker_symbol(-d, m // t)
* ( 1 if (m // t) % 2 != 0 else
( 4 if (m // t) % 4 == 0 else 2 * kronecker_symbol(-d, 2) ))
for d in (4 * m).divisors() )
tmp = tmp / len(ts)
cusp_dimension -= 1 / ZZ(2) * tmp
print "5: ", -1 / ZZ(2) * tmp
if k == 2 :
cusp_dimension += len( (m // f // (m // f).squarefree_part()).isqrt().divisors() )
return eis_dimension + cusp_dimension