def _qr(mat):
r"""
Compute the R matrix in QR decomposition using Housholder reflections.
This is an adoption of the implementation in mpmath by Andreas Strombergson.
"""
CC = mat.base_ring()
mat = copy(mat)
m = mat.nrows()
n = mat.ncols()
cur_row = 0
for j in range(0, n):
if all(mat[i, j].contains_zero() for i in xrange(cur_row + 1, m)):
if not mat[cur_row, j].contains_zero():
cur_row += 1
continue
s = sum((abs(mat[i, j]))**2 for i in xrange(cur_row, m))
if s.contains_zero():
raise RuntimeError(
"Cannot handle imprecise sums of elements that are too precise"
)
p = sqrt(s)
if (s - p * mat[cur_row, j]).contains_zero():
raise RuntimeError(
"Cannot handle imprecise sums of elements that are too precise"
)
kappa = 1 / (s - p * mat[cur_row, j])
mat[cur_row, j] -= p
for k in range(j + 1, n):
y = sum(mat[i, j].conjugate() * mat[i, k]
for i in xrange(cur_row, m)) * kappa
for i in range(cur_row, m):
mat[i, k] -= mat[i, j] * y
mat[cur_row, j] = p
for i in range(cur_row + 1, m):
mat[i, j] = CC(0)
cur_row += 1
return mat
def dimension__jacobi_scalar(k, m) :
raise RuntimeError( "There is a bug in the implementation" )
m = ZZ(m)
dimension = 0
for d in (m // m.squarefree_part()).isqrt().divisors() :
m_d = m // d**2
dimension += sum ( dimension__jacobi_scalar_f(k, m_d, f)
for f in m_d.divisors() )
return dimension
def dimension__jacobi_scalar(k, m):
raise RuntimeError("There is a bug in the implementation")
m = ZZ(m)
dimension = 0
for d in (m // m.squarefree_part()).isqrt().divisors():
m_d = m // d**2
dimension += sum(
dimension__jacobi_scalar_f(k, m_d, f) for f in m_d.divisors())
return dimension
def _qr(mat) :
r"""
Compute the R matrix in QR decomposition using Housholder reflections.
This is an adoption of the implementation in mpmath by Andreas Strombergson.
"""
CC = mat.base_ring()
mat = copy(mat)
m = mat.nrows()
n = mat.ncols()
cur_row = 0
for j in range(0, n) :
if all( mat[i,j].contains_zero() for i in xrange(cur_row + 1, m) ) :
if not mat[cur_row,j].contains_zero() :
cur_row += 1
continue
s = sum( (abs(mat[i,j]))**2 for i in xrange(cur_row, m) )
if s.contains_zero() :
raise RuntimeError( "Cannot handle imprecise sums of elements that are too precise" )
p = sqrt(s)
if (s - p * mat[cur_row,j]).contains_zero() :
raise RuntimeError( "Cannot handle imprecise sums of elements that are too precise" )
kappa = 1 / (s - p * mat[cur_row,j])
mat[cur_row,j] -= p
for k in range(j + 1, n) :
y = sum(mat[i,j].conjugate() * mat[i,k] for i in xrange(cur_row, m)) * kappa
for i in range(cur_row, m):
mat[i,k] -= mat[i,j] * y
mat[cur_row,j] = p
for i in range(cur_row + 1, m) :
mat[i,j] = CC(0)
cur_row += 1
return mat
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__vector_valued(k, L, conjugate = False) :
r"""
Compute the dimension of the space of weight `k` vector valued modular forms
for the Weil representation (or its conjugate) attached to the lattice `L`.
See [Borcherds, Borcherds - Reflection groups of Lorentzian lattices] for a proof
of the formula that we use here.
INPUT:
- `k` -- A half-integer.
- ``L`` -- An quadratic form.
- ``conjugate`` -- A boolean; If ``True``, then compute the dimension for
the conjugated Weil representation.
OUTPUT:
An integer.
TESTS::
sage: dimension__vector_valued(3, QuadraticForm(-matrix(2, [2, 1, 1, 2])))
1
sage: dimension__vector_valued(3, QuadraticForm(-matrix(2, [2, 0, 0, 2])))
1
sage: dimension__vector_valued(3, QuadraticForm(-matrix(2, [2, 0, 0, 4])))
1
"""
if 2 * k not in ZZ :
raise ValueError( "Weight must be half-integral" )
if k <= 0 :
return 0
if k < 2 :
raise NotImplementedError( "Weight <2 is not implemented." )
if L.matrix().rank() != L.matrix().nrows() :
raise ValueError( "The lattice (={0}) must be non-degenerate.".format(L) )
L_dimension = L.matrix().nrows()
if L_dimension % 2 != ZZ(2 * k) % 2 :
return 0
plus_basis = ZZ(L_dimension + 2 * k) % 4 == 0
## The bilinear and the quadratic form attached to L
quadratic = lambda x: L(x) // 2
bilinear = lambda x,y: L(x + y) - L(x) - L(y)
## A dual basis for L
(elementary_divisors, dual_basis_pre, _) = L.matrix().smith_form()
elementary_divisors = elementary_divisors.diagonal()
dual_basis = map(operator.div, list(dual_basis_pre), elementary_divisors)
L_level = ZZ(lcm([ b.denominator() for b in dual_basis ]))
(elementary_divisors, _, discriminant_basis_pre) = (L_level * matrix(dual_basis)).change_ring(ZZ).smith_form()
elementary_divisors = filter( lambda d: d not in ZZ, (elementary_divisors / L_level).diagonal() )
elementary_divisors_inv = map(ZZ, [ed**-1 for ed in elementary_divisors])
discriminant_basis = matrix(map( operator.mul,
discriminant_basis_pre.inverse().rows()[:len(elementary_divisors)],
elementary_divisors )).transpose()
## This is a form over QQ, so that we cannot use an instance of QuadraticForm
discriminant_form = discriminant_basis.transpose() * L.matrix() * discriminant_basis
if conjugate :
discriminant_form = - discriminant_form
if prod(elementary_divisors_inv) > 100 :
disc_den = discriminant_form.denominator()
disc_bilinear_pre = \
cython_lambda( ', '.join( ['int a{0}'.format(i) for i in range(discriminant_form.nrows())]
+ ['int b{0}'.format(i) for i in range(discriminant_form.nrows())] ),
' + '.join('{0} * a{1} * b{2}'.format(disc_den * discriminant_form[i,j], i, j)
for i in range(discriminant_form.nrows())
for j in range(discriminant_form.nrows())) )
disc_bilinear = lambda *a: disc_bilinear_pre(*a) / disc_den
else :
disc_bilinear = lambda *xy: vector(ZZ, xy[:discriminant_form.nrows()]) * discriminant_form * vector(ZZ, xy[discriminant_form.nrows():])
disc_quadratic = lambda *a: disc_bilinear(*(2 * a)) / 2
## red gives a normal form for elements in the discriminant group
red = lambda x : map(operator.mod, x, elementary_divisors_inv)
def is_singl(x) :
y = red(map(operator.neg, x))
for (e, f) in zip(x, y) :
if e < f :
return -1
elif e > f :
return 1
return 0
## singls and pairs are elements of the discriminant group that are, respectively,
## fixed and not fixed by negation.
singls = list()
pairs = list()
for x in mrange(elementary_divisors_inv) :
si = is_singl(x)
if si == 0 :
singls.append(x)
elif si == 1 :
pairs.append(x)
if plus_basis :
subspace_dimension = len(singls + pairs)
else :
subspace_dimension = len(pairs)
## 200 bits are, by far, sufficient to distinguish 12-th roots of unity
## by increasing the precision by 4 for each additional dimension, we
## compensate, by far, the errors introduced by the QR decomposition,
## which are of the size of (absolute error) * dimension
CC = ComplexIntervalField(200 + subspace_dimension * 4)
zeta_order = ZZ(lcm([8, 12] + map(lambda ed: 2 * ed, elementary_divisors_inv)))
zeta = CC(exp(2 * pi * I / zeta_order))
sqrt2 = CC(sqrt(2))
drt = CC(sqrt(L.det()))
Tmat = diagonal_matrix(CC, [zeta**(zeta_order*disc_quadratic(*a)) for a in (singls + pairs if plus_basis else pairs)])
if plus_basis :
Smat = zeta**(zeta_order / 8 * L_dimension) / drt \
* matrix( CC, [ [zeta**(-zeta_order * disc_bilinear(*(gamma + delta))) for delta in singls]
+ [sqrt2 * zeta**(-zeta_order * disc_bilinear(*(gamma + delta))) for delta in pairs]
for gamma in singls] \
+ [ [sqrt2 * zeta**(-zeta_order * disc_bilinear(*(gamma + delta))) for delta in singls]
+ [zeta**(-zeta_order * disc_bilinear(*(gamma + delta))) + zeta**(-zeta_order * disc_bilinear(*(gamma + map(operator.neg, delta)))) for delta in pairs]
for gamma in pairs] )
else :
Smat = zeta**(zeta_order / 8 * L_dimension) / drt \
* matrix( CC, [ [zeta**(-zeta_order * disc_bilinear(*(gamma + delta))) - zeta**(-zeta_order * disc_bilinear(*(gamma + map(operator.neg,delta)))) for delta in pairs]
for gamma in pairs ] )
STmat = Smat * Tmat
## This function overestimates the number of eigenvalues, if it is not correct
def eigenvalue_multiplicity(mat, ev) :
mat = matrix(CC, mat - ev * identity_matrix(subspace_dimension))
return len(filter( lambda row: all( e.contains_zero() for e in row), _qr(mat).rows() ))
rti = CC(exp(2 * pi * I / 8))
S_ev_multiplicity = [eigenvalue_multiplicity(Smat, rti**n) for n in range(8)]
## Together with the fact that eigenvalue_multiplicity overestimates the multiplicities
## this asserts that the computed multiplicities are correct
assert sum(S_ev_multiplicity) == subspace_dimension
rho = CC(exp(2 * pi * I / 12))
ST_ev_multiplicity = [eigenvalue_multiplicity(STmat, rho**n) for n in range(12)]
## Together with the fact that eigenvalue_multiplicity overestimates the multiplicities
## this asserts that the computed multiplicities are correct
assert sum(ST_ev_multiplicity) == subspace_dimension
T_evs = [ ZZ((zeta_order * disc_quadratic(*a)) % zeta_order) / zeta_order
for a in (singls + pairs if plus_basis else pairs) ]
return subspace_dimension * (1 + QQ(k) / 12) \
- ZZ(sum( (ST_ev_multiplicity[n] * ((-2 * k - n) % 12)) for n in range(12) )) / 12 \
- ZZ(sum( (S_ev_multiplicity[n] * ((2 * k + n) % 8)) for n in range(8) )) / 8 \
- sum(T_evs)
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__vector_valued(k, L, conjugate=False):
r"""
Compute the dimension of the space of weight `k` vector valued modular forms
for the Weil representation (or its conjugate) attached to the lattice `L`.
See [Borcherds, Borcherds - Reflection groups of Lorentzian lattices] for a proof
of the formula that we use here.
INPUT:
- `k` -- A half-integer.
- ``L`` -- An quadratic form.
- ``conjugate`` -- A boolean; If ``True``, then compute the dimension for
the conjugated Weil representation.
OUTPUT:
An integer.
TESTS::
sage: dimension__vector_valued(3, QuadraticForm(-matrix(2, [2, 1, 1, 2])))
1
sage: dimension__vector_valued(3, QuadraticForm(-matrix(2, [2, 0, 0, 2])))
1
sage: dimension__vector_valued(3, QuadraticForm(-matrix(2, [2, 0, 0, 4])))
1
"""
if 2 * k not in ZZ:
raise ValueError("Weight must be half-integral")
if k <= 0:
return 0
if k < 2:
raise NotImplementedError("Weight <2 is not implemented.")
if L.matrix().rank() != L.matrix().nrows():
raise ValueError(
"The lattice (={0}) must be non-degenerate.".format(L))
L_dimension = L.matrix().nrows()
if L_dimension % 2 != ZZ(2 * k) % 2:
return 0
plus_basis = ZZ(L_dimension + 2 * k) % 4 == 0
## The bilinear and the quadratic form attached to L
quadratic = lambda x: L(x) // 2
bilinear = lambda x, y: L(x + y) - L(x) - L(y)
## A dual basis for L
(elementary_divisors, dual_basis_pre, _) = L.matrix().smith_form()
elementary_divisors = elementary_divisors.diagonal()
dual_basis = map(operator.div, list(dual_basis_pre), elementary_divisors)
L_level = ZZ(lcm([b.denominator() for b in dual_basis]))
(elementary_divisors, _, discriminant_basis_pre) = (
L_level * matrix(dual_basis)).change_ring(ZZ).smith_form()
elementary_divisors = filter(lambda d: d not in ZZ,
(elementary_divisors / L_level).diagonal())
elementary_divisors_inv = map(ZZ, [ed**-1 for ed in elementary_divisors])
discriminant_basis = matrix(
map(operator.mul,
discriminant_basis_pre.inverse().rows()[:len(elementary_divisors)],
elementary_divisors)).transpose()
## This is a form over QQ, so that we cannot use an instance of QuadraticForm
discriminant_form = discriminant_basis.transpose() * L.matrix(
) * discriminant_basis
if conjugate:
discriminant_form = -discriminant_form
if prod(elementary_divisors_inv) > 100:
disc_den = discriminant_form.denominator()
disc_bilinear_pre = \
cython_lambda( ', '.join( ['int a{0}'.format(i) for i in range(discriminant_form.nrows())]
+ ['int b{0}'.format(i) for i in range(discriminant_form.nrows())] ),
' + '.join('{0} * a{1} * b{2}'.format(disc_den * discriminant_form[i,j], i, j)
for i in range(discriminant_form.nrows())
for j in range(discriminant_form.nrows())) )
disc_bilinear = lambda *a: disc_bilinear_pre(*a) / disc_den
else:
disc_bilinear = lambda *xy: vector(ZZ, xy[:discriminant_form.nrows(
)]) * discriminant_form * vector(ZZ, xy[discriminant_form.nrows():])
disc_quadratic = lambda *a: disc_bilinear(*(2 * a)) / 2
## red gives a normal form for elements in the discriminant group
red = lambda x: map(operator.mod, x, elementary_divisors_inv)
def is_singl(x):
y = red(map(operator.neg, x))
for (e, f) in zip(x, y):
if e < f:
return -1
elif e > f:
return 1
return 0
## singls and pairs are elements of the discriminant group that are, respectively,
## fixed and not fixed by negation.
singls = list()
pairs = list()
for x in mrange(elementary_divisors_inv):
si = is_singl(x)
if si == 0:
singls.append(x)
elif si == 1:
pairs.append(x)
if plus_basis:
subspace_dimension = len(singls + pairs)
else:
subspace_dimension = len(pairs)
## 200 bits are, by far, sufficient to distinguish 12-th roots of unity
## by increasing the precision by 4 for each additional dimension, we
## compensate, by far, the errors introduced by the QR decomposition,
## which are of the size of (absolute error) * dimension
CC = ComplexIntervalField(200 + subspace_dimension * 4)
zeta_order = ZZ(
lcm([8, 12] + map(lambda ed: 2 * ed, elementary_divisors_inv)))
zeta = CC(exp(2 * pi * I / zeta_order))
sqrt2 = CC(sqrt(2))
drt = CC(sqrt(L.det()))
Tmat = diagonal_matrix(CC, [
zeta**(zeta_order * disc_quadratic(*a))
for a in (singls + pairs if plus_basis else pairs)
])
if plus_basis:
Smat = zeta**(zeta_order / 8 * L_dimension) / drt \
* matrix( CC, [ [zeta**(-zeta_order * disc_bilinear(*(gamma + delta))) for delta in singls]
+ [sqrt2 * zeta**(-zeta_order * disc_bilinear(*(gamma + delta))) for delta in pairs]
for gamma in singls] \
+ [ [sqrt2 * zeta**(-zeta_order * disc_bilinear(*(gamma + delta))) for delta in singls]
+ [zeta**(-zeta_order * disc_bilinear(*(gamma + delta))) + zeta**(-zeta_order * disc_bilinear(*(gamma + map(operator.neg, delta)))) for delta in pairs]
for gamma in pairs] )
else:
Smat = zeta**(zeta_order / 8 * L_dimension) / drt \
* matrix( CC, [ [zeta**(-zeta_order * disc_bilinear(*(gamma + delta))) - zeta**(-zeta_order * disc_bilinear(*(gamma + map(operator.neg,delta)))) for delta in pairs]
for gamma in pairs ] )
STmat = Smat * Tmat
## This function overestimates the number of eigenvalues, if it is not correct
def eigenvalue_multiplicity(mat, ev):
mat = matrix(CC, mat - ev * identity_matrix(subspace_dimension))
return len(
filter(lambda row: all(e.contains_zero() for e in row),
_qr(mat).rows()))
rti = CC(exp(2 * pi * I / 8))
S_ev_multiplicity = [
eigenvalue_multiplicity(Smat, rti**n) for n in range(8)
]
## Together with the fact that eigenvalue_multiplicity overestimates the multiplicities
## this asserts that the computed multiplicities are correct
assert sum(S_ev_multiplicity) == subspace_dimension
rho = CC(exp(2 * pi * I / 12))
ST_ev_multiplicity = [
eigenvalue_multiplicity(STmat, rho**n) for n in range(12)
]
## Together with the fact that eigenvalue_multiplicity overestimates the multiplicities
## this asserts that the computed multiplicities are correct
assert sum(ST_ev_multiplicity) == subspace_dimension
T_evs = [
ZZ((zeta_order * disc_quadratic(*a)) % zeta_order) / zeta_order
for a in (singls + pairs if plus_basis else pairs)
]
return subspace_dimension * (1 + QQ(k) / 12) \
- ZZ(sum( (ST_ev_multiplicity[n] * ((-2 * k - n) % 12)) for n in range(12) )) / 12 \
- ZZ(sum( (S_ev_multiplicity[n] * ((2 * k + n) % 8)) for n in range(8) )) / 8 \
- sum(T_evs)