def eisenstein_basis(N, k, verbose=False):
r"""
Find spanning list of 'easy' generators for the subspace of
`M_k(\Gamma_0(N))` generated by level 1 Eisenstein series and
their images of even integer weights up to `k`.
INPUT:
- N -- positive integer
- k -- positive integer
- ``verbose`` -- bool (default: False)
OUTPUT:
- list of monomials in images of level 1 Eisenstein series
- prec of q-expansions needed to determine element of
`M_k(\Gamma_0(N))`.
EXAMPLES::
sage: from psage.modform.rational.special import eisenstein_basis
sage: eisenstein_basis(5,4)
([E4(q^5)^1, E4(q^1)^1, E2^*(q^5)^2], 3)
sage: eisenstein_basis(11,2,verbose=True) # warning below because of verbose
Warning -- not enough series.
([E2^*(q^11)^1], 2)
sage: eisenstein_basis(11,2,verbose=False)
([E2^*(q^11)^1], 2)
"""
assert N > 1
if k % 2 != 0:
return []
# Make list E of Eisenstein series, to enough precision to
# determine them, until we span space.
M = ModularForms(N, k)
prec = M.echelon_basis()[-1].valuation() + 1
gens = eisenstein_gens(N, k, prec)
R = PolynomialRing(ZZ, len(gens), ['E%sq%s'%(g[1],g[0]) for g in gens])
z = [(R.gen(i), g[1]) for i, g in enumerate(gens)]
m = monomials(z, k)
A = QQ**prec
V = A.zero_subspace()
E = []
for i, z in enumerate(m):
d = z.degrees()
f = prod(g[2]**d[i] for i, g in enumerate(gens) if d[i])
v = A(f.padded_list(prec))
if v not in V:
V = V + A.span([v])
w = [(gens[i][0],gens[i][1],d[i]) for i in range(len(d)) if d[i]]
E.append(EisensteinMonomial(w))
if V.dimension() == M.dimension():
return E, prec
if verbose: print "Warning -- not enough series."
return E, prec
def _rank(self, K):
"""
Return the dimension of the level N space of given weight.
"""
if not K is QQ:
raise NotImplementedError
if not self.__level.is_prime():
raise NotImplementedError
if self.__weight % 2 != 0:
raise NotImplementedError("Only even weights available")
N = self.__level
if N == 1:
## By Igusa's theorem on the generators of the graded ring
P = PowerSeriesRing(ZZ, 't')
t = P.gen(0)
dims = 1 / ((1 - t**4) * (1 - t**6) * (1 - t**10) *
(1 - t**12)).add_bigoh(self.__weight + 1)
return dims[self.__weight]
if N == 2:
## As in Ibukiyama, Onodera - On the graded ring of modular forms of the Siegel
## paramodular group of level 2, Proposition 2
P = PowerSeriesRing(ZZ, 't')
t = P.gen(0)
dims = ((1 + t**10) * (1 + t**12) * (1 + t**11))
dims = dims / ((1 - t**4) * (1 - t**6) * (1 - t**8) *
(1 - t**12)).add_bigoh(self.__weight + 1)
return dims[self.__weight]
if N == 3:
## By Dern, Paramodular forms of degree 2 and level 3, Corollary 5.6
P = PowerSeriesRing(ZZ, 't')
t = P.gen(0)
dims = ((1 + t**12) * (1 + t**8 + t**9 + t**10 + t**11 + t**19))
dims = dims / ((1 - t**4) * (1 - t**6)**2 *
(1 - t**12)).add_bigoh(self.__weight + 1)
return dims[self.__weight]
if N == 5:
## By Marschner, Paramodular forms of degree 2 with particular emphasis on level t = 5,
## Corollary 7.3.4. PhD thesis electronically available via the library of
## RWTH University, Aachen, Germany
P = PowerSeriesRing(ZZ, 't')
t = P.gen(0)
dims = t**30 + t**24 + t**23 + 2*t**22 + t**21 + 2*t**20 + t**19 + 2*t**18 \
+ 2*t**16 + 2*t**14 + 2*t**12 + t**11 + 2*t**10 + t**9 + 2*t**8 + t**7 + t**6 + 1
dims = dims / ((1 - t**4) * (1 - t**5) * (1 - t**6) *
(1 - t**12)).add_bigoh(self.__weight + 1)
return dims[self.__weight]
if self.__weight == 2:
## There are only cuspforms, since there is no elliptic modular form
## of weight 2.
if N < 277:
## Poor, Yuen - Paramodular cusp forms tells us that all forms are
## Gritsenko lifts
return JacobiFormD1NN_Gamma(self.__level, 2)._rank(QQ)
raise NotImplementedError
elif self.__weight == 4:
## This is the formula cited by Poor and Yuen in Paramodular cusp forms
cuspidal_dim = Integer((N**2 - 143) / Integer(576) +
N / Integer(8) + kronecker_symbol(-1, N) *
(N - 12) / Integer(96) +
kronecker_symbol(2, N) / Integer(8) +
kronecker_symbol(3, N) / Integer(12) +
kronecker_symbol(-3, N) * N / Integer(36))
else:
## This is the formula given by Ibukiyama in
## Relations of dimension of automorphic forms of Sp(2,R) and its compact twist Sp(2),
## Theorem 4
p = N
k = self.__weight
## This is the reversed Ibukiyama symbol [.., .., ..; ..]
def ibukiyama_symbol(modulus, *args):
return args[k % modulus]
## if p == 2 this formula is wrong. If the weight is even it differs by
## -3/16 from the true dimension and if the weight is odd it differs by
## -1/16 from the true dimension.
H1 = (p**2 + 1) * (2 * k - 2) * (2 * k - 3) * (
2 * k - 4) / Integer(2**9 * 3**3 * 5)
H2 = (-1)**k * (2 * k - 2) * (2 * k - 4) / Integer(2**8 * 3**2) \
+ ( (-1)**k * (2 * k - 2) * (2 * k - 4) / Integer(2**7 * 3)
if p !=2 else
(-1)**k * (2 * k - 2) * (2 * k - 4) / Integer(2**9) )
H3 = (ibukiyama_symbol(4, k - 2, -k + 1, -k + 2, k - 1) /
Integer(2**4 * 3) if p != 3 else 5 *
ibukiyama_symbol(4, k - 2, -k + 1, -k + 2, k - 1) /
Integer(2**5 * 3))
H4 = (ibukiyama_symbol(3, 2 * k - 3, -k + 1, -k + 2) /
Integer(2**2 * 3**3) if p != 2 else 5 *
ibukiyama_symbol(3, 2 * k - 3, -k + 1, -k + 2) /
Integer(2**2 * 3**3))
H5 = ibukiyama_symbol(6, -1, -k + 1, -k + 2, 1, k - 1,
k - 2) / Integer(2**2 * 3**2)
if p % 4 == 1:
H6 = 5 * (2 * k - 3) * (p + 1) / Integer(
2**7 * 3) + (-1)**k * (p + 1) / Integer(2**7)
elif p % 4 == 3:
H6 = (2 * k - 3) * (p - 1) / Integer(
2**7) + 5 * (-1)**k * (p - 1) / Integer(2**7 * 3)
else:
H6 = 3 * (2 * k - 3) / Integer(2**7) + 7 * (-1)**k / Integer(
2**7 * 3)
if p % 3 == 1:
H7 = (2 * k - 3) * (p + 1) / Integer(2 * 3**3) \
+ (p + 1) * ibukiyama_symbol(3, 0, -1, 1) / Integer(2**2 * 3**3)
elif p % 3 == 2:
H7 = (2 * k - 3) * (p - 1) / Integer(2**2 * 3**3) \
+ (p - 1) * ibukiyama_symbol(3, 0, -1, 1) / Integer(2 * 3**3)
else:
H7 = 5 * (2 * k - 3) / Integer(2**2 * 3**3) \
+ ibukiyama_symbol(3, 0, -1, 1) / Integer(3**3)
H8 = ibukiyama_symbol(12, 1, 0, 0, -1, -1, -1, -1, 0, 0, 1, 1,
1) / Integer(2 * 3)
H9 = (2 * ibukiyama_symbol(6, 1, 0, 0, -1, 0, 0) / Integer(3**2)
if p != 2 else ibukiyama_symbol(6, 1, 0, 0, -1, 0, 0) /
Integer(2 * 3**2))
H10 = (1 + kronecker_symbol(5, p)) * ibukiyama_symbol(
5, 1, 0, 0, -1, 0) / Integer(5)
H11 = (1 + kronecker_symbol(2, p)) * ibukiyama_symbol(
4, 1, 0, 0, -1) / Integer(2**3)
if p % 12 == 1:
H12 = ibukiyama_symbol(3, 0, 1, -1) / Integer(2 * 3)
elif p % 12 == 11:
H12 = (-1)**k / Integer(2 * 3)
elif p == 2 or p == 3:
H12 = (-1)**k / Integer(2**2 * 3)
else:
H12 = 0
I1 = ibukiyama_symbol(6, 0, 1, 1, 0, -1, -1) / Integer(6)
I2 = ibukiyama_symbol(3, -2, 1, 1) / Integer(2 * 3**2)
if p == 3:
I3 = ibukiyama_symbol(3, -2, 1, 1) / Integer(3**2)
elif p % 3 == 1:
I3 = 2 * ibukiyama_symbol(3, -1, 1, 0) / Integer(3**2)
else:
I3 = 2 * ibukiyama_symbol(3, -1, 0, 1) / Integer(3**2)
I4 = ibukiyama_symbol(4, -1, 1, 1, -1) / Integer(2**2)
I5 = (-1)**k / Integer(2**3)
I6 = (-1)**k * (2 - kronecker_symbol(-1, p)) / Integer(2**4)
I7 = -(-1)**k * (2 * k - 3) / Integer(2**3 * 3)
I8 = -p * (2 * k - 3) / Integer(2**4 * 3**2)
I9 = -1 / Integer(2**3 * 3)
I10 = (p + 1) / Integer(2**3 * 3)
I11 = -(1 + kronecker_symbol(-1, p)) / Integer(8)
I12 = -(1 + kronecker_symbol(-3, p)) / Integer(6)
cuspidal_dim = H1 + H2 + H3 + H4 + H5 + H6 + H7 + H8 + H9 + H10 + H11 + H12 \
+ I1 + I2 + I3 + I4 + I5 + I6 + I7 + I8 + I9 + I10 + I11 + I12
mfs = ModularForms(1, self.__weight)
return cuspidal_dim + mfs.dimension() + mfs.cuspidal_subspace(
).dimension()
def _rank(self, K):
"""
Return the dimension of the level N space of given weight.
"""
if not K is QQ:
raise NotImplementedError
if not self.__level.is_prime():
raise NotImplementedError
if self.__weight % 2 != 0:
raise NotImplementedError("Only even weights available")
N = self.__level
if N == 1:
## By Igusa's theorem on the generators of the graded ring
P = PowerSeriesRing(ZZ, "t")
t = P.gen(0)
dims = 1 / ((1 - t ** 4) * (1 - t ** 6) * (1 - t ** 10) * (1 - t ** 12)).add_bigoh(self.__weight + 1)
return dims[self.__weight]
if N == 2:
## As in Ibukiyama, Onodera - On the graded ring of modular forms of the Siegel
## paramodular group of level 2, Proposition 2
P = PowerSeriesRing(ZZ, "t")
t = P.gen(0)
dims = (1 + t ** 10) * (1 + t ** 12) * (1 + t ** 11)
dims = dims / ((1 - t ** 4) * (1 - t ** 6) * (1 - t ** 8) * (1 - t ** 12)).add_bigoh(self.__weight + 1)
return dims[self.__weight]
if N == 3:
## By Dern, Paramodular forms of degree 2 and level 3, Corollary 5.6
P = PowerSeriesRing(ZZ, "t")
t = P.gen(0)
dims = (1 + t ** 12) * (1 + t ** 8 + t ** 9 + t ** 10 + t ** 11 + t ** 19)
dims = dims / ((1 - t ** 4) * (1 - t ** 6) ** 2 * (1 - t ** 12)).add_bigoh(self.__weight + 1)
return dims[self.__weight]
if N == 5:
## By Marschner, Paramodular forms of degree 2 with particular emphasis on level t = 5,
## Corollary 7.3.4. PhD thesis electronically available via the library of
## RWTH University, Aachen, Germany
P = PowerSeriesRing(ZZ, "t")
t = P.gen(0)
dims = (
t ** 30
+ t ** 24
+ t ** 23
+ 2 * t ** 22
+ t ** 21
+ 2 * t ** 20
+ t ** 19
+ 2 * t ** 18
+ 2 * t ** 16
+ 2 * t ** 14
+ 2 * t ** 12
+ t ** 11
+ 2 * t ** 10
+ t ** 9
+ 2 * t ** 8
+ t ** 7
+ t ** 6
+ 1
)
dims = dims / ((1 - t ** 4) * (1 - t ** 5) * (1 - t ** 6) * (1 - t ** 12)).add_bigoh(self.__weight + 1)
return dims[self.__weight]
if self.__weight == 2:
## There are only cuspforms, since there is no elliptic modular form
## of weight 2.
if N < 277:
## Poor, Yuen - Paramodular cusp forms tells us that all forms are
## Gritsenko lifts
return JacobiFormD1NN_Gamma(self.__level, 2)._rank(QQ)
raise NotImplementedError
elif self.__weight == 4:
## This is the formula cited by Poor and Yuen in Paramodular cusp forms
cuspidal_dim = Integer(
(N ** 2 - 143) / Integer(576)
+ N / Integer(8)
+ kronecker_symbol(-1, N) * (N - 12) / Integer(96)
+ kronecker_symbol(2, N) / Integer(8)
+ kronecker_symbol(3, N) / Integer(12)
+ kronecker_symbol(-3, N) * N / Integer(36)
)
else:
## This is the formula given by Ibukiyama in
## Relations of dimension of automorphic forms of Sp(2,R) and its compact twist Sp(2),
## Theorem 4
p = N
k = self.__weight
## This is the reversed Ibukiyama symbol [.., .., ..; ..]
def ibukiyama_symbol(modulus, *args):
return args[k % modulus]
## if p == 2 this formula is wrong. If the weight is even it differs by
## -3/16 from the true dimension and if the weight is odd it differs by
## -1/16 from the true dimension.
H1 = (p ** 2 + 1) * (2 * k - 2) * (2 * k - 3) * (2 * k - 4) / Integer(2 ** 9 * 3 ** 3 * 5)
H2 = (-1) ** k * (2 * k - 2) * (2 * k - 4) / Integer(2 ** 8 * 3 ** 2) + (
(-1) ** k * (2 * k - 2) * (2 * k - 4) / Integer(2 ** 7 * 3)
if p != 2
else (-1) ** k * (2 * k - 2) * (2 * k - 4) / Integer(2 ** 9)
)
H3 = (
ibukiyama_symbol(4, k - 2, -k + 1, -k + 2, k - 1) / Integer(2 ** 4 * 3)
if p != 3
else 5 * ibukiyama_symbol(4, k - 2, -k + 1, -k + 2, k - 1) / Integer(2 ** 5 * 3)
)
H4 = (
ibukiyama_symbol(3, 2 * k - 3, -k + 1, -k + 2) / Integer(2 ** 2 * 3 ** 3)
if p != 2
else 5 * ibukiyama_symbol(3, 2 * k - 3, -k + 1, -k + 2) / Integer(2 ** 2 * 3 ** 3)
)
H5 = ibukiyama_symbol(6, -1, -k + 1, -k + 2, 1, k - 1, k - 2) / Integer(2 ** 2 * 3 ** 2)
if p % 4 == 1:
H6 = 5 * (2 * k - 3) * (p + 1) / Integer(2 ** 7 * 3) + (-1) ** k * (p + 1) / Integer(2 ** 7)
elif p % 4 == 3:
H6 = (2 * k - 3) * (p - 1) / Integer(2 ** 7) + 5 * (-1) ** k * (p - 1) / Integer(2 ** 7 * 3)
else:
H6 = 3 * (2 * k - 3) / Integer(2 ** 7) + 7 * (-1) ** k / Integer(2 ** 7 * 3)
if p % 3 == 1:
H7 = (2 * k - 3) * (p + 1) / Integer(2 * 3 ** 3) + (p + 1) * ibukiyama_symbol(3, 0, -1, 1) / Integer(
2 ** 2 * 3 ** 3
)
elif p % 3 == 2:
H7 = (2 * k - 3) * (p - 1) / Integer(2 ** 2 * 3 ** 3) + (p - 1) * ibukiyama_symbol(
3, 0, -1, 1
) / Integer(2 * 3 ** 3)
else:
H7 = 5 * (2 * k - 3) / Integer(2 ** 2 * 3 ** 3) + ibukiyama_symbol(3, 0, -1, 1) / Integer(3 ** 3)
H8 = ibukiyama_symbol(12, 1, 0, 0, -1, -1, -1, -1, 0, 0, 1, 1, 1) / Integer(2 * 3)
H9 = (
2 * ibukiyama_symbol(6, 1, 0, 0, -1, 0, 0) / Integer(3 ** 2)
if p != 2
else ibukiyama_symbol(6, 1, 0, 0, -1, 0, 0) / Integer(2 * 3 ** 2)
)
H10 = (1 + kronecker_symbol(5, p)) * ibukiyama_symbol(5, 1, 0, 0, -1, 0) / Integer(5)
H11 = (1 + kronecker_symbol(2, p)) * ibukiyama_symbol(4, 1, 0, 0, -1) / Integer(2 ** 3)
if p % 12 == 1:
H12 = ibukiyama_symbol(3, 0, 1, -1) / Integer(2 * 3)
elif p % 12 == 11:
H12 = (-1) ** k / Integer(2 * 3)
elif p == 2 or p == 3:
H12 = (-1) ** k / Integer(2 ** 2 * 3)
else:
H12 = 0
I1 = ibukiyama_symbol(6, 0, 1, 1, 0, -1, -1) / Integer(6)
I2 = ibukiyama_symbol(3, -2, 1, 1) / Integer(2 * 3 ** 2)
if p == 3:
I3 = ibukiyama_symbol(3, -2, 1, 1) / Integer(3 ** 2)
elif p % 3 == 1:
I3 = 2 * ibukiyama_symbol(3, -1, 1, 0) / Integer(3 ** 2)
else:
I3 = 2 * ibukiyama_symbol(3, -1, 0, 1) / Integer(3 ** 2)
I4 = ibukiyama_symbol(4, -1, 1, 1, -1) / Integer(2 ** 2)
I5 = (-1) ** k / Integer(2 ** 3)
I6 = (-1) ** k * (2 - kronecker_symbol(-1, p)) / Integer(2 ** 4)
I7 = -(-1) ** k * (2 * k - 3) / Integer(2 ** 3 * 3)
I8 = -p * (2 * k - 3) / Integer(2 ** 4 * 3 ** 2)
I9 = -1 / Integer(2 ** 3 * 3)
I10 = (p + 1) / Integer(2 ** 3 * 3)
I11 = -(1 + kronecker_symbol(-1, p)) / Integer(8)
I12 = -(1 + kronecker_symbol(-3, p)) / Integer(6)
cuspidal_dim = (
H1
+ H2
+ H3
+ H4
+ H5
+ H6
+ H7
+ H8
+ H9
+ H10
+ H11
+ H12
+ I1
+ I2
+ I3
+ I4
+ I5
+ I6
+ I7
+ I8
+ I9
+ I10
+ I11
+ I12
)
mfs = ModularForms(1, self.__weight)
return cuspidal_dim + mfs.dimension() + mfs.cuspidal_subspace().dimension()
def _siegel_modular_forms_generators(parent, prec=None, degree=0):
r"""
Compute the four Igusa generators of the ring of Siegel modular forms
of degree 2 and level 1 and even weight (this happens if weights='even').
If weights = 'all' you get the Siegel modular forms of degree 2, level 1
and even and odd weight.
EXAMPLES::
sage: A, B, C, D = SiegelModularFormsAlgebra().gens()
sage: C[(2, 0, 9)]
-390420
sage: D[(4, 2, 5)]
17689760
sage: A2, B2, C2, D2 = SiegelModularFormsAlgebra().gens(prec=50)
sage: A[(1, 0, 1)] == A2[(1, 0, 1)]
True
sage: A3, B3, C3, D3 = SiegelModularFormsAlgebra().gens(prec=500)
sage: B2[(2, 1, 3)] == B3[(2, 1, 3)]
True
TESTS::
sage: from sage.modular.siegel.siegel_modular_forms_algebra import _siegel_modular_forms_generators
sage: S = SiegelModularFormsAlgebra()
sage: S.gens() == _siegel_modular_forms_generators(S)
True
"""
group = parent.group()
weights = parent.weights()
if prec is None:
prec = parent.default_prec()
if group == 'Sp(4,Z)' and 0 == degree:
from sage.modular.all import ModularForms
E4 = ModularForms(1, 4).gen(0)
M6 = ModularForms(1, 6)
E6 = ModularForms(1, 6).gen(0)
M8 = ModularForms(1, 8)
M10 = ModularForms(1, 10)
Delta = ModularForms(1, 12).cuspidal_subspace().gen(0)
M14 = ModularForms(1, 14)
A = SiegelModularForm(60 * E4, M6(0), prec=prec, name='Igusa_4')
B = SiegelModularForm(-84 * E6, M8(0), prec=prec, name='Igusa_6')
C = SiegelModularForm(M10(0), -Delta, prec=prec, name='Igusa_10')
D = SiegelModularForm(Delta, M14(0), prec=prec, name='Igusa_12')
# TODO: the base_ring of A, B, ... should be ZZ
# Here a hack for now:
a = [A, B, C, D]
b = []
from sage.rings.all import ZZ
for F in a:
c = F.coeffs()
for f in c.iterkeys():
c[f] = ZZ(c[f])
F = parent.element_class(parent=parent,
weight=F.weight(),
coeffs=c,
prec=prec,
name=F.name())
b.append(F)
if weights == 'even': return b
if weights == 'all':
from fastmult import chi35
coeffs35 = chi35(prec, b[0], b[1], b[2], b[3])
from sage.groups.all import KleinFourGroup
G = KleinFourGroup()
from sage.algebras.all import GroupAlgebra
R = GroupAlgebra(G)
det = R(G.gen(0))
E = parent.element_class(parent=parent,
weight=35,
coeffs=coeffs35,
prec=prec,
name='Delta_35')
E = E * det
b.append(E)
return b
raise ValueError, "weights = '%s': should be 'all' or 'even'" % weights
if group == 'Sp(4,Z)' and weights == 'even' and 2 == degree:
b = _siegel_modular_forms_generators(parent=parent)
c = []
for F in b:
i = b.index(F)
for G in b[(i + 1):]:
c.append(F.satoh_bracket(G))
return c
raise NotImplementedError, "Not yet implemented"