def gritsenko_lift(self, f, k, is_integral=False):
"""
INPUT:
- `f` -- the Fourier expansion of a Jacobi form as a dictionary
"""
N = self.level()
frN = 4 * N
p1list = self.precision()._p1list()
divisor_dict = self._divisor_dict()
##TODO: Precalculate disc_coeffs or at least use a recursive definition
disc_coeffs = dict()
coeffs = dict()
f_index = lambda d, b: ((d + b**2) // frN, b)
for (
((a, b, c), l), eps, disc
) in self.__precision._iter_positive_forms_with_content_and_discriminant(
):
(_, bp, _) = apply_GL_to_form(p1list[l], (a, b, c))
try:
coeffs[((a, b, c), l)] = disc_coeffs[(disc, bp, eps)]
except KeyError:
disc_coeffs[(disc, bp, eps)] = \
sum( t**(k-1) * f[ f_index(disc//t**2, (bp // t) % (2 * N)) ]
for t in divisor_dict[eps] )
if disc_coeffs[(disc, bp, eps)] != 0:
coeffs[((a, b, c), l)] = disc_coeffs[(disc, bp, eps)]
for ((a, b, c), l) in self.__precision.iter_indefinite_forms():
if l == 0:
coeffs[((a, b, c), l)] = (
sigma(c, k - 1) * f[(0, 0)] if c != 0 else
(Integer(-bernoulli(k) / Integer(2 * k) *
f[(0, 0)]) if is_integral else -bernoulli(k) /
Integer(2 * k) * f[(0, 0)]))
else:
coeffs[((a, b, c), l)] = (
sigma(c // self.level(), k - 1) * f[(0, 0)] if c != 0 else
(Integer(-bernoulli(k) / Integer(2 * k) *
f[(0, 0)]) if is_integral else -bernoulli(k) /
Integer(2 * k) * f[(0, 0)]))
return coeffs
def _sz_s( k, m, l, n):
r"""
OUTPUT
The function $s_{k,m}(l,n)$, i.e.~the trace
of $T(l) \circ W_n$ on the "certain" space
$\mathcal{M}_{2k-2}^{\text{cusp}}(m)$ of modular forms
as defined in [S-Z].
INPUT
l -- index of the Hecke operator, rel. prime to the level $m$
n -- index of the Atkin-Lehner involution (exact divisor of $m$)
k -- integer ($2k-2$ is the weight)
m -- level
"""
k = Integer(k)
m = Integer(m)
l = Integer(l)
n = Integer(n)
if k < 2:
return 0
if 1 != m.gcd(l):
raise ValueError, \
"gcd(%s,%s) != 1: not yet implemented" % (m,l)
ellT = 0
for np in n.divisors():
ellT -= sum( l**(k-2) \
* _gegenbauer_pol( k-2, s*s*np/l) \
* hurwitz_kronecker_class_no_x( m/n, s*s*np*np - 4*l*np) \
for s in range( -(Integer(4*l*np).isqrt()//np), \
1+(Integer(4*l*np).isqrt()//np)) \
if (n//np).gcd(s*s).is_squarefree())/2
parT = -sum( min( lp, l//lp)**(2*k-3) \
* _ab(n)[0].gcd( lp+l//lp) \
* _ab(m/n)[0].gcd( lp-l//lp) \
for lp in l.divisors())/2
if 2 == k and (m//n).is_square():
corT = sigma(n,0)*sigma(l,1)
else:
corT = 0
return Integer( ellT + parT + corT)
def gritsenko_lift(self, f, k, is_integral = False) :
"""
INPUT:
- `f` -- the Fourier expansion of a Jacobi form as a dictionary
"""
N = self.level()
frN = 4 * N
p1list = self.precision()._p1list()
divisor_dict = self._divisor_dict()
##TODO: Precalculate disc_coeffs or at least use a recursive definition
disc_coeffs = dict()
coeffs = dict()
f_index = lambda d,b : ((d + b**2)//frN, b)
for (((a,b,c),l), eps, disc) in self.__precision._iter_positive_forms_with_content_and_discriminant() :
(_,bp,_) = apply_GL_to_form(p1list[l], (a,b,c))
try :
coeffs[((a,b,c),l)] = disc_coeffs[(disc, bp, eps)]
except KeyError :
disc_coeffs[(disc, bp, eps)] = \
sum( t**(k-1) * f[ f_index(disc//t**2, (bp // t) % (2 * N)) ]
for t in divisor_dict[eps] )
if disc_coeffs[(disc, bp, eps)] != 0 :
coeffs[((a,b,c),l)] = disc_coeffs[(disc, bp, eps)]
for ((a,b,c), l) in self.__precision.iter_indefinite_forms() :
if l == 0 :
coeffs[((a,b,c),l)] = ( sigma(c, k-1) * f[(0,0)]
if c != 0 else
( Integer(-bernoulli(k) / Integer(2 * k) * f[(0,0)])
if is_integral else
-bernoulli(k) / Integer(2 * k) * f[(0,0)] ) )
else :
coeffs[((a,b,c),l)] = ( sigma(c//self.level(), k-1) * f[(0,0)]
if c != 0 else
( Integer(-bernoulli(k) / Integer(2 * k) * f[(0,0)])
if is_integral else
-bernoulli(k) / Integer(2 * k) * f[(0,0)] ) )
return coeffs
def _cWP(self, d, r):
#\wp(z) & = \frac{1}{12} + \frac{p}{(1-p)^2} + \sum_{k,r \geq 1} k (p^k - 2 + p^{-k}) q^{kr}
if d < 0: return 0
if d == 0:
if r < 0: return 0
elif r == 0: return QQ(1/12)
else: return r
if d > 0:
if r == 0: return -2*sigma(d,1)
elif d % r == 0:
return abs(r)
else:
return 0
def coeff(m):
m = ZZ(m)
if m < 0:
return ZZ(0)
elif m == 0:
return ZZ(1)
factor = -2*k / QQ(bernoulli(k)) / lamk
sum1 = sigma(m, k-1)
if M.divides(m):
sum2 = (lamk-1) * sigma(ZZ(m/M), k-1)
else:
sum2 = ZZ(0)
if (M == 1):
sum3 = ZZ(0)
else:
if (m == 1):
N = ZZ(1)
else:
N = ZZ(m / M**ZZ(m.valuation(M)))
sum3 = -sigma(ZZ(N), k-1) * ZZ(m/N)**(k-1) / (lamk + 1)
return factor * (sum1 + sum2 + sum3) * dval**m
def coeff(m):
m = ZZ(m)
if m < 0:
return ZZ(0)
elif m == 0:
return ZZ(1)
factor = -2 * k / QQ(bernoulli(k)) / lamk
sum1 = sigma(m, k - 1)
if M.divides(m):
sum2 = (lamk - 1) * sigma(ZZ(m / M), k - 1)
else:
sum2 = ZZ(0)
if (M == 1):
sum3 = ZZ(0)
else:
if (m == 1):
N = ZZ(1)
else:
N = ZZ(m / M**ZZ(m.valuation(M)))
sum3 = -sigma(ZZ(N), k - 1) * ZZ(m / N)**(k - 1) / (lamk + 1)
return factor * (sum1 + sum2 + sum3) * dval**m
def maass_form(self, f, g, k=None, is_integral=False):
r"""
Return the Siegel modular form `I(f,g)` (Notation as in [Sko]).
INPUT:
- `f` -- modular form of level `1`
- `g` -- cusp form of level `1` and weight = ``weight of f + 2``
- ``is_integral`` -- ``True`` if the result is garanteed to have integer
coefficients
"""
## we introduce an abbreviations
if is_integral:
PS = self.integral_power_series_ring()
else:
PS = self.power_series_ring()
fismodular = isinstance(f, ModularFormElement)
gismodular = isinstance(g, ModularFormElement)
## We only check the arguments if f and g are ModularFormElements.
## Otherwise we trust in the user
if fismodular and gismodular:
assert( f.weight() + 2 == g.weight() | (f==0) | (g==0)), \
"incorrect weights!"
assert (
g.q_expansion(1) == 0), "second argument is not a cusp form"
qexp_prec = self._get_maass_form_qexp_prec()
if qexp_prec is None: # there are no forms below prec
return dict()
if fismodular:
k = f.weight()
if f == f.parent()(0):
f = PS(0, qexp_prec)
else:
f = PS(f.qexp(qexp_prec), qexp_prec)
elif f == 0:
f = PS(0, qexp_prec)
else:
f = PS(f(qexp_prec), qexp_prec)
if gismodular:
k = g.weight() - 2
if g == g.parent()(0):
g = PS(0, qexp_prec)
else:
g = PS(g.qexp(qexp_prec), qexp_prec)
elif g == 0:
g = PS(0, qexp_prec)
else:
g = PS(g(qexp_prec), qexp_prec)
if k is None:
raise ValueError, "if neither f nor g are not ModularFormElements " + \
"you must pass k"
fderiv = f.derivative().shift(1)
f *= Integer(k / 2)
gfderiv = g - fderiv
## Form A and B - the Jacobi forms used in [Sko]'s I map.
## This is not necessary if we multiply Ifg0 and Ifg1 by etapow
# (A0,A1,B0,B1) = (a0*etapow, a1*etapow, b0*etapow, b1*etapow)
## Calculate the image of the pair of modular forms (f,g) under
## [Sko]'s isomorphism I : M_{k} \oplus S_{k+2} -> J_{k,1}.
# Multiplication of big polynomials may take > 60 GB, so wie have
# to do it in small parts; This is only implemented for integral
# coefficients.
"""
Create the Jacobi form I(f,g) as in [Sko].
It suffices to construct for all Jacobi forms phi only the part
sum_{r=0,1;n} c_phi(r^2-4n) q^n zeta^r.
When, in this code part, we speak of Jacobi form we only mean this part.
We need to compute Ifg = \sum_{r=0,1; n} c(r^2-4n) q^n zeta^r up to
4n-r^2 <= Dtop, i.e. n < precision
"""
## Create the Jacobi forms A=a*etapow and B=b*etapow in stages.
## Recall a = sum_{s != r mod 2} s^2*(-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## b = sum_{s != r mod 2} (-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## r, s run over ZZ but with opposite parities.
## For r=0, we need s odd, (s^2-1)/4 < precision, with s=2t+1 hence t^2+t < precision.
## For r=1, we need s even, s^2/4 < precision, with s=2t hence t^2 < precision.
## we use a slightly overestimated ab_prec
ab_prec = isqrt(qexp_prec + 1)
a1dict = dict()
a0dict = dict()
b1dict = dict()
b0dict = dict()
for t in xrange(1, ab_prec + 1):
tmp = t**2
a1dict[tmp] = -8 * tmp
b1dict[tmp] = -2
tmp += t
a0dict[tmp] = 8 * tmp + 2
b0dict[tmp] = 2
b1dict[0] = -1
a0dict[0] = 2
b0dict[0] = 2
a1 = PS(a1dict)
b1 = PS(b1dict)
a0 = PS(a0dict)
b0 = PS(b0dict)
## Finally: I(f,g) is given by the formula below:
## We multiply by etapow explecitely and save two multiplications
# Ifg0 = k/2*f*A0 - fderiv*B0 + g*B0 + O(q^precision)
# Ifg1 = k/2*f*A1 - fderiv*B1 + g*B1 + O(q^precision)
Ifg0 = (self._eta_power() * (f * a0 + gfderiv * b0)).list()
Ifg1 = (self._eta_power() * (f * a1 + gfderiv * b1)).list()
if len(Ifg0) < qexp_prec:
Ifg0 += [0] * (qexp_prec - len(Ifg0))
if len(Ifg1) < qexp_prec:
Ifg1 += [0] * (qexp_prec - len(Ifg1))
## For applying the Maass' lifting to genus 2 modular forms.
## we put the coefficients of Ifg into a dictionary Chi
## so that we can access the coefficient corresponding to
## discriminant D by going Chi[D].
Cphi = dict([(0, 0)])
for i in xrange(qexp_prec):
Cphi[-4 * i] = Ifg0[i]
Cphi[1 - 4 * i] = Ifg1[i]
del Ifg0[:], Ifg1[:]
"""
Create the Maas lift F := VI(f,g) as in [Sko].
"""
## The constant term is given by -Cphi[0]*B_{2k}/(4*k)
## (note in [Sko] this coeff has typos).
## For nonconstant terms,
## The Siegel coefficient of q^n * zeta^r * qdash^m is given
## by the formula \sum_{ a | gcd(n,r,m) } Cphi[D/a^2] where
## D = r^2-4*n*m is the discriminant.
## Hence in either case the coefficient
## is fully deterimined by the pair (D,gcd(n,r,m)).
## Put (D,t) -> \sum_{ a | t } Cphi[D/a^2]
## in a dictionary (hash table) maassc.
maass_coeffs = dict()
divisor_dict = self._divisor_dict()
## First calculate maass coefficients corresponding to strictly positive definite matrices:
for disc in self._negative_fundamental_discriminants():
for s in xrange(
1,
isqrt((-self.__precision.discriminant()) // disc) + 1):
## add (disc*s^2,t) as a hash key, for each t that divides s
for t in divisor_dict[s]:
maass_coeffs[(disc * s**2,t)] = \
sum( a**(k-1) * Cphi[disc * s**2 / a**2]
for a in divisor_dict[t] )
## Compute the coefficients of the Siegel form $F$:
siegel_coeffs = dict()
for (n, r,
m), g in self.__precision.iter_positive_forms_with_content():
siegel_coeffs[(n, r, m)] = maass_coeffs[(r**2 - 4 * m * n, g)]
## Secondly, deal with the singular part.
## Include the coeff corresponding to (0,0,0):
## maass_coeffs = {(0,0): -bernoulli(k)/(2*k)*Cphi[0]}
siegel_coeffs[(0, 0, 0)] = -bernoulli(k) / (2 * k) * Cphi[0]
if is_integral:
siegel_coeffs[(0, 0, 0)] = Integer(siegel_coeffs[(0, 0, 0)])
## Calculate the other discriminant-zero maass coefficients.
## Since sigma is quite cheap it is faster to estimate the bound and
## save the time for repeated calculation
for i in xrange(1, self.__precision._indefinite_content_bound()):
## maass_coeffs[(0,i)] = sigma(i, k-1) * Cphi[0]
siegel_coeffs[(0, 0, i)] = sigma(i, k - 1) * Cphi[0]
return siegel_coeffs
def additive_lift(self,
forms,
weight,
with_character=False,
is_integral=False):
"""
Borcherds additive lift to hermitian modular forms of
degree `2`. This coinsides with Gritsenko's arithmetic lift after
using the theta decomposition.
INPUT:
- ``forms`` -- A list of functions accepting an integer and
returning a q-expansion.
- ``weight`` -- A positive integer; The weight of the lift.
- ``with_character`` -- A boolean (default: ``False``); Whether the
lift has nontrivial character.
- ``is_integral`` -- A boolean (default: ``False``); If ``True``
use rings of integral q-expansions over `\Z`.
ALGORITHME:
We use the explicite formulas in [D].
TESTS::
sage: from hermitianmodularforms.hermitianmodularformd2_fegenerators import HermitianModularFormD2AdditiveLift
sage: HermitianModularFormD2AdditiveLift(4, [1,0,0], -3, 4).coefficients()
{(2, 3, 2, 2): 720, (1, 1, 1, 1): 27, (1, 0, 0, 2): 270, (3, 3, 3, 3): 2943, (2, 1, 1, 3): 2592, (0, 0, 0, 2): 9, (2, 2, 2, 2): 675, (2, 3, 2, 3): 2160, (1, 1, 1, 2): 216, (3, 0, 0, 3): 8496, (2, 0, 0, 3): 2214, (1, 0, 0, 3): 720, (2, 1, 1, 2): 1080, (0, 0, 0, 1): 1, (3, 3, 2, 3): 4590, (3, 1, 1, 3): 4590, (1, 1, 1, 3): 459, (2, 0, 0, 2): 1512, (1, 0, 0, 1): 72, (0, 0, 0, 0): 1/240, (3, 4, 3, 3): 2808, (0, 0, 0, 3): 28, (3, 2, 2, 3): 4752, (2, 2, 2, 3): 1350}
sage: HermitianModularFormD2AdditiveLift(4, [0,1,0], -3, 6).coefficients()
{(2, 3, 2, 2): -19680, (1, 1, 1, 1): -45, (1, 0, 0, 2): -3690, (3, 3, 3, 3): -306225, (2, 1, 1, 3): -250560, (0, 0, 0, 2): 33, (2, 2, 2, 2): -13005, (2, 3, 2, 3): -153504, (1, 1, 1, 2): -1872, (3, 0, 0, 3): -1652640, (2, 0, 0, 3): -295290, (1, 0, 0, 3): -19680, (2, 1, 1, 2): -43920, (0, 0, 0, 1): 1, (3, 3, 2, 3): -948330, (3, 1, 1, 3): -1285290, (1, 1, 1, 3): -11565, (2, 0, 0, 2): -65520, (1, 0, 0, 1): -240, (0, 0, 0, 0): -1/504, (3, 4, 3, 3): -451152, (0, 0, 0, 3): 244, (3, 2, 2, 3): -839520, (2, 2, 2, 3): -108090}
"""
if with_character and self.__D % 4 != 0:
raise ValueError(
"Characters are only possible for even discriminants.")
## This will be needed if characters are implemented
if with_character:
if (Integer(self.__D / 4) % 4) in [-2, 2]:
alpha = (-self.__D / 4, 1 / 2)
else:
alpha = (-self.__D / 8, 1 / 2)
#minv = 1/2 if with_character else 1
R = self.power_series_ring()
q = R.gen(0)
(vv_expfactor, vv_basis) = self._additive_lift_vector_valued_basis()
vvform = dict(
(self._reduce_vector_valued_index(k), R(0)) for (k, _) in self.
_semireduced_vector_valued_indices_with_discriminant_offset(1))
for (f, b) in zip(forms, vv_basis):
## We have to apply the scaling of exponents to the form
f = R( f(self._qexp_precision()) ).add_bigoh(self._qexp_precision()) \
.subs({q : q**vv_expfactor})
if not f.is_zero():
for (k, e) in b.iteritems():
vvform[k] = vvform[k] + e * f
## the T = matrix(2,[*, t / 2, \bar t / 2, *] th fourier coefficients of the lift
## only depends on (- 4 * D * det(T), eps = gcd(T), \theta \cong t / eps)
## if m != 1 we consider 2*T
maass_coeffs = dict()
## TODO: use divisor dictionaries
if not with_character:
## The factor for the exponent of the basis of vector valued forms
## and the factor D in the formula for the discriminant are combined
## here
vv_expfactor = vv_expfactor // (-self.__D)
for eps in self._iterator_content():
for (
theta, offset
) in self._semireduced_vector_valued_indices_with_discriminant_offset(
eps):
for disc in self._iterator_discriminant(eps, offset):
maass_coeffs[(disc, eps, theta)] = \
sum( a**(weight-1) *
vvform[self._reduce_vector_valued_index((theta[0]//a, theta[1]//a))][vv_expfactor * disc // a**2]
for a in divisors(eps))
else:
## The factor for the exponent of the basis of vector valued forms
## and the factor D in the formula for the discriminant are combined
## here
vv_expfactor = (2 * vv_expfactor) // (-self.__D)
if self.__D // 4 % 2 == 0:
for eps in self._iterator_content():
for (
theta, offset
) in self._semireduced_vector_valued_indices_with_discriminant_offset(
eps):
for disc in self._iter_discriminant(eps, offset):
maass_coeffs[(disc, eps, theta)] = \
sum( a**(weight-1) * (1 if (theta[0] + theta[1] - 1) % 4 == 0 else -1) *
vvform[self._reduce_vector_valued_index((theta[0]//a, theta[1]//a))][vv_expfactor * disc // a**2]
for a in divisors(eps))
else:
for eps in self._iterator_content():
for (
theta, offset
) in self._semireduced_vector_valued_indices_with_discriminant_offset(
eps):
for disc in self._iter_discriminant(eps, offset):
maass_coeffs[(disc, eps, theta)] = \
sum( a**(weight-1) * (1 if (theta[1] - 1) % 4 == 0 else -1) *
vvform[self._reduce_vector_valued_index((theta[0]//a, theta[1]//a))][vv_expfactor * disc // a**2]
for a in divisors(eps) )
lift_coeffs = dict()
## TODO: Check whether this is correct. Add the character as an argument.
for ((a, b1, b2, c), eps, disc) in self.precision(
).iter_positive_forms_for_character_with_content_and_discriminant(
for_character=with_character):
(theta1, theta2) = self._reduce_vector_valued_index(
(b1 / eps, b2 / eps))
theta = (eps * theta1, eps * theta2)
try:
lift_coeffs[(a, b1, b2, c)] = maass_coeffs[(disc, eps, theta)]
except:
raise RuntimeError(
str((a, b1, b2, c)) + " ; " + str((disc, eps, theta)))
# Eisenstein component
for (_, _, _,
c) in self.precision().iter_semidefinite_forms_for_character(
for_character=with_character):
if c != 0:
lift_coeffs[(0, 0, 0,
c)] = vvform[(0, 0)][0] * sigma(c, weight - 1)
lift_coeffs[(
0, 0, 0,
0)] = -vvform[(0, 0)][0] * bernoulli(weight) / Integer(2 * weight)
if is_integral:
lift_coeffs[(0, 0, 0, 0)] = ZZ(lift_coeffs[(0, 0, 0, 0)])
return lift_coeffs
def maass_form( self, f, g, k = None, is_integral = False) :
r"""
Return the Siegel modular form `I(f,g)` (Notation as in [Sko]).
INPUT:
- `f` -- modular form of level `1`
- `g` -- cusp form of level `1` and weight = ``weight of f + 2``
- ``is_integral`` -- ``True`` if the result is garanteed to have integer
coefficients
"""
## we introduce an abbreviations
if is_integral :
PS = self.integral_power_series_ring()
else :
PS = self.power_series_ring()
fismodular = isinstance(f, ModularFormElement)
gismodular = isinstance(g, ModularFormElement)
## We only check the arguments if f and g are ModularFormElements.
## Otherwise we trust in the user
if fismodular and gismodular :
assert( f.weight() + 2 == g.weight() | (f==0) | (g==0)), \
"incorrect weights!"
assert( g.q_expansion(1) == 0), "second argument is not a cusp form"
qexp_prec = self._get_maass_form_qexp_prec()
if qexp_prec is None : # there are no forms below prec
return dict()
if fismodular :
k = f.weight()
if f == f.parent()(0) :
f = PS(0, qexp_prec)
else :
f = PS(f.qexp(qexp_prec), qexp_prec)
elif f == 0 :
f = PS(0, qexp_prec)
else :
f = PS(f(qexp_prec), qexp_prec)
if gismodular :
k = g.weight() - 2
if g == g.parent()(0) :
g = PS(0, qexp_prec)
else :
g = PS(g.qexp(qexp_prec), qexp_prec)
elif g == 0 :
g = PS(0, qexp_prec)
else :
g = PS(g(qexp_prec), qexp_prec)
if k is None :
raise ValueError, "if neither f nor g are not ModularFormElements " + \
"you must pass k"
fderiv = f.derivative().shift(1)
f *= Integer(k/2)
gfderiv = g - fderiv
## Form A and B - the Jacobi forms used in [Sko]'s I map.
## This is not necessary if we multiply Ifg0 and Ifg1 by etapow
# (A0,A1,B0,B1) = (a0*etapow, a1*etapow, b0*etapow, b1*etapow)
## Calculate the image of the pair of modular forms (f,g) under
## [Sko]'s isomorphism I : M_{k} \oplus S_{k+2} -> J_{k,1}.
# Multiplication of big polynomials may take > 60 GB, so wie have
# to do it in small parts; This is only implemented for integral
# coefficients.
"""
Create the Jacobi form I(f,g) as in [Sko].
It suffices to construct for all Jacobi forms phi only the part
sum_{r=0,1;n} c_phi(r^2-4n) q^n zeta^r.
When, in this code part, we speak of Jacobi form we only mean this part.
We need to compute Ifg = \sum_{r=0,1; n} c(r^2-4n) q^n zeta^r up to
4n-r^2 <= Dtop, i.e. n < precision
"""
## Create the Jacobi forms A=a*etapow and B=b*etapow in stages.
## Recall a = sum_{s != r mod 2} s^2*(-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## b = sum_{s != r mod 2} (-1)^r*q^((s^2+r^2-1)/4)*zeta^r
## r, s run over ZZ but with opposite parities.
## For r=0, we need s odd, (s^2-1)/4 < precision, with s=2t+1 hence t^2+t < precision.
## For r=1, we need s even, s^2/4 < precision, with s=2t hence t^2 < precision.
## we use a slightly overestimated ab_prec
ab_prec = isqrt(qexp_prec + 1)
a1dict = dict(); a0dict = dict()
b1dict = dict(); b0dict = dict()
for t in xrange(1, ab_prec + 1) :
tmp = t**2
a1dict[tmp] = -8*tmp
b1dict[tmp] = -2
tmp += t
a0dict[tmp] = 8*tmp + 2
b0dict[tmp] = 2
b1dict[0] = -1
a0dict[0] = 2; b0dict[0] = 2
a1 = PS(a1dict); b1 = PS(b1dict)
a0 = PS(a0dict); b0 = PS(b0dict)
## Finally: I(f,g) is given by the formula below:
## We multiply by etapow explecitely and save two multiplications
# Ifg0 = k/2*f*A0 - fderiv*B0 + g*B0 + O(q^precision)
# Ifg1 = k/2*f*A1 - fderiv*B1 + g*B1 + O(q^precision)
Ifg0 = (self._eta_power() * (f*a0 + gfderiv*b0)).list()
Ifg1 = (self._eta_power() * (f*a1 + gfderiv*b1)).list()
if len(Ifg0) < qexp_prec :
Ifg0 += [0]*(qexp_prec - len(Ifg0))
if len(Ifg1) < qexp_prec :
Ifg1 += [0]*(qexp_prec - len(Ifg1))
## For applying the Maass' lifting to genus 2 modular forms.
## we put the coefficients of Ifg into a dictionary Chi
## so that we can access the coefficient corresponding to
## discriminant D by going Chi[D].
Cphi = dict([(0,0)])
for i in xrange(qexp_prec) :
Cphi[-4*i] = Ifg0[i]
Cphi[1-4*i] = Ifg1[i]
del Ifg0[:], Ifg1[:]
"""
Create the Maas lift F := VI(f,g) as in [Sko].
"""
## The constant term is given by -Cphi[0]*B_{2k}/(4*k)
## (note in [Sko] this coeff has typos).
## For nonconstant terms,
## The Siegel coefficient of q^n * zeta^r * qdash^m is given
## by the formula \sum_{ a | gcd(n,r,m) } Cphi[D/a^2] where
## D = r^2-4*n*m is the discriminant.
## Hence in either case the coefficient
## is fully deterimined by the pair (D,gcd(n,r,m)).
## Put (D,t) -> \sum_{ a | t } Cphi[D/a^2]
## in a dictionary (hash table) maassc.
maass_coeffs = dict()
divisor_dict = self._divisor_dict()
## First calculate maass coefficients corresponding to strictly positive definite matrices:
for disc in self._negative_fundamental_discriminants() :
for s in xrange(1, isqrt((-self.__precision.discriminant()) // disc) + 1) :
## add (disc*s^2,t) as a hash key, for each t that divides s
for t in divisor_dict[s] :
maass_coeffs[(disc * s**2,t)] = \
sum( a**(k-1) * Cphi[disc * s**2 / a**2]
for a in divisor_dict[t] )
## Compute the coefficients of the Siegel form $F$:
siegel_coeffs = dict()
for (n,r,m), g in self.__precision.iter_positive_forms_with_content() :
siegel_coeffs[(n,r,m)] = maass_coeffs[(r**2 - 4*m*n, g)]
## Secondly, deal with the singular part.
## Include the coeff corresponding to (0,0,0):
## maass_coeffs = {(0,0): -bernoulli(k)/(2*k)*Cphi[0]}
siegel_coeffs[(0,0,0)] = -bernoulli(k)/(2*k)*Cphi[0]
if is_integral :
siegel_coeffs[(0,0,0)] = Integer(siegel_coeffs[(0,0,0)])
## Calculate the other discriminant-zero maass coefficients.
## Since sigma is quite cheap it is faster to estimate the bound and
## save the time for repeated calculation
for i in xrange(1, self.__precision._indefinite_content_bound()) :
## maass_coeffs[(0,i)] = sigma(i, k-1) * Cphi[0]
siegel_coeffs[(0,0,i)] = sigma(i, k-1) * Cphi[0]
return siegel_coeffs
def additive_lift(self, forms, weight, with_character = False, is_integral = False) :
"""
Borcherds additive lift to hermitian modular forms of
degree `2`. This coinsides with Gritsenko's arithmetic lift after
using the theta decomposition.
INPUT:
- ``forms`` -- A list of functions accepting an integer and
returning a q-expansion.
- ``weight`` -- A positive integer; The weight of the lift.
- ``with_character`` -- A boolean (default: ``False``); Whether the
lift has nontrivial character.
- ``is_integral`` -- A boolean (default: ``False``); If ``True``
use rings of integral q-expansions over `\Z`.
ALGORITHME:
We use the explicite formulas in [D].
TESTS::
sage: from hermitianmodularforms.hermitianmodularformd2_fegenerators import HermitianModularFormD2AdditiveLift
sage: HermitianModularFormD2AdditiveLift(4, [1,0,0], -3, 4).coefficients()
{(2, 3, 2, 2): 720, (1, 1, 1, 1): 27, (1, 0, 0, 2): 270, (3, 3, 3, 3): 2943, (2, 1, 1, 3): 2592, (0, 0, 0, 2): 9, (2, 2, 2, 2): 675, (2, 3, 2, 3): 2160, (1, 1, 1, 2): 216, (3, 0, 0, 3): 8496, (2, 0, 0, 3): 2214, (1, 0, 0, 3): 720, (2, 1, 1, 2): 1080, (0, 0, 0, 1): 1, (3, 3, 2, 3): 4590, (3, 1, 1, 3): 4590, (1, 1, 1, 3): 459, (2, 0, 0, 2): 1512, (1, 0, 0, 1): 72, (0, 0, 0, 0): 1/240, (3, 4, 3, 3): 2808, (0, 0, 0, 3): 28, (3, 2, 2, 3): 4752, (2, 2, 2, 3): 1350}
sage: HermitianModularFormD2AdditiveLift(4, [0,1,0], -3, 6).coefficients()
{(2, 3, 2, 2): -19680, (1, 1, 1, 1): -45, (1, 0, 0, 2): -3690, (3, 3, 3, 3): -306225, (2, 1, 1, 3): -250560, (0, 0, 0, 2): 33, (2, 2, 2, 2): -13005, (2, 3, 2, 3): -153504, (1, 1, 1, 2): -1872, (3, 0, 0, 3): -1652640, (2, 0, 0, 3): -295290, (1, 0, 0, 3): -19680, (2, 1, 1, 2): -43920, (0, 0, 0, 1): 1, (3, 3, 2, 3): -948330, (3, 1, 1, 3): -1285290, (1, 1, 1, 3): -11565, (2, 0, 0, 2): -65520, (1, 0, 0, 1): -240, (0, 0, 0, 0): -1/504, (3, 4, 3, 3): -451152, (0, 0, 0, 3): 244, (3, 2, 2, 3): -839520, (2, 2, 2, 3): -108090}
"""
if with_character and self.__D % 4 != 0 :
raise ValueError( "Characters are only possible for even discriminants." )
## This will be needed if characters are implemented
if with_character :
if (Integer(self.__D / 4) % 4) in [-2,2] :
alpha = (-self.__D / 4, 1/2)
else :
alpha = (-self.__D / 8, 1/2)
#minv = 1/2 if with_character else 1
R = self.power_series_ring()
q = R.gen(0)
(vv_expfactor, vv_basis) = self._additive_lift_vector_valued_basis()
vvform = dict((self._reduce_vector_valued_index(k), R(0)) for (k,_) in self._semireduced_vector_valued_indices_with_discriminant_offset(1))
for (f,b) in zip(forms, vv_basis) :
## We have to apply the scaling of exponents to the form
f = R( f(self._qexp_precision()) ).add_bigoh(self._qexp_precision()) \
.subs({q : q**vv_expfactor})
if not f.is_zero() :
for (k,e) in b.iteritems() :
vvform[k] = vvform[k] + e * f
## the T = matrix(2,[*, t / 2, \bar t / 2, *] th fourier coefficients of the lift
## only depends on (- 4 * D * det(T), eps = gcd(T), \theta \cong t / eps)
## if m != 1 we consider 2*T
maass_coeffs = dict()
## TODO: use divisor dictionaries
if not with_character :
## The factor for the exponent of the basis of vector valued forms
## and the factor D in the formula for the discriminant are combined
## here
vv_expfactor = vv_expfactor // (- self.__D)
for eps in self._iterator_content() :
for (theta, offset) in self._semireduced_vector_valued_indices_with_discriminant_offset(eps) :
for disc in self._iterator_discriminant(eps, offset) :
maass_coeffs[(disc, eps, theta)] = \
sum( a**(weight-1) *
vvform[self._reduce_vector_valued_index((theta[0]//a, theta[1]//a))][vv_expfactor * disc // a**2]
for a in divisors(eps))
else :
## The factor for the exponent of the basis of vector valued forms
## and the factor D in the formula for the discriminant are combined
## here
vv_expfactor = (2 * vv_expfactor) // (- self.__D)
if self.__D // 4 % 2 == 0 :
for eps in self._iterator_content() :
for (theta, offset) in self._semireduced_vector_valued_indices_with_discriminant_offset(eps) :
for disc in self._iter_discriminant(eps, offset) :
maass_coeffs[(disc, eps, theta)] = \
sum( a**(weight-1) * (1 if (theta[0] + theta[1] - 1) % 4 == 0 else -1) *
vvform[self._reduce_vector_valued_index((theta[0]//a, theta[1]//a))][vv_expfactor * disc // a**2]
for a in divisors(eps))
else :
for eps in self._iterator_content() :
for (theta, offset) in self._semireduced_vector_valued_indices_with_discriminant_offset(eps) :
for disc in self._iter_discriminant(eps, offset) :
maass_coeffs[(disc, eps, theta)] = \
sum( a**(weight-1) * (1 if (theta[1] - 1) % 4 == 0 else -1) *
vvform[self._reduce_vector_valued_index((theta[0]//a, theta[1]//a))][vv_expfactor * disc // a**2]
for a in divisors(eps) )
lift_coeffs = dict()
## TODO: Check whether this is correct. Add the character as an argument.
for ((a,b1,b2,c), eps, disc) in self.precision().iter_positive_forms_for_character_with_content_and_discriminant(for_character = with_character) :
(theta1, theta2) = self._reduce_vector_valued_index((b1/eps, b2/eps))
theta = (eps * theta1, eps * theta2)
try:
lift_coeffs[(a,b1,b2,c)] = maass_coeffs[(disc, eps, theta)]
except :
raise RuntimeError(str((a,b1,b2,c)) + " ; " + str((disc, eps, theta)))
# Eisenstein component
for (_,_,_,c) in self.precision().iter_semidefinite_forms_for_character(for_character = with_character) :
if c != 0 :
lift_coeffs[(0,0,0,c)] = vvform[(0,0)][0] * sigma(c, weight - 1)
lift_coeffs[(0,0,0,0)] = - vvform[(0,0)][0] * bernoulli(weight) / Integer(2 * weight)
if is_integral :
lift_coeffs[(0,0,0,0)] = ZZ(lift_coeffs[(0,0,0,0)])
return lift_coeffs
