def zeta__exact(n):
r"""
Returns the exact value of the Riemann Zeta function
The argument must be a critical value, namely either positive even
or negative odd.
See for example [Iwasawa]_, p13, Special value of `\zeta(2k)`
EXAMPLES:
Let us test the accuracy for negative special values::
sage: RR = RealField(100)
sage: for i in range(1,10):
... print "zeta(" + str(1-2*i) + "): ", RR(zeta__exact(1-2*i)) - zeta(RR(1-2*i))
zeta(-1): 0.00000000000000000000000000000
zeta(-3): 0.00000000000000000000000000000
zeta(-5): 0.00000000000000000000000000000
zeta(-7): 0.00000000000000000000000000000
zeta(-9): 0.00000000000000000000000000000
zeta(-11): 0.00000000000000000000000000000
zeta(-13): 0.00000000000000000000000000000
zeta(-15): 0.00000000000000000000000000000
zeta(-17): 0.00000000000000000000000000000
Let us test the accuracy for positive special values::
sage: all(abs(RR(zeta__exact(2*i))-zeta(RR(2*i))) < 10**(-28) for i in range(1,10))
True
TESTS::
sage: zeta__exact(5)
Traceback (most recent call last):
...
TypeError: n must be a critical value (i.e. even > 0 or odd < 0)
REFERENCES:
.. [Iwasawa] Iwasawa, *Lectures on p-adic L-functions*
.. [IreRos] Ireland and Rosen, *A Classical Introduction to Modern Number Theory*
.. [WashCyc] Washington, *Cyclotomic Fields*
"""
if n < 0:
return bernoulli(1 - n) / (n - 1)
elif n > 1:
if (n % 2 == 0):
return ZZ(-1)**(n / 2 + 1) * ZZ(2)**(
n - 1) * pi**n * bernoulli(n) / factorial(n)
else:
raise TypeError(
"n must be a critical value (i.e. even > 0 or odd < 0)")
elif n == 1:
return infinity
elif n == 0:
return -1 / 2
def zeta__exact(n):
r"""
Returns the exact value of the Riemann Zeta function
The argument must be a critical value, namely either positive even
or negative odd.
See for example [Iwasawa]_, p13, Special value of `\zeta(2k)`
EXAMPLES:
Let us test the accuracy for negative special values::
sage: RR = RealField(100)
sage: for i in range(1,10):
... print "zeta(" + str(1-2*i) + "): ", RR(zeta__exact(1-2*i)) - zeta(RR(1-2*i))
zeta(-1): 0.00000000000000000000000000000
zeta(-3): 0.00000000000000000000000000000
zeta(-5): 0.00000000000000000000000000000
zeta(-7): 0.00000000000000000000000000000
zeta(-9): 0.00000000000000000000000000000
zeta(-11): 0.00000000000000000000000000000
zeta(-13): 0.00000000000000000000000000000
zeta(-15): 0.00000000000000000000000000000
zeta(-17): 0.00000000000000000000000000000
Let us test the accuracy for positive special values::
sage: all(abs(RR(zeta__exact(2*i))-zeta(RR(2*i))) < 10**(-28) for i in range(1,10))
True
TESTS::
sage: zeta__exact(5)
Traceback (most recent call last):
...
TypeError: n must be a critical value (i.e. even > 0 or odd < 0)
REFERENCES:
.. [Iwasawa] Iwasawa, *Lectures on p-adic L-functions*
.. [IreRos] Ireland and Rosen, *A Classical Introduction to Modern Number Theory*
.. [WashCyc] Washington, *Cyclotomic Fields*
"""
if n < 0:
return bernoulli(1-n)/(n-1)
elif n > 1:
if (n % 2 == 0):
return ZZ(-1)**(n/2 + 1) * ZZ(2)**(n-1) * pi**n * bernoulli(n) / factorial(n)
else:
raise TypeError("n must be a critical value (i.e. even > 0 or odd < 0)")
elif n==1:
return infinity
elif n==0:
return -1/2
def zeta__exact(n):
r"""
Returns the exact value of the Riemann Zeta function
References:
- Iwasawa's "Lectures on p-adic L-functions", p13, "Special value of `\zeta(2k)`"
- Ireland and Rosen's "A Classical Introduction to Modern Number Theory"
- Washington's "Cyclotomic Fields"
EXAMPLES::
sage: ## Testing the accuracy of the negative special values
sage: RR = RealField(100)
sage: for i in range(1,10):
... print "zeta(" + str(1-2*i) + "): ", RR(zeta__exact(1-2*i)) - zeta(RR(1-2*i))
zeta(-1): 0.00000000000000000000000000000
zeta(-3): 0.00000000000000000000000000000
zeta(-5): 0.00000000000000000000000000000
zeta(-7): 0.00000000000000000000000000000
zeta(-9): 0.00000000000000000000000000000
zeta(-11): 0.00000000000000000000000000000
zeta(-13): 0.00000000000000000000000000000
zeta(-15): 0.00000000000000000000000000000
zeta(-17): 0.00000000000000000000000000000
sage: RR = RealField(100)
sage: for i in range(1,10):
... print "zeta(" + str(1-2*i) + "): ", RR(zeta__exact(1-2*i)) - zeta(RR(1-2*i))
zeta(-1): 0.00000000000000000000000000000
zeta(-3): 0.00000000000000000000000000000
zeta(-5): 0.00000000000000000000000000000
zeta(-7): 0.00000000000000000000000000000
zeta(-9): 0.00000000000000000000000000000
zeta(-11): 0.00000000000000000000000000000
zeta(-13): 0.00000000000000000000000000000
zeta(-15): 0.00000000000000000000000000000
zeta(-17): 0.00000000000000000000000000000
"""
if n < 0:
k = 1 - n
return -bernoulli(k) / k
elif n > 1:
if (n % 2 == 0):
return ZZ(-1)**(n / 2 + 1) * ZZ(2)**(
n - 1) * pi**n * bernoulli(n) / factorial(n)
else:
raise TypeError, "n must be a critical value! (I.e. even > 0 or odd < 0.)"
elif n == 1:
return infinity
elif n == 0:
return -1 / 2
def zeta__exact(n):
r"""
Returns the exact value of the Riemann Zeta function
References:
- Iwasawa's "Lectures on p-adic L-functions", p13, "Special value of `\zeta(2k)`"
- Ireland and Rosen's "A Classical Introduction to Modern Number Theory"
- Washington's "Cyclotomic Fields"
EXAMPLES::
sage: ## Testing the accuracy of the negative special values
sage: RR = RealField(100)
sage: for i in range(1,10):
... print "zeta(" + str(1-2*i) + "): ", RR(zeta__exact(1-2*i)) - zeta(RR(1-2*i))
zeta(-1): 0.00000000000000000000000000000
zeta(-3): 0.00000000000000000000000000000
zeta(-5): 0.00000000000000000000000000000
zeta(-7): 0.00000000000000000000000000000
zeta(-9): 0.00000000000000000000000000000
zeta(-11): 0.00000000000000000000000000000
zeta(-13): 0.00000000000000000000000000000
zeta(-15): 0.00000000000000000000000000000
zeta(-17): 0.00000000000000000000000000000
sage: RR = RealField(100)
sage: for i in range(1,10):
... print "zeta(" + str(1-2*i) + "): ", RR(zeta__exact(1-2*i)) - zeta(RR(1-2*i))
zeta(-1): 0.00000000000000000000000000000
zeta(-3): 0.00000000000000000000000000000
zeta(-5): 0.00000000000000000000000000000
zeta(-7): 0.00000000000000000000000000000
zeta(-9): 0.00000000000000000000000000000
zeta(-11): 0.00000000000000000000000000000
zeta(-13): 0.00000000000000000000000000000
zeta(-15): 0.00000000000000000000000000000
zeta(-17): 0.00000000000000000000000000000
"""
if n<0:
k = 1-n
return -bernoulli(k)/k
elif n>1:
if (n % 2 == 0):
return ZZ(-1)**(n/2 + 1) * ZZ(2)**(n-1) * pi**n * bernoulli(n) / factorial(n)
else:
raise TypeError, "n must be a critical value! (I.e. even > 0 or odd < 0.)"
elif n==1:
return infinity
elif n==0:
return -1/2
def gritsenko_lift_subspace(N, weight, precision):
"""
Return a list of data, which can be used by the Fourier expansion
generator, for the space of Gritsenko lifts.
INPUT:
- `N` -- the level of the paramodular group
- ``weight`` -- the weight of the space, that we consider
- ``precision`` -- A precision class
NOTE:
The lifts returned have to have integral Fourier coefficients.
"""
jf = JacobiFormsD1NN(QQ, JacobiFormD1NN_Gamma(
N,
weight), (4 * N**2 + precision._enveloping_discriminant_bound() - 1) //
(4 * N) + 1)
return Sequence([
gritsenko_lift_fourier_expansion(
g if i != 0 else (bernoulli(weight) /
(2 * weight)).denominator() * g, precision, True)
for (i, g) in enumerate(jf.gens())
],
universe=ParamodularFormD2FourierExpansionRing(ZZ, N),
immutable=True,
check=False)
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 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 gritsenko_lift_subspace(N, weight, precision) :
"""
Return a list of data, which can be used by the Fourier expansion
generator, for the space of Gritsenko lifts.
INPUT:
- `N` -- the level of the paramodular group
- ``weight`` -- the weight of the space, that we consider
- ``precision`` -- A precision class
NOTE:
The lifts returned have to have integral Fourier coefficients.
"""
jf = JacobiFormsD1NN( QQ, JacobiFormD1NN_Gamma(N, weight),
(4 * N**2 + precision._enveloping_discriminant_bound() - 1)//(4 * N) + 1)
return Sequence( [ gritsenko_lift_fourier_expansion( g if i != 0 else (bernoulli(weight) / (2 * weight)).denominator() * g,
precision, True )
for (i,g) in enumerate(jf.gens()) ],
universe = ParamodularFormD2FourierExpansionRing(ZZ, N), immutable = True,
check = False )
def __maass_lifts(self, k, precision, return_value) :
r"""
Return the Fourier expansion of all Maass forms of weight `k`.
"""
result = []
if k < 4 or k % 2 != 0 :
return []
mf = ModularForms(1,k).echelon_basis()
cf = ModularForms(1,k + 2).echelon_basis()[1:]
integrality_factor = 2*k * bernoulli(k).denominator()
for c in [(integrality_factor * mf[0],0)] \
+ [ (f,0) for f in mf[1:] ] + [ (0,g) for g in cf ] :
if return_value == "lifts" :
result.append(SiegelModularFormG2MaassLift(c[0],c[1], precision, True))
else :
result.append(c)
return result
def __maass_lifts(self, k, precision, return_value):
r"""
Return the Fourier expansion of all Maass forms of weight `k`.
"""
result = []
if k < 4 or k % 2 != 0:
return []
mf = ModularForms(1, k).echelon_basis()
cf = ModularForms(1, k + 2).echelon_basis()[1:]
integrality_factor = 2 * k * bernoulli(k).denominator()
for c in [(integrality_factor * mf[0],0)] \
+ [ (f,0) for f in mf[1:] ] + [ (0,g) for g in cf ] :
if return_value == "lifts":
result.append(
SiegelModularFormG2MaassLift(c[0], c[1], precision, True))
else:
result.append(c)
return result
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 eta_char(i,p,k,j,M): v=[0 for j in range(0,i)]+[binomial(j,i)*bernoulli(j-i) for j in range(i,M)] return dist_char(p,k,j,v)
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 eta(i,p,k,M): """helper function in solving the difference equation -- see Lemma 4.4 of [PS]""" v=[0 for j in range(0,i)]+[binomial(j,i)*bernoulli(j-i) for j in range(i,M)] return dist(p,k,vector(v))
def siegel_product(self, u):
"""
Computes the infinite product of local densities of the quadratic
form for the number `u`.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.theta_series(11)
1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + 104*q^9 + 144*q^10 + O(q^11)
sage: Q.siegel_product(1)
8
sage: Q.siegel_product(2) ## This one is wrong -- expect 24, and the higher powers of 2 don't work... =(
24
sage: Q.siegel_product(3)
32
sage: Q.siegel_product(5)
48
sage: Q.siegel_product(6)
96
sage: Q.siegel_product(7)
64
sage: Q.siegel_product(9)
104
sage: Q.local_density(2,1)
1
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3
1
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3 # long time (41s on sage.math, 2011)
1
sage: Q.local_density(2,2)
3/2
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3
3/2
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3 # long time (41s on sage.math, 2011)
3/2
TESTS::
sage: [1] + [Q.siegel_product(ZZ(a)) for a in range(1,11)] == Q.theta_series(11).list()
True
"""
## Protect u (since it fails often if it's an just an int!)
u = ZZ(u)
n = self.dim()
d = self.det() ## ??? Warning: This is a factor of 2^n larger than it should be!
## DIAGNOSTIC
verbose("n = " + str(n))
verbose("d = " + str(d))
verbose("In siegel_product: d = ", d, "\n");
## Product of "bad" places to omit
S = 2 * d * u
## DIAGNOSTIC
verbose("siegel_product Break 1. \n")
verbose(" u = ", u, "\n")
## Make the odd generic factors
if ((n % 2) == 1):
m = (n-1) / 2
d1 = fundamental_discriminant(((-1)**m) * 2*d * u) ## Replaced d by 2d here to compensate for the determinant
f = abs(d1) ## gaining an odd power of 2 by using the matrix of 2Q instead
## of the matrix of Q.
## --> Old d1 = CoreDiscriminant((mpz_class(-1)^m) * d * u);
## Make the ratio of factorials factor: [(2m)! / m!] * prod_{i=1}^m (2*i-1)
factor1 = 1
for i in range(1, m+1):
factor1 *= 2*i - 1
for i in range(m+1, 2*m + 1):
factor1 *= i
genericfactor = factor1 * ((u / f) ** m) \
* QQ(sqrt((2 ** n) * f) / (u * d)) \
* abs(QuadraticBernoulliNumber(m, d1) / bernoulli(2*m))
## DIAGNOSTIC
verbose("siegel_product Break 2. \n")
## Make the even generic factor
if ((n % 2) == 0):
m = n / 2
d1 = fundamental_discriminant(((-1)**m) * d)
f = abs(d1)
## DIAGNOSTIC
#cout << " mpz_class(-1)^m = " << (mpz_class(-1)^m) << " and d = " << d << endl;
#cout << " f = " << f << " and d1 = " << d1 << endl;
genericfactor = m / QQ(sqrt(f*d)) \
* ((u/2) ** (m-1)) * (f ** m) \
/ abs(QuadraticBernoulliNumber(m, d1)) \
* (2 ** m) ## This last factor compensates for using the matrix of 2*Q
##return genericfactor
## Omit the generic factors in S and compute them separately
omit = 1
include = 1
S_divisors = prime_divisors(S)
## DIAGNOSTIC
#cout << "\n S is " << S << endl;
#cout << " The Prime divisors of S are :";
#PrintV(S_divisors);
for p in S_divisors:
Q_normal = self.local_normal_form(p)
## DIAGNOSTIC
verbose(" p = " + str(p) + " and its Kronecker symbol (d1/p) = (" + str(d1) + "/" + str(p) + ") is " + str(kronecker_symbol(d1, p)) + "\n")
omit *= 1 / (1 - (kronecker_symbol(d1, p) / (p**m)))
## DIAGNOSTIC
verbose(" omit = " + str(omit) + "\n")
verbose(" Q_normal is \n" + str(Q_normal) + "\n")
verbose(" Q_normal = \n" + str(Q_normal))
verbose(" p = " + str(p) + "\n")
verbose(" u = " +str(u) + "\n")
verbose(" include = " + str(include) + "\n")
include *= Q_normal.local_density(p, u)
## DIAGNOSTIC
#cout << " Including the p = " << p << " factor: " << local_density(Q_normal, p, u) << endl;
## DIAGNSOTIC
verbose(" --- Exiting loop \n")
#// **************** Important *******************
#// Additional fix (only included for n=4) to deal
#// with the power of 2 introduced at the real place
#// by working with Q instead of 2*Q. This needs to
#// be done for all other n as well...
#/*
#if (n==4)
# genericfactor = 4 * genericfactor;
#*/
## DIAGNSOTIC
#cout << endl;
#cout << " generic factor = " << genericfactor << endl;
#cout << " omit = " << omit << endl;
#cout << " include = " << include << endl;
#cout << endl;
## DIAGNSOTIC
#// cout << "siegel_product Break 3. " << endl;
## Return the final factor (and divide by 2 if n=2)
if (n == 2):
return (genericfactor * omit * include / 2)
else:
return (genericfactor * omit * include)
def siegel_product(self, u):
"""
Computes the infinite product of local densities of the quadratic
form for the number `u`.
EXAMPLES::
sage: Q = DiagonalQuadraticForm(ZZ, [1,1,1,1])
sage: Q.theta_series(11)
1 + 8*q + 24*q^2 + 32*q^3 + 24*q^4 + 48*q^5 + 96*q^6 + 64*q^7 + 24*q^8 + 104*q^9 + 144*q^10 + O(q^11)
sage: Q.siegel_product(1)
8
sage: Q.siegel_product(2) ## This one is wrong -- expect 24, and the higher powers of 2 don't work... =(
24
sage: Q.siegel_product(3)
32
sage: Q.siegel_product(5)
48
sage: Q.siegel_product(6)
96
sage: Q.siegel_product(7)
64
sage: Q.siegel_product(9)
104
sage: Q.local_density(2,1)
1
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3
1
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 1]) / M^3 # long time (41s on sage.math, 2011)
1
sage: Q.local_density(2,2)
3/2
sage: M = 4; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3
3/2
sage: M = 16; len([v for v in mrange([M,M,M,M]) if Q(v) % M == 2]) / M^3 # long time (41s on sage.math, 2011)
3/2
TESTS::
sage: [1] + [Q.siegel_product(ZZ(a)) for a in range(1,11)] == Q.theta_series(11).list()
True
"""
## Protect u (since it fails often if it's an just an int!)
u = ZZ(u)
n = self.dim()
d = self.det(
) ## ??? Warning: This is a factor of 2^n larger than it should be!
## DIAGNOSTIC
verbose("n = " + str(n))
verbose("d = " + str(d))
verbose("In siegel_product: d = ", d, "\n")
## Product of "bad" places to omit
S = 2 * d * u
## DIAGNOSTIC
verbose("siegel_product Break 1. \n")
verbose(" u = ", u, "\n")
## Make the odd generic factors
if ((n % 2) == 1):
m = (n - 1) / 2
d1 = fundamental_discriminant(
((-1)**m) * 2 * d *
u) ## Replaced d by 2d here to compensate for the determinant
f = abs(
d1) ## gaining an odd power of 2 by using the matrix of 2Q instead
## of the matrix of Q.
## --> Old d1 = CoreDiscriminant((mpz_class(-1)^m) * d * u);
## Make the ratio of factorials factor: [(2m)! / m!] * prod_{i=1}^m (2*i-1)
factor1 = 1
for i in range(1, m + 1):
factor1 *= 2 * i - 1
for i in range(m + 1, 2 * m + 1):
factor1 *= i
genericfactor = factor1 * ((u / f) ** m) \
* QQ(sqrt((2 ** n) * f) / (u * d)) \
* abs(QuadraticBernoulliNumber(m, d1) / bernoulli(2*m))
## DIAGNOSTIC
verbose("siegel_product Break 2. \n")
## Make the even generic factor
if ((n % 2) == 0):
m = n / 2
d1 = fundamental_discriminant(((-1)**m) * d)
f = abs(d1)
## DIAGNOSTIC
#cout << " mpz_class(-1)^m = " << (mpz_class(-1)^m) << " and d = " << d << endl;
#cout << " f = " << f << " and d1 = " << d1 << endl;
genericfactor = m / QQ(sqrt(f*d)) \
* ((u/2) ** (m-1)) * (f ** m) \
/ abs(QuadraticBernoulliNumber(m, d1)) \
* (2 ** m) ## This last factor compensates for using the matrix of 2*Q
##return genericfactor
## Omit the generic factors in S and compute them separately
omit = 1
include = 1
S_divisors = prime_divisors(S)
## DIAGNOSTIC
#cout << "\n S is " << S << endl;
#cout << " The Prime divisors of S are :";
#PrintV(S_divisors);
for p in S_divisors:
Q_normal = self.local_normal_form(p)
## DIAGNOSTIC
verbose(" p = " + str(p) + " and its Kronecker symbol (d1/p) = (" +
str(d1) + "/" + str(p) + ") is " +
str(kronecker_symbol(d1, p)) + "\n")
omit *= 1 / (1 - (kronecker_symbol(d1, p) / (p**m)))
## DIAGNOSTIC
verbose(" omit = " + str(omit) + "\n")
verbose(" Q_normal is \n" + str(Q_normal) + "\n")
verbose(" Q_normal = \n" + str(Q_normal))
verbose(" p = " + str(p) + "\n")
verbose(" u = " + str(u) + "\n")
verbose(" include = " + str(include) + "\n")
include *= Q_normal.local_density(p, u)
## DIAGNOSTIC
#cout << " Including the p = " << p << " factor: " << local_density(Q_normal, p, u) << endl;
## DIAGNSOTIC
verbose(" --- Exiting loop \n")
#// **************** Important *******************
#// Additional fix (only included for n=4) to deal
#// with the power of 2 introduced at the real place
#// by working with Q instead of 2*Q. This needs to
#// be done for all other n as well...
#/*
#if (n==4)
# genericfactor = 4 * genericfactor;
#*/
## DIAGNSOTIC
#cout << endl;
#cout << " generic factor = " << genericfactor << endl;
#cout << " omit = " << omit << endl;
#cout << " include = " << include << endl;
#cout << endl;
## DIAGNSOTIC
#// cout << "siegel_product Break 3. " << endl;
## Return the final factor (and divide by 2 if n=2)
if (n == 2):
return (genericfactor * omit * include / 2)
else:
return (genericfactor * omit * include)
def solve_diff_eqn(self):
r"""
Solves the difference equation.
See Theorem 4.5 and Lemma 4.4 of [PS].
INPUT:
- ``self`` - an overconvergent distribution `\mu` of absolute
precision `M`
OUTPUT:
- an overconvergent distribution `\nu` of absolute precision
`M - \lfloor\log_p(M)\rfloor - 1` such that
.. math::
\nu|\Delta = \mu,\text{ where }\Delta=\begin{pmatrix}1&1\\0&1
\end{pmatrix} - 1.
EXAMPLES::
sage: D = OverconvergentDistributions(0,7,base=ZpCA(7,5))
sage: D10 = D.change_precision(10)
sage: mu10 = D10((O(7^10), 4 + 6*7 + 5*7^3 + 2*7^4 + 5*7^5 + O(7^9), 5 + 7^3 + 5*7^4 + 6*7^5 + 7^6 + 6*7^7 + O(7^8), 2 + 7 + 6*7^2 + 6*7^4 + 7^5 + 7^6 + O(7^7), 3*7 + 4*7^2 + 4*7^3 + 3*7^4 + 3*7^5 + O(7^6), 5 + 3*7 + 2*7^2 + 7^3 + 3*7^4 + O(7^5), 1 + 7^2 + 7^3 + O(7^4), 6*7 + 6*7^2 + O(7^3), 2 + 3*7 + O(7^2), 1 + O(7)))
sage: nu10 = mu10.solve_diff_eqn()
sage: MS = OverconvergentModularSymbols(14, coefficients=D)
sage: MR = MS.source()
sage: Id = MR.gens()[0]
sage: nu10 * MR.gammas[Id] - nu10 - mu10
7^8 * ()
sage: D = OverconvergentDistributions(0,7,base=Qp(7,5))
sage: D10 = D.change_precision(10)
sage: mu10 = D10((O(7^10), 4 + 6*7 + 5*7^3 + 2*7^4 + 5*7^5 + O(7^9), 5 + 7^3 + 5*7^4 + 6*7^5 + 7^6 + 6*7^7 + O(7^8), 2 + 7 + 6*7^2 + 6*7^4 + 7^5 + 7^6 + O(7^7), 3*7 + 4*7^2 + 4*7^3 + 3*7^4 + 3*7^5 + O(7^6), 5 + 3*7 + 2*7^2 + 7^3 + 3*7^4 + O(7^5), 1 + 7^2 + 7^3 + O(7^4), 6*7 + 6*7^2 + O(7^3), 2 + 3*7 + O(7^2), 1 + O(7)))
sage: nu10 = mu10.solve_diff_eqn()
sage: MS = OverconvergentModularSymbols(14, coefficients=D);
sage: MR = MS.source();
sage: Id = MR.gens()[0]
sage: nu10 * MR.gammas[Id] - nu10 - mu10
7^8 * ()
sage: R = ZpCA(5, 5); D = OverconvergentDistributions(0,base=R);
sage: nu = D((R(O(5^5)), R(5 + 3*5^2 + 4*5^3 + O(5^4)), R(5 + O(5^3)), R(2*5 + O(5^2), 2 + O(5))));
sage: nu.solve_diff_eqn()
5 * (1 + 3*5 + O(5^2), O(5))
Check input of relative precision 2::
sage: from sage.modular.pollack_stevens.coeffmod_OMS_element import CoeffMod_OMS_element
sage: R = ZpCA(3, 9)
sage: D = OverconvergentDistributions(0, base=R, prec_cap=4)
sage: V = D.approx_module(2)
sage: nu = CoeffMod_OMS_element(V([R(0, 9), R(2*3^2 + 2*3^4 + 2*3^7 + 3^8 + O(3^9))]), D, ordp=0, check=False)
sage: mu = nu.solve_diff_eqn()
sage: mu
3 * ()
"""
#RH: see tests.sage for randomized verification that this function works correctly
p = self.parent().prime()
abs_prec = ZZ(self.precision_absolute())
if self.is_zero():
mu = self.parent()(0)
mu.ordp = abs_prec - abs_prec.exact_log(p) - 1
return mu
if self._unscaled_moment(0) != 0:
raise ValueError("Distribution must have total measure 0 to be in image of difference operator.")
M = ZZ(len(self._moments))
## RP: This should never happen -- the distribution must be 0 at this point if M==1
if M == 1:
return self.parent()(0)
if M == 2:
if p == 2:
raise ValueError("Not enough accuracy to return anything")
else:
out_prec = abs_prec - abs_prec.exact_log(p) - 1
if self.ordp >= out_prec:
mu = self.parent()(0)
mu.ordp = out_prec
return mu
mu = self.parent()(self._unscaled_moment(1))
mu.ordp = self.ordp
return mu
R = self.parent().base_ring()
K = R.fraction_field()
bern = [bernoulli(i) for i in range(0,M-1,2)]
minhalf = ~K(-2) #bernoulli(1)
# bernoulli(1) = -1/2; the only nonzero odd bernoulli number
v = [minhalf * self.moment(m) for m in range(M-1)] #(m choose m-1) * B_1 * mu[m]/m
for m in range(1,M):
scalar = K(self.moment(m)) * (~K(m))
for j in range(m-1,M-1,2):
v[j] += binomial(j,m-1) * bern[(j-m+1)//2] * scalar
ordp = min(a.valuation() for a in v)
#Is this correct in ramified extensions of QQp?
verbose("abs_prec: %s, ordp: %s"%(abs_prec, ordp), level=2)
if ordp != 0:
new_M = abs_prec - 1 - (abs_prec).exact_log(p) - ordp
verbose("new_M: %s"%(new_M), level=2)
V = self.parent().approx_module(new_M)
v = V([R(v[i] >> ordp) for i in range(new_M)])
else:
new_M = abs_prec - abs_prec.exact_log(p) - 1
verbose("new_M: %s"%(new_M), level=2)
V = self.parent().approx_module(new_M)
v = V([R(v[i]) for i in range(new_M)])
v[new_M-1] = v[new_M-1].add_bigoh(1) #To force normalize to deal with this properly. May not be necessary any more.
mu = CoeffMod_OMS_element(v, self.parent(), ordp=ordp, check=False)
verbose("mu.ordp: %s, mu._moments: %s"%(mu.ordp, mu._moments), level=2)
return mu.normalize()
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