class AlgebraicWeight(WeightCharacter):
r"""
A point in weight space corresponding to a locally algebraic character, of
the form `x \mapsto \chi(x) x^k` where `k` is an integer and `\chi` is a
Dirichlet character modulo `p^n` for some `n`.
TESTS::
sage: w = pAdicWeightSpace(23)(12, DirichletGroup(23, QQ).0) # exact
sage: w == loads(dumps(w))
True
sage: w = pAdicWeightSpace(23)(12, DirichletGroup(23, Qp(23)).0) # inexact
sage: w == loads(dumps(w))
True
sage: w is loads(dumps(w)) # elements are not globally unique
False
"""
def __init__(self, parent, k, chi=None):
r"""
Create a locally algebraic weight-character.
EXAMPLES::
sage: pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0)
(13, 29, [2 + 2*29 + ... + O(29^20)])
"""
WeightCharacter.__init__(self, parent)
k = ZZ(k)
self._k = k
if chi is None:
chi = trivial_character(self._p, QQ)
n = ZZ(chi.conductor())
if n == 1:
n = self._p
if not n.is_power_of(self._p):
raise ValueError, "Character must have %s-power conductor" % p
self._chi = DirichletGroup(n, chi.base_ring())(chi)
def __call__(self, x):
r"""
Evaluate this character at an element of `\ZZ_p^\times`.
EXAMPLES:
Exact answers are returned when this is possible::
sage: kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, QQ).0)
sage: kappa(1)
1
sage: kappa(0)
0
sage: kappa(12)
-106993205379072
sage: kappa(-1)
-1
sage: kappa(13 + 4*29 + 11*29^2 + O(29^3))
9 + 21*29 + 27*29^2 + O(29^3)
When the character chi is defined over a p-adic field, the results returned are inexact::
sage: kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0^14)
sage: kappa(1)
1 + O(29^20)
sage: kappa(0)
0
sage: kappa(12)
17 + 11*29 + 7*29^2 + 4*29^3 + 5*29^4 + 2*29^5 + 13*29^6 + 3*29^7 + 18*29^8 + 21*29^9 + 28*29^10 + 28*29^11 + 28*29^12 + 28*29^13 + 28*29^14 + 28*29^15 + 28*29^16 + 28*29^17 + 28*29^18 + 28*29^19 + O(29^20)
sage: kappa(12) == -106993205379072
True
sage: kappa(-1) == -1
True
sage: kappa(13 + 4*29 + 11*29^2 + O(29^3))
9 + 21*29 + 27*29^2 + O(29^3)
"""
if isinstance(x, pAdicGenericElement):
if x.parent().prime() != self._p:
raise TypeError, "x must be an integer or a %s-adic integer" % self._p
if self._p**(x.precision_absolute()) < self._chi.conductor():
raise Exception, "Precision too low"
xint = x.lift()
else:
xint = x
if (xint % self._p == 0): return 0
return self._chi(xint) * x**self._k
def k(self):
r"""
If this character is `x \mapsto x^k \chi(x)` for an integer `k` and a
Dirichlet character `\chi`, return `k`.
EXAMPLE::
sage: kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0^14)
sage: kappa.k()
13
"""
return self._k
def chi(self):
r"""
If this character is `x \mapsto x^k \chi(x)` for an integer `k` and a
Dirichlet character `\chi`, return `\chi`.
EXAMPLE::
sage: kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0^14)
sage: kappa.chi()
Dirichlet character modulo 29 of conductor 29 mapping 2 |--> 28 + 28*29 + 28*29^2 + ... + O(29^20)
"""
return self._chi
def _repr_(self):
r"""
String representation of self.
EXAMPLES::
sage: pAdicWeightSpace(17)(2)._repr_()
'2'
sage: pAdicWeightSpace(17)(2, DirichletGroup(17, QQ).0)._repr_()
'(2, 17, [-1])'
sage: pAdicWeightSpace(17)(2, DirichletGroup(17, QQ).0^2)._repr_()
'2'
"""
if self._chi.is_trivial():
return "%s" % self._k
else:
return "(%s, %s, %s)" % (self._k, self._chi.modulus(), self._chi._repr_short_())
def teichmuller_type(self):
r"""
Return the Teichmuller type of this weight-character `\kappa`, which is
the unique `t \in \ZZ/(p-1)\ZZ` such that `\kappa(\mu) =
\mu^t` for \mu a `(p-1)`-st root of 1.
For `p = 2` this doesn't make sense, but we still want the Teichmuller
type to correspond to the index of the component of weight space in
which `\kappa` lies, so we return 1 if `\kappa` is odd and 0 otherwise.
EXAMPLE::
sage: pAdicWeightSpace(11)(2, DirichletGroup(11,QQ).0).teichmuller_type()
7
sage: pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0).teichmuller_type()
14
sage: pAdicWeightSpace(2)(3, DirichletGroup(4,QQ).0).teichmuller_type()
0
"""
# Special case p == 2
if self._p == 2:
if self.is_even():
return IntegerModRing(2)(0)
else:
return IntegerModRing(2)(1)
m = IntegerModRing(self._p).multiplicative_generator()
x = [y for y in IntegerModRing(self._chi.modulus()) if y == m and y**(self._p - 1) == 1]
if len(x) != 1: raise ArithmeticError
x = x[0]
f = IntegerModRing(self._p)(self._chi(x)).log(m)
return IntegerModRing(self._p - 1)(self._k + f)
def Lvalue(self):
r"""
Return the value of the p-adic L-function of `\QQ` evaluated at
this weight-character. If the character is `x \mapsto x^k \chi(x)`
where `k > 0` and `\chi` has conductor a power of `p`, this is an
element of the number field generated by the values of `\chi`, equal to
the value of the complex L-function `L(1-k, \chi)`. If `\chi` is
trivial, it is equal to `(1 - p^{k-1})\zeta(1-k)`.
At present this is not implemented in any other cases, except the
trivial character (for which the value is `\infty`).
TODO: Implement this more generally using the Amice transform machinery
in sage/schemes/elliptic_curves/padic_lseries.py, which should clearly
be factored out into a separate class.
EXAMPLES::
sage: pAdicWeightSpace(7)(4).Lvalue() == (1 - 7^3)*zeta__exact(-3)
True
sage: pAdicWeightSpace(7)(5, DirichletGroup(7, Qp(7)).0^4).Lvalue()
0
sage: pAdicWeightSpace(7)(6, DirichletGroup(7, Qp(7)).0^4).Lvalue()
1 + 2*7 + 7^2 + 3*7^3 + 3*7^5 + 4*7^6 + 2*7^7 + 5*7^8 + 2*7^9 + 3*7^10 + 6*7^11 + 2*7^12 + 3*7^13 + 5*7^14 + 6*7^15 + 5*7^16 + 3*7^17 + 6*7^18 + O(7^19)
"""
if self._k > 0:
return -self._chi.bernoulli(self._k)/self._k
if self.is_trivial():
return Infinity
else:
raise NotImplementedError, "Don't know how to compute value of this L-function"
def dimension_of_ordinary_subspace(self, p=None, cusp=False):
"""
If ``cusp`` is ``True``, return dimension of cuspidal ordinary
subspace. This does a weight 2 computation with sage's ModularSymbols.
EXAMPLES::
sage: M = OverconvergentModularSymbols(11, 0, sign=-1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
2
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M.dimension_of_ordinary_subspace()
4
sage: M = OverconvergentModularSymbols(11, 0, sign=0, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
6
sage: M.dimension_of_ordinary_subspace(cusp=True)
4
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
sage: M = OverconvergentModularSymbols(11, 2, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
0
sage: M.dimension_of_ordinary_subspace()
1
sage: M = OverconvergentModularSymbols(11, 10, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
An example with odd weight and hence non-trivial character::
sage: K = Qp(11, 6)
sage: DG = DirichletGroup(11, K)
sage: chi = DG([K(378703)])
sage: MM = FamiliesOfOMS(chi, 1, p=11, prec_cap=[4, 4], base_coeffs=ZpCA(11, 4), sign=-1)
sage: MM.dimension_of_ordinary_subspace()
1
"""
try:
p = self.prime()
except AttributeError:
if p is None:
raise ValueError(
"If self doesn't have a prime, must specify p.")
try:
return self._ord_dim_dict[(p, cusp)]
except AttributeError:
self._ord_dim_dict = {}
except KeyError:
pass
from sage.modular.dirichlet import DirichletGroup
from sage.rings.finite_rings.constructor import GF
try:
chi = self.character()
except AttributeError:
chi = DirichletGroup(self.level(), GF(p))[0]
if chi is None:
chi = DirichletGroup(self.level(), GF(p))[0]
from sage.modular.modsym.modsym import ModularSymbols
r = self.weight() % (p - 1)
if chi.is_trivial():
N = chi.modulus()
if N % p != 0:
N *= p
else:
e = N.valuation(p)
N.divide_knowing_divisible_by(p**(e - 1))
chi = DirichletGroup(N, GF(p))[0]
elif chi.modulus() % p != 0:
chi = DirichletGroup(chi.modulus() * p, GF(p))(chi)
DG = DirichletGroup(chi.modulus(), GF(p))
if r == 0:
from sage.modular.arithgroup.congroup_gamma0 import Gamma0_constructor as Gamma0
verbose("in dim: %s, %s, %s" % (self.sign(), chi, p))
M = ModularSymbols(DG(chi), 2, self.sign(), GF(p))
else:
psi = [GF(p)(u)**r for u in DG.unit_gens()] #mod p Teichmuller^r
psi = DG(psi)
M = ModularSymbols(DG(chi) * psi, 2, self.sign(), GF(p))
if cusp:
M = M.cuspidal_subspace()
hecke_poly = M.hecke_polynomial(p)
verbose("in dim: %s" % (hecke_poly))
x = hecke_poly.parent().gen()
d = hecke_poly.degree() - hecke_poly.ord(x)
self._ord_dim_dict[(p, cusp)] = d
return d
class AlgebraicWeight(WeightCharacter):
r"""
A point in weight space corresponding to a locally algebraic character, of
the form `x \mapsto \chi(x) x^k` where `k` is an integer and `\chi` is a
Dirichlet character modulo `p^n` for some `n`.
TESTS::
sage: w = pAdicWeightSpace(23)(12, DirichletGroup(23, QQ).0) # exact
sage: w == loads(dumps(w))
True
sage: w = pAdicWeightSpace(23)(12, DirichletGroup(23, Qp(23)).0) # inexact
sage: w == loads(dumps(w))
True
sage: w is loads(dumps(w)) # elements are not globally unique
False
"""
def __init__(self, parent, k, chi=None):
r"""
Create a locally algebraic weight-character.
EXAMPLES::
sage: pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0)
(13, 29, [2 + 2*29 + ... + O(29^20)])
"""
WeightCharacter.__init__(self, parent)
k = ZZ(k)
self._k = k
if chi is None:
chi = trivial_character(self._p, QQ)
n = ZZ(chi.conductor())
if n == 1:
n = self._p
if not n.is_power_of(self._p):
raise ValueError("Character must have %s-power conductor" % p)
self._chi = DirichletGroup(n, chi.base_ring())(chi)
def __call__(self, x):
r"""
Evaluate this character at an element of `\ZZ_p^\times`.
EXAMPLES:
Exact answers are returned when this is possible::
sage: kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, QQ).0)
sage: kappa(1)
1
sage: kappa(0)
0
sage: kappa(12)
-106993205379072
sage: kappa(-1)
-1
sage: kappa(13 + 4*29 + 11*29^2 + O(29^3))
9 + 21*29 + 27*29^2 + O(29^3)
When the character chi is defined over a p-adic field, the results returned are inexact::
sage: kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0^14)
sage: kappa(1)
1 + O(29^20)
sage: kappa(0)
0
sage: kappa(12)
17 + 11*29 + 7*29^2 + 4*29^3 + 5*29^4 + 2*29^5 + 13*29^6 + 3*29^7 + 18*29^8 + 21*29^9 + 28*29^10 + 28*29^11 + 28*29^12 + 28*29^13 + 28*29^14 + 28*29^15 + 28*29^16 + 28*29^17 + 28*29^18 + 28*29^19 + O(29^20)
sage: kappa(12) == -106993205379072
True
sage: kappa(-1) == -1
True
sage: kappa(13 + 4*29 + 11*29^2 + O(29^3))
9 + 21*29 + 27*29^2 + O(29^3)
"""
if isinstance(x, pAdicGenericElement):
if x.parent().prime() != self._p:
raise TypeError("x must be an integer or a %s-adic integer" %
self._p)
if self._p**(x.precision_absolute()) < self._chi.conductor():
raise PrecisionError("Precision too low")
xint = x.lift()
else:
xint = x
if (xint % self._p == 0): return 0
return self._chi(xint) * x**self._k
def k(self):
r"""
If this character is `x \mapsto x^k \chi(x)` for an integer `k` and a
Dirichlet character `\chi`, return `k`.
EXAMPLES::
sage: kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0^14)
sage: kappa.k()
13
"""
return self._k
def chi(self):
r"""
If this character is `x \mapsto x^k \chi(x)` for an integer `k` and a
Dirichlet character `\chi`, return `\chi`.
EXAMPLES::
sage: kappa = pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0^14)
sage: kappa.chi()
Dirichlet character modulo 29 of conductor 29 mapping 2 |--> 28 + 28*29 + 28*29^2 + ... + O(29^20)
"""
return self._chi
def __hash__(self):
r"""
TESTS::
sage: w = pAdicWeightSpace(23)(12, DirichletGroup(23, QQ).0)
sage: hash(w)
2363715643371367891 # 64-bit
-1456525869 # 32-bit
"""
if self._chi.is_trivial():
return hash(self._k)
else:
return hash((self._k, self._chi.modulus(), self._chi))
def _repr_(self):
r"""
String representation of self.
EXAMPLES::
sage: pAdicWeightSpace(17)(2)._repr_()
'2'
sage: pAdicWeightSpace(17)(2, DirichletGroup(17, QQ).0)._repr_()
'(2, 17, [-1])'
sage: pAdicWeightSpace(17)(2, DirichletGroup(17, QQ).0^2)._repr_()
'2'
"""
if self._chi.is_trivial():
return "%s" % self._k
else:
return "(%s, %s, %s)" % (self._k, self._chi.modulus(),
self._chi._repr_short_())
def teichmuller_type(self):
r"""
Return the Teichmuller type of this weight-character `\kappa`, which is
the unique `t \in \ZZ/(p-1)\ZZ` such that `\kappa(\mu) =
\mu^t` for \mu a `(p-1)`-st root of 1.
For `p = 2` this doesn't make sense, but we still want the Teichmuller
type to correspond to the index of the component of weight space in
which `\kappa` lies, so we return 1 if `\kappa` is odd and 0 otherwise.
EXAMPLES::
sage: pAdicWeightSpace(11)(2, DirichletGroup(11,QQ).0).teichmuller_type()
7
sage: pAdicWeightSpace(29)(13, DirichletGroup(29, Qp(29)).0).teichmuller_type()
14
sage: pAdicWeightSpace(2)(3, DirichletGroup(4,QQ).0).teichmuller_type()
0
"""
# Special case p == 2
if self._p == 2:
if self.is_even():
return IntegerModRing(2)(0)
else:
return IntegerModRing(2)(1)
m = IntegerModRing(self._p).multiplicative_generator()
x = [
y for y in IntegerModRing(self._chi.modulus())
if y == m and y**(self._p - 1) == 1
]
if len(x) != 1: raise ArithmeticError
x = x[0]
f = IntegerModRing(self._p)(self._chi(x)).log(m)
return IntegerModRing(self._p - 1)(self._k + f)
def Lvalue(self):
r"""
Return the value of the p-adic L-function of `\QQ` evaluated at
this weight-character. If the character is `x \mapsto x^k \chi(x)`
where `k > 0` and `\chi` has conductor a power of `p`, this is an
element of the number field generated by the values of `\chi`, equal to
the value of the complex L-function `L(1-k, \chi)`. If `\chi` is
trivial, it is equal to `(1 - p^{k-1})\zeta(1-k)`.
At present this is not implemented in any other cases, except the
trivial character (for which the value is `\infty`).
TODO: Implement this more generally using the Amice transform machinery
in sage/schemes/elliptic_curves/padic_lseries.py, which should clearly
be factored out into a separate class.
EXAMPLES::
sage: pAdicWeightSpace(7)(4).Lvalue() == (1 - 7^3)*zeta__exact(-3)
True
sage: pAdicWeightSpace(7)(5, DirichletGroup(7, Qp(7)).0^4).Lvalue()
0
sage: pAdicWeightSpace(7)(6, DirichletGroup(7, Qp(7)).0^4).Lvalue()
1 + 2*7 + 7^2 + 3*7^3 + 3*7^5 + 4*7^6 + 2*7^7 + 5*7^8 + 2*7^9 + 3*7^10 + 6*7^11 + 2*7^12 + 3*7^13 + 5*7^14 + 6*7^15 + 5*7^16 + 3*7^17 + 6*7^18 + O(7^19)
"""
if self._k > 0:
return -self._chi.bernoulli(self._k) / self._k
if self.is_trivial():
return Infinity
else:
raise NotImplementedError(
"Don't know how to compute value of this L-function")
def dimension_of_ordinary_subspace(self, p=None, cusp=False):
"""
If ``cusp`` is ``True``, return dimension of cuspidal ordinary
subspace. This does a weight 2 computation with sage's ModularSymbols.
EXAMPLES::
sage: M = OverconvergentModularSymbols(11, 0, sign=-1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
2
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
2
sage: M.dimension_of_ordinary_subspace()
4
sage: M = OverconvergentModularSymbols(11, 0, sign=0, p=3, prec_cap=4, base=ZpCA(3, 8))
sage: M.dimension_of_ordinary_subspace()
6
sage: M.dimension_of_ordinary_subspace(cusp=True)
4
sage: M = OverconvergentModularSymbols(11, 0, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
sage: M = OverconvergentModularSymbols(11, 2, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
0
sage: M.dimension_of_ordinary_subspace()
1
sage: M = OverconvergentModularSymbols(11, 10, sign=1, p=11, prec_cap=4, base=ZpCA(11, 8))
sage: M.dimension_of_ordinary_subspace(cusp=True)
1
sage: M.dimension_of_ordinary_subspace()
2
An example with odd weight and hence non-trivial character::
sage: K = Qp(11, 6)
sage: DG = DirichletGroup(11, K)
sage: chi = DG([K(378703)])
sage: MM = FamiliesOfOMS(chi, 1, p=11, prec_cap=[4, 4], base_coeffs=ZpCA(11, 4), sign=-1)
sage: MM.dimension_of_ordinary_subspace()
1
"""
try:
p = self.prime()
except AttributeError:
if p is None:
raise ValueError("If self doesn't have a prime, must specify p.")
try:
return self._ord_dim_dict[(p, cusp)]
except AttributeError:
self._ord_dim_dict = {}
except KeyError:
pass
from sage.modular.dirichlet import DirichletGroup
from sage.rings.finite_rings.constructor import GF
try:
chi = self.character()
except AttributeError:
chi = DirichletGroup(self.level(), GF(p))[0]
if chi is None:
chi = DirichletGroup(self.level(), GF(p))[0]
from sage.modular.modsym.modsym import ModularSymbols
r = self.weight() % (p-1)
if chi.is_trivial():
N = chi.modulus()
if N % p != 0:
N *= p
else:
e = N.valuation(p)
N.divide_knowing_divisible_by(p ** (e-1))
chi = DirichletGroup(N, GF(p))[0]
elif chi.modulus() % p != 0:
chi = DirichletGroup(chi.modulus() * p, GF(p))(chi)
DG = DirichletGroup(chi.modulus(), GF(p))
if r == 0:
from sage.modular.arithgroup.congroup_gamma0 import Gamma0_constructor as Gamma0
verbose("in dim: %s, %s, %s"%(self.sign(), chi, p))
M = ModularSymbols(DG(chi), 2, self.sign(), GF(p))
else:
psi = [GF(p)(u) ** r for u in DG.unit_gens()] #mod p Teichmuller^r
psi = DG(psi)
M = ModularSymbols(DG(chi) * psi, 2, self.sign(), GF(p))
if cusp:
M = M.cuspidal_subspace()
hecke_poly = M.hecke_polynomial(p)
verbose("in dim: %s"%(hecke_poly))
x = hecke_poly.parent().gen()
d = hecke_poly.degree() - hecke_poly.ord(x)
self._ord_dim_dict[(p, cusp)] = d
return d
