def time_is_semistable(self):
R = PolynomialRing(QQ, 'x')
x = R.gen()
Y = SuperellipticCurve(x**4 - 1, 3)
v_2 = QQ.valuation(2)
Y2 = SemistableModel(Y, v_2)
return Y2.is_semistable()
def _test(self):
from sage.all import QQ, PolynomialRing, GaussValuation
R = PolynomialRing(QQ, 'x')
x = R.gen()
v = QQ.valuation(2)
v = GaussValuation(R, v).augmentation(x + 1, QQ(1) / 2)
f = x**4 - 30 * x**2 - 75
v.mac_lane_step(f)
def _test(self):
from sage.all import QQ, PolynomialRing, GaussValuation, FunctionField
R = PolynomialRing(QQ, 'x')
x = R.gen()
v = GaussValuation(R, QQ.valuation(2))
K = FunctionField(QQ, 'x')
x = K.gen()
v = K.valuation(v)
K.valuation((v, K.hom(x/2), K.hom(2*x)))
def time_is_semistable(self):
K = FunctionField(QQ, 'x')
x = K.gen()
R = PolynomialRing(K, 'T')
T = R.gen()
f = 64 * x**3 * T - 64 * x**3 + 36 * x**2 * T**2 + 208 * x**2 * T + 192 * x**2 + 9 * x * T**3 + 72 * x * T**2 + 240 * x * T + 64 * x + T**4 + 9 * T**3 + 52 * T**2 + 48 * T
L = K.extension(f, 'y')
Y = SmoothProjectiveCurve(L)
v = QQ.valuation(13)
M = SemistableModel(Y, v)
return M.is_semistable()
def __init__(self, K0, vK):
assert K0.is_absolute(), "K0 must be an absolute number field"
assert vK.domain() == K0, "K0 must be the domain of vK"
p = vK.p()
# we want vK to be normalized such that vK(p)=1
vK = vK / vK(p)
assert vK(p) == 1
R = PolynomialRing(K0, 'x')
x = R.gen()
n = K0.degree()
if n > 1:
piK = K0.gen()
e = ZZ(1 / vK(vK.uniformizer()))
F = vK.residue_field() # should be a finite field, therefore:
f = F.cardinality().log(
F.characteristic()) # should be the absolute degree of F
assert vK(
piK
) == 1 / e, "the generator of K must be a uniformizer for vK"
P = K0.polynomial()
G = P(x + piK).shift(-1)
S = matrix(QQ, n)
for i in range(n):
S[i, i] = p**(i / e).floor()
# the matrix S has the property that S^(-1)*a.matrix()*S is p-integral,
# for an element a in O_K; this follows from the fact that
# pi^i/p^(i/e).floor(), i=0,..,n-1, is a p-integral basis
is_Qp = False
else:
K0 = QQ
piK = K0(1)
e = ZZ(1)
f = ZZ(1)
P = x - 1
G = R(1)
S = matrix(K0, 1, [1])
is_Qp = True
assert n == e * f
assert n == P.degree()
self._number_field = K0
self._v_p = QQ.valuation(p)
self._valuation = vK
self._p = p
self._uniformizer = piK
self._degree = n
self._ramification_degree = e
self._inertia_degree = f
self._polynomial = P
self._G = G
self._S = S
self._is_Qp = is_Qp
def extension(self, f, embedding=False):
r"""
Return a `p`-adic extension such that ``f`` has a root in an unramified
extension of it.
INPUT:
- ``f`` -- a monic univariate polynomial over the number field `K_0`
underlying this p-adic extension `K`, which is irreducible over `K`.
- ``embedding`` -- a boolean (default: ``False``)
OUTPUT:
A `p`-adic extension `L` of `K` such that ``f`` has a root
in an unramified extension of `L`, or (if ``embedding`` is ``True``)
the pair `(L,\phi)`, where `\phi` is the canonical embedding of
`K` into `L`.
If `K_0` is the underlying number field, then `f\in K_0[x]` is irreducible
over the completion `K`; the resulting finite extension `L/K`
is a subextension of the extension of `K`
.. MATH::
K[x]/(f).
However, the number field `L_0` underlying `L` is in general not
equal to `K_0[x]/(f)`, and there may not exist any embedding of `K_0`
into `L_0`.
"""
# print("entering extension with ")
# print("K = ", self)
# print("f = ", f)
# print()
K0 = self.number_field()
assert K0.has_coerce_map_from(f.parent().base_ring())
f = f.change_ring(K0)
vK = self.valuation()
if embedding:
raise NotImplementedError
V = vK.mac_lane_approximants(f)
assert len(V) == 1, "f is not irreducible"
v = LimitValuation(V[0], f)
pix = v.uniformizer()
r = v(pix)
for m in range(2,10):
pix1 = pix.map_coefficients(lambda c:self.reduce(c, m))
if v(pix1) == r:
break
assert m < 9, "m too small"
# now pix1 is a polynomial over `K_0` which maps to a uniformizer for the
# unique extension of vK to the relative extension L=K[x]/(f)/K
g = self.characteristic_polynomial(f, pix1)
# now g is the relative characteristic polynomial of the image pi of pix1 in L
# we need the minimal polynomial, so we do a square free factorization
g = g.squarefree_decomposition()[0][0]
# now we compute the absolute characteristic polynomial of a root
# of g
done = False
N = m+5
while not done:
Pmod = self.characteristic_polynomial_mod(g, N)
# this is very heuristic; there should be a conclusive test whether
# the result is correct, or the choice of the precision should be made correctly and
# optimal from the start
P = Pmod.map_coefficients(lambda c:c.lift(), QQ)
P = P.squarefree_decomposition()[0][0]
L0 = NumberField(P, 'pi%s'%P.degree())
V = QQ.valuation(self.p()).extensions(L0)
vL = V[0]
e = ZZ(1/vL(vL.uniformizer()))
f = vL.residue_field().degree()
if 1/e == r:
if len(V) == 1:
return FakepAdicCompletion(L0, vL)
else:
QQp = FakepAdicCompletion(QQ, self.base_valuation())
g = QQp.approximate_irreducible_factor(P)
return QQp.extension(g)
N = N + 5
