32 KiB
32 KiB
N = 2
p = 3
RQ = PolynomialRing(QQ, 'X', 2*N)
X = RQ.gens()[:N]
Y = RQ.gens()[N:]
Rpx.<x> = PolynomialRing(GF(p), 1)
#RQx.<x> = PolynomialRing(QQ, 1)
def witt_pol(lista):
n = len(lista)
return sum(p^i*lista[i]^(p^(n-i-1)) for i in range(0, n))
def witt_sum(n):
if n == 0:
return X[0] + Y[0]
return 1/p^n*(witt_pol(X[:n+1]) + witt_pol(Y[:n+1]) - sum(p^k*witt_sum(k)^(p^(n-k)) for k in range(0, n)))
class witt:
def __init__(self, coordinates):
self.coordinates = coordinates
def __repr__(self):
lista = [Rpx(a) for a in self.coordinates]
return str(lista)
def __add__(self, other):
lista = []
for i in range(0, N):
lista+= [witt_sum(i)(self.coordinates + other.coordinates)]
return witt(lista)
def __rmul__(self, constant):
if constant<0:
m = (-constant)*(p^N-1)
return m*self
if constant == 0:
return witt(N*[0])
return self + (constant - 1)*self
def __sub__(self, other):
return self + (-1)*other
def fraction_pol(g):
RxX.<x, X> = PolynomialRing(QQ)
g = RxX(g)
result = 0
for a in g.monomials():
c = g.monomial_coefficient(a)
c = c%(p^2)
dX = a.degree(X)
dx = a.degree(x)
if dX%p == 0:
result += c*x^(dX//p + dx)
else:
result += c*X^(dX + dx*p)
return result
def teichmuller(f):
Rx.<x> = PolynomialRing(QQ)
RxX.<x, X> = PolynomialRing(QQ, 2)
f = Rx(f)
ff = witt([f, 0])
coeffs = f.coefficients(sparse=false)
for i in range(0, len(coeffs)):
ff -= coeffs[i]*witt([Rx(x^i), 0])
f1 = sum(coeffs[i]*RxX(x^(i)) for i in range(0, len(coeffs))) + p*RxX(ff.coordinates[1](x = X))
f1 = fraction_pol(f1)
#RXp.<Xp> = PolynomialRing(Integers(p^2))
#f1 = RXp(f1)
return f1def basis_de_rham_degrees(f, m, p):
r = f.degree()
delta = GCD(r, m)
Rx.<x> = PolynomialRing(QQ)
Rxy.<x, y> = PolynomialRing(QQ, 2)
Fxy = FractionField(Rxy)
basis_holo = holomorphic_differentials_basis(f, m, p)
basis = []
for k in range(0, len(basis_holo)):
basis += [(basis_holo[k], Rx(0))]
## non-holomorphic elts of H^1_dR
t = len(basis)
degrees0 = {}
degrees1 = {}
for j in range(1, m):
for i in range(1, r):
if (r*(m-j) - m*i >= delta):
s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f
psi = Rx(cut(s, i))
basis += [(Fxy(psi/y^j), Fxy(m*y^(m-j)/x^i))]
degrees0[t] = (psi.degree(), j)
degrees1[t] = (-i, m-j)
t += 1
return basis, degrees0, degrees1
def de_rham_basis(f, m, p):
basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)
return basis
def degrees_de_rham0(f, m, p):
basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)
return degrees0
def degrees_de_rham1(f, m, p):
basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)
return degrees1
class superelliptic:
def __init__(self, f, m, p):
Rx.<x> = PolynomialRing(QQ)
Rxy.<x, y> = PolynomialRing(QQ, 2)
Fxy = FractionField(Rxy)
self.polynomial = Rx(f)
self.exponent = m
self.characteristic = p
r = Rx(f).degree()
delta = GCD(r, m)
self.degree_holo = degrees_holomorphic_differentials(f, m, p)
self.degree_de_rham0 = degrees_de_rham0(f, m, p)
self.degree_de_rham1 = degrees_de_rham1(f, m, p)
holo_basis = holomorphic_differentials_basis(f, m, p)
holo_basis_converted = []
for a in holo_basis:
holo_basis_converted += [superelliptic_form(self, a)]
self.basis_holomorphic_differentials = holo_basis_converted
dr_basis = de_rham_basis(f, m, p)
dr_basis_converted = []
for (a, b) in dr_basis:
dr_basis_converted += [superelliptic_cech(self, superelliptic_form(self, a), superelliptic_function(self, b))]
self.basis_de_rham = dr_basis_converted
def __repr__(self):
f = self.polynomial
m = self.exponent
p = self.characteristic
return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over finite field with ' + str(p) + ' elements.'
def is_smooth(self):
f = self.polynomial
if f.discriminant() == 0:
return 0
return 1
def genus(self):
r = self.polynomial.degree()
m = self.exponent
delta = GCD(r, m)
return 1/2*((r-1)*(m-1) - delta + 1)
class superelliptic_function:
def __init__(self, C, g):
p = C.characteristic
RxXy.<x, X, y> = PolynomialRing(QQ, 3)
FxXy = FractionField(RXy)
f = C.polynomial
r = f.degree()
m = C.exponent
self.curve = C
self.function = g
def __repr__(self):
return str(self.function)
def jth_component(self, j):
g = self.function
C = self.curve
p = C.characteristic
RxX.<x,X> = PolynomialRing(QQ, 2)
FxX.<x> = FractionField(RxX)
FxXRy.<y> = PolynomialRing(FxX)
g = FxXRy(g)
return coff(g, j)
def __add__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 + g2)
return superelliptic_function(C, g)
def __sub__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 - g2)
return superelliptic_function(C, g)
def __mul__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
g = reduction(C, g1 * g2)
return superelliptic_function(C, g)
def __truediv__(self, other):
C = self.curve
g1 = self.function
g2 = other.function
return superelliptic_function(C, g1 / g2)
def diffn(self):
C = self.curve
f = C.polynomial
m = C.exponent
p = C.characteristic
g = self.function
RXy.<X, y> = PolynomialRing(QQ, 2)
FXy = FractionField(RXy)
g = RXy(g)
A = g.derivative(X)*X^(-(p-1))/p
t = teichmuller(f)
B = g.derivative(y)*t.derivative()/(m*y^(m-1))*X^(-(p-1))/p
return superelliptic_form(C, A+B)
class superelliptic_form:
def __init__(self, C, g):
p = C.characteristic
Rxy.<x, y> = PolynomialRing(QQ, 2)
Fxy = FractionField(Rxy)
g = Fxy(reduction_form(C, g))
self.form = g
self.curve = C
def __add__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
g = reduction(C, g1 + g2)
return superelliptic_form(C, g)
def __sub__(self, other):
C = self.curve
g1 = self.form
g2 = other.form
g = reduction(C, g1 - g2)
return superelliptic_form(C, g)
def __repr__(self):
g = self.form
if len(str(g)) == 1:
return str(g) + ' dx'
return '('+str(g) + ') dx'
def __rmul__(self, constant):
C = self.curve
omega = self.form
return superelliptic_form(C, constant*omega)
def coordinates(self):
C = self.curve
p = C.characteristic
m = C.exponent
Rx.<x> = PolynomialRing(QQ)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = C.genus()
degrees_holo = C.degree_holo
degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
basis = C.basis_holomorphic_differentials
for j in range(1, m):
omega_j = Fx(self.jth_component(j))
if omega_j != Fx(0):
d = degree_of_rational_fctn(omega_j, p)
index = degrees_holo_inv[(d, j)]
a = coeff_of_rational_fctn(omega_j, p)
a1 = coeff_of_rational_fctn(basis[index].jth_component(j), p)
elt = self - (a/a1)*basis[index]
return elt.coordinates() + a/a1*vector([QQ(i == index) for i in range(0, g)])
return vector(g*[0])
def jth_component(self, j):
g = self.form
C = self.curve
p = C.characteristic
Rx.<x> = PolynomialRing(QQ)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
Fxy = FractionField(FxRy)
Ryinv.<y_inv> = PolynomialRing(Fx)
g = Fxy(g)
g = g(y = 1/y_inv)
g = Ryinv(g)
return coff(g, j)
def is_regular_on_U0(self):
C = self.curve
p = C.characteristic
m = C.exponent
Rx.<x> = PolynomialRing(QQ)
for j in range(1, m):
if self.jth_component(j) not in Rx:
return 0
return 1
def is_regular_on_Uinfty(self):
C = self.curve
p = C.characteristic
m = C.exponent
f = C.polynomial
r = f.degree()
delta = GCD(m, r)
M = m/delta
R = r/delta
for j in range(1, m):
A = self.jth_component(j)
d = degree_of_rational_fctn(A, p)
if(-d*M + j*R -(M+1)<0):
return 0
return 1
class superelliptic_cech:
def __init__(self, C, omega, fct):
self.omega0 = omega
self.omega8 = omega - diffn(fct)
self.f = fct
self.curve = C
def __add__(self, other):
C = self.curve
return superelliptic_cech(C, self.omega0 + other.omega0, self.f + other.f)
def __sub__(self, other):
C = self.curve
return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)
def __rmul__(self, constant):
C = self.curve
w1 = self.omega0.form
f1 = self.f.function
w2 = superelliptic_form(C, constant*w1)
f2 = superelliptic_function(C, constant*f1)
return superelliptic_cech(C, w2, f2)
def __repr__(self):
return "(" + str(self.omega0) + ", " + str(self.f) + ", " + str(self.omega8) + ")"
def coordinates(self):
C = self.curve
p = C.characteristic
m = C.exponent
Rx.<x> = PolynomialRing(QQ)
Fx = FractionField(Rx)
FxRy.<y> = PolynomialRing(Fx)
g = C.genus()
degrees_holo = C.degree_holo
degrees_holo_inv = {b:a for a, b in degrees_holo.items()}
degrees0 = C.degree_de_rham0
degrees0_inv = {b:a for a, b in degrees0.items()}
degrees1 = C.degree_de_rham1
degrees1_inv = {b:a for a, b in degrees1.items()}
basis = C.basis_de_rham
omega = self.omega0
fct = self.f
if fct.function == Rx(0) and omega.form != Rx(0):
for j in range(1, m):
omega_j = Fx(omega.jth_component(j))
if omega_j != Fx(0):
d = degree_of_rational_fctn(omega_j, p)
index = degrees_holo_inv[(d, j)]
a = coeff_of_rational_fctn(omega_j, p)
a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), p)
elt = self - (a/a1)*basis[index]
return elt.coordinates() + a/a1*vector([QQ(i == index) for i in range(0, 2*g)])
for j in range(1, m):
fct_j = Fx(fct.jth_component(j))
if (fct_j != Rx(0)):
d = degree_of_rational_fctn(fct_j, p)
if (d, j) in degrees1.values():
index = degrees1_inv[(d, j)]
a = coeff_of_rational_fctn(fct_j, p)
elt = self - (a/m)*basis[index]
return elt.coordinates() + a/m*vector([QQ(i == index) for i in range(0, 2*g)])
if d<0:
a = coeff_of_rational_fctn(fct_j, p)
h = superelliptic_function(C, FxRy(a*y^j*x^d))
elt = superelliptic_cech(C, self.omega0, self.f - h)
return elt.coordinates()
if (fct_j != Rx(0)):
G = superelliptic_function(C, y^j*x^d)
a = coeff_of_rational_fctn(fct_j, p)
elt =self - a*superelliptic_cech(C, diffn(G), G)
return elt.coordinates()
return vector(2*g*[0])
def is_cocycle(self):
w0 = self.omega0
w8 = self.omega8
fct = self.f
if not w0.is_regular_on_U0() and not w8.is_regular_on_Uinfty():
return('w0 & w8')
if not w0.is_regular_on_U0():
return('w0')
if not w8.is_regular_on_Uinfty():
return('w8')
if w0.is_regular_on_U0() and w8.is_regular_on_Uinfty():
return 1
return 0
def degree_of_rational_fctn(f, p):
Rx.<x> = PolynomialRing(QQ)
Fx = FractionField(Rx)
f = Fx(f)
f1 = f.numerator()
f2 = f.denominator()
d1 = f1.degree()
d2 = f2.degree()
return(d1 - d2)
def coeff_of_rational_fctn(f, p):
Rx.<x> = PolynomialRing(QQ)
Fx = FractionField(Rx)
f = Fx(f)
if f == Rx(0):
return 0
f1 = f.numerator()
f2 = f.denominator()
d1 = f1.degree()
d2 = f2.degree()
a1 = f1.coefficients(sparse = false)[d1]
a2 = f2.coefficients(sparse = false)[d2]
return(a1/a2)
def coff(f, d):
lista = f.coefficients(sparse = false)
if len(lista) <= d:
return 0
return lista[d]
def cut(f, i):
R = f.parent()
coeff = f.coefficients(sparse = false)
return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))
def polynomial_part(p, h):
Rx.<x> = PolynomialRing(QQ)
h = Rx(h)
result = Rx(0)
for i in range(0, h.degree()+1):
if (i%p) == p-1:
power = Integer((i-(p-1))/p)
result += Integer(h[i]) * x^(power)
return resultRx.<x> = PolynomialRing(QQ)
f = Rx(x^3 - x)
teichmuller(f)-720*X^9 - 15312*X^7 - 107160*X^5 + x^3 - 249984*X^3 - x
6*X^7 + 3*X^5 + x^3 + 8*x
Rx.<x> = PolynomialRing(QQ)
C = superelliptic(x^3 - x, 2, 3)[0;31m---------------------------------------------------------------------------[0m [0;31mTypeError[0m Traceback (most recent call last) [0;32m/tmp/ipykernel_1111/3447231159.py[0m in [0;36m<module>[0;34m[0m [1;32m 1[0m [0mRx[0m [0;34m=[0m [0mPolynomialRing[0m[0;34m([0m[0mQQ[0m[0;34m,[0m [0mnames[0m[0;34m=[0m[0;34m([0m[0;34m'x'[0m[0;34m,[0m[0;34m)[0m[0;34m)[0m[0;34m;[0m [0;34m([0m[0mx[0m[0;34m,[0m[0;34m)[0m [0;34m=[0m [0mRx[0m[0;34m.[0m[0m_first_ngens[0m[0;34m([0m[0;36m1[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m----> 2[0;31m [0mC[0m [0;34m=[0m [0msuperelliptic[0m[0;34m([0m[0mx[0m[0;34m**[0m[0mInteger[0m[0;34m([0m[0;36m3[0m[0;34m)[0m [0;34m-[0m [0mx[0m[0;34m,[0m [0mInteger[0m[0;34m([0m[0;36m2[0m[0;34m)[0m[0;34m,[0m [0mInteger[0m[0;34m([0m[0;36m3[0m[0;34m)[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m [0;32m/tmp/ipykernel_1111/3436947063.py[0m in [0;36m__init__[0;34m(self, f, m, p)[0m [1;32m 68[0m [0mdr_basis_converted[0m [0;34m=[0m [0;34m[[0m[0;34m][0m[0;34m[0m[0;34m[0m[0m [1;32m 69[0m [0;32mfor[0m [0;34m([0m[0ma[0m[0;34m,[0m [0mb[0m[0;34m)[0m [0;32min[0m [0mdr_basis[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m---> 70[0;31m [0mdr_basis_converted[0m [0;34m+=[0m [0;34m[[0m[0msuperelliptic_cech[0m[0;34m([0m[0mself[0m[0;34m,[0m [0msuperelliptic_form[0m[0;34m([0m[0mself[0m[0;34m,[0m [0ma[0m[0;34m)[0m[0;34m,[0m [0msuperelliptic_function[0m[0;34m([0m[0mself[0m[0;34m,[0m [0mb[0m[0;34m)[0m[0;34m)[0m[0;34m][0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 71[0m [0;34m[0m[0m [1;32m 72[0m [0mself[0m[0;34m.[0m[0mbasis_de_rham[0m [0;34m=[0m [0mdr_basis_converted[0m[0;34m[0m[0;34m[0m[0m [0;32m/tmp/ipykernel_1111/3436947063.py[0m in [0;36m__init__[0;34m(self, C, omega, fct)[0m [1;32m 260[0m [0;32mdef[0m [0m__init__[0m[0;34m([0m[0mself[0m[0;34m,[0m [0mC[0m[0;34m,[0m [0momega[0m[0;34m,[0m [0mfct[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [1;32m 261[0m [0mself[0m[0;34m.[0m[0momega0[0m [0;34m=[0m [0momega[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 262[0;31m [0mself[0m[0;34m.[0m[0momega8[0m [0;34m=[0m [0momega[0m [0;34m-[0m [0mdiffn[0m[0;34m([0m[0mfct[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 263[0m [0mself[0m[0;34m.[0m[0mf[0m [0;34m=[0m [0mfct[0m[0;34m[0m[0;34m[0m[0m [1;32m 264[0m [0mself[0m[0;34m.[0m[0mcurve[0m [0;34m=[0m [0mC[0m[0;34m[0m[0;34m[0m[0m [0;32m/tmp/ipykernel_1111/3436947063.py[0m in [0;36mdiffn[0;34m(self)[0m [1;32m 153[0m [0mA[0m [0;34m=[0m [0mg[0m[0;34m.[0m[0mderivative[0m[0;34m([0m[0mXp[0m[0;34m)[0m[0;34m*[0m[0mXp[0m[0;34m**[0m[0;34m([0m[0;34m-[0m[0;34m([0m[0mp[0m[0;34m-[0m[0mInteger[0m[0;34m([0m[0;36m1[0m[0;34m)[0m[0;34m)[0m[0;34m)[0m[0;34m/[0m[0mp[0m[0;34m[0m[0;34m[0m[0m [1;32m 154[0m [0mt[0m [0;34m=[0m [0mteichmuller[0m[0;34m([0m[0mf[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0;32m--> 155[0;31m [0mB[0m [0;34m=[0m [0mg[0m[0;34m.[0m[0mderivative[0m[0;34m([0m[0my[0m[0;34m)[0m[0;34m*[0m[0mt[0m[0;34m.[0m[0mderivative[0m[0;34m([0m[0;34m)[0m[0;34m/[0m[0;34m([0m[0mm[0m[0;34m*[0m[0my[0m[0;34m**[0m[0;34m([0m[0mm[0m[0;34m-[0m[0mInteger[0m[0;34m([0m[0;36m1[0m[0;34m)[0m[0;34m)[0m[0;34m)[0m[0;34m*[0m[0mXp[0m[0;34m**[0m[0;34m([0m[0;34m-[0m[0;34m([0m[0mp[0m[0;34m-[0m[0mInteger[0m[0;34m([0m[0;36m1[0m[0;34m)[0m[0;34m)[0m[0;34m)[0m[0;34m/[0m[0mp[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 156[0m [0;32mreturn[0m [0msuperelliptic_form[0m[0;34m([0m[0mC[0m[0;34m,[0m [0mA[0m[0;34m+[0m[0mB[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 157[0m [0;34m[0m[0m [0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/element.pyx[0m in [0;36msage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:12253)[0;34m()[0m [1;32m 1514[0m [0;32mreturn[0m [0;34m([0m[0;34m<[0m[0mElement[0m[0;34m>[0m[0mleft[0m[0;34m)[0m[0;34m.[0m[0m_mul_[0m[0;34m([0m[0mright[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [1;32m 1515[0m [0;32mif[0m [0mBOTH_ARE_ELEMENT[0m[0;34m([0m[0mcl[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;32m-> 1516[0;31m [0;32mreturn[0m [0mcoercion_model[0m[0;34m.[0m[0mbin_op[0m[0;34m([0m[0mleft[0m[0;34m,[0m [0mright[0m[0;34m,[0m [0mmul[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 1517[0m [0;34m[0m[0m [1;32m 1518[0m [0mcdef[0m [0mlong[0m [0mvalue[0m[0;34m[0m[0;34m[0m[0m [0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/coerce.pyx[0m in [0;36msage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:11751)[0;34m()[0m [1;32m 1246[0m [0;31m# We should really include the underlying error.[0m[0;34m[0m[0;34m[0m[0m [1;32m 1247[0m [0;31m# This causes so much headache.[0m[0;34m[0m[0;34m[0m[0m [0;32m-> 1248[0;31m [0;32mraise[0m [0mbin_op_exception[0m[0;34m([0m[0mop[0m[0;34m,[0m [0mx[0m[0;34m,[0m [0my[0m[0;34m)[0m[0;34m[0m[0;34m[0m[0m [0m[1;32m 1249[0m [0;34m[0m[0m [1;32m 1250[0m [0mcpdef[0m [0mcanonical_coercion[0m[0;34m([0m[0mself[0m[0;34m,[0m [0mx[0m[0;34m,[0m [0my[0m[0;34m)[0m[0;34m:[0m[0;34m[0m[0;34m[0m[0m [0;31mTypeError[0m: unsupported operand parent(s) for *: 'Multivariate Polynomial Ring in Xp, y over Rational Field' and 'Univariate Polynomial Ring in Xp over Ring of integers modulo 9'
C.basis_de_rham[((1/y) dx, 0, (1/y) dx), ((x/y) dx, 2/x*y, ((-1)/(x*y)) dx)]
g = basis_de_rham_degrees(x^3 - x, 2, 3)RXpy.<Xp, y> = PolynomialRing(QQ, 2)
FXpy = FractionField(RXpy)g(x = Xp^p, y = y)2*y/Xp^3
t = teichmuller(x^3 - x)t.derivative()6*Xp^6 + 6*Xp^4 + 6*Xp^2
RxX.<x, X> = PolynomialRing(QQ)g = RxX(x*X + 2*x*X^3)g.monomials()[0].degree(X)3
type(g.monomial_coefficient(X))<class 'sage.rings.rational.Rational'>