jgarnek/DeRhamComputation
1
0
Fork 0
Code Issues Pull Requests Projects Releases Wiki Activity

deleted ipynb files

Browse Source
This commit is contained in:
jgarnek 2023-09-22 06:48:13 +00:00
parent 599aa94db7
commit ec58dc244e
6 changed files with 20 additions and 5923 deletions
Show Stats Download Patch File Download Diff File Expand all files Collapse all files

755
crystalline_p2.ipynb
View File

@ -1,755 +0,0 @@
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"N = 2\n",
"p = 3\n",
"RQ = PolynomialRing(QQ, 'X', 2*N)\n",
"X = RQ.gens()[:N]\n",
"Y = RQ.gens()[N:]\n",
"Rpx.<x> = PolynomialRing(GF(p), 1)\n",
"#RQx.<x> = PolynomialRing(QQ, 1)\n",
"\n",
"def witt_pol(lista):\n",
" n = len(lista)\n",
" return sum(p^i*lista[i]^(p^(n-i-1)) for i in range(0, n))\n",
"\n",
"def witt_sum(n):\n",
" if n == 0:\n",
" return X[0] + Y[0]\n",
" 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)))\n",
"\n",
"class witt:\n",
" def __init__(self, coordinates):\n",
" self.coordinates = coordinates\n",
" def __repr__(self):\n",
" lista = [Rpx(a) for a in self.coordinates]\n",
" return str(lista)\n",
" def __add__(self, other):\n",
" lista = []\n",
" for i in range(0, N):\n",
" lista+= [witt_sum(i)(self.coordinates + other.coordinates)]\n",
" return witt(lista)\n",
" def __rmul__(self, constant):\n",
" if constant<0:\n",
" m = (-constant)*(p^N-1)\n",
" return m*self\n",
" if constant == 0:\n",
" return witt(N*[0])\n",
" return self + (constant - 1)*self\n",
" def __sub__(self, other):\n",
" return self + (-1)*other\n",
"\n",
"def fraction_pol(g):\n",
" RxX.<x, X> = PolynomialRing(QQ)\n",
" g = RxX(g)\n",
" result = 0\n",
" for a in g.monomials():\n",
" c = g.monomial_coefficient(a)\n",
" c = c%(p^2)\n",
" dX = a.degree(X)\n",
" dx = a.degree(x)\n",
" if dX%p == 0:\n",
" result += c*x^(dX//p + dx)\n",
" else:\n",
" result += c*X^(dX + dx*p)\n",
" return result\n",
" \n",
"def teichmuller(f):\n",
" Rx.<x> = PolynomialRing(QQ)\n",
" RxX.<x, X> = PolynomialRing(QQ, 2)\n",
" f = Rx(f)\n",
" ff = witt([f, 0])\n",
" coeffs = f.coefficients(sparse=false)\n",
" for i in range(0, len(coeffs)):\n",
" ff -= coeffs[i]*witt([Rx(x^i), 0])\n",
" f1 = sum(coeffs[i]*RxX(x^(i)) for i in range(0, len(coeffs))) + p*RxX(ff.coordinates[1](x = X))\n",
" f1 = fraction_pol(f1)\n",
" #RXp.<Xp> = PolynomialRing(Integers(p^2))\n",
" #f1 = RXp(f1)\n",
" return f1"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"def basis_de_rham_degrees(f, m, p):\n",
" r = f.degree()\n",
" delta = GCD(r, m)\n",
" Rx.<x> = PolynomialRing(QQ)\n",
" Rxy.<x, y> = PolynomialRing(QQ, 2)\n",
" Fxy = FractionField(Rxy)\n",
" basis_holo = holomorphic_differentials_basis(f, m, p)\n",
" basis = []\n",
" for k in range(0, len(basis_holo)):\n",
" basis += [(basis_holo[k], Rx(0))]\n",
"\n",
" ## non-holomorphic elts of H^1_dR\n",
" t = len(basis)\n",
" degrees0 = {}\n",
" degrees1 = {}\n",
" for j in range(1, m):\n",
" for i in range(1, r):\n",
" if (r*(m-j) - m*i >= delta): \n",
" s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f\n",
" psi = Rx(cut(s, i))\n",
" basis += [(Fxy(psi/y^j), Fxy(m*y^(m-j)/x^i))]\n",
" degrees0[t] = (psi.degree(), j)\n",
" degrees1[t] = (-i, m-j)\n",
" t += 1\n",
" return basis, degrees0, degrees1\n",
"\n",
"def de_rham_basis(f, m, p):\n",
" basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)\n",
" return basis\n",
"\n",
"def degrees_de_rham0(f, m, p):\n",
" basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)\n",
" return degrees0\n",
"\n",
"def degrees_de_rham1(f, m, p):\n",
" basis, degrees0, degrees1 = basis_de_rham_degrees(f, m, p)\n",
" return degrees1\n",
"\n",
"\n",
"class superelliptic:\n",
" \n",
" def __init__(self, f, m, p):\n",
" Rx.<x> = PolynomialRing(QQ)\n",
" Rxy.<x, y> = PolynomialRing(QQ, 2)\n",
" Fxy = FractionField(Rxy)\n",
" self.polynomial = Rx(f)\n",
" self.exponent = m\n",
" self.characteristic = p\n",
" \n",
" r = Rx(f).degree()\n",
" delta = GCD(r, m)\n",
" self.degree_holo = degrees_holomorphic_differentials(f, m, p)\n",
" self.degree_de_rham0 = degrees_de_rham0(f, m, p)\n",
" self.degree_de_rham1 = degrees_de_rham1(f, m, p)\n",
" \n",
" holo_basis = holomorphic_differentials_basis(f, m, p)\n",
" holo_basis_converted = []\n",
" for a in holo_basis:\n",
" holo_basis_converted += [superelliptic_form(self, a)]\n",
" \n",
" self.basis_holomorphic_differentials = holo_basis_converted\n",
" \n",
"\n",
" dr_basis = de_rham_basis(f, m, p)\n",
" dr_basis_converted = []\n",
" for (a, b) in dr_basis:\n",
" dr_basis_converted += [superelliptic_cech(self, superelliptic_form(self, a), superelliptic_function(self, b))]\n",
" \n",
" self.basis_de_rham = dr_basis_converted\n",
" \n",
" def __repr__(self):\n",
" f = self.polynomial\n",
" m = self.exponent\n",
" p = self.characteristic\n",
" return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over finite field with ' + str(p) + ' elements.'\n",
" \n",
" def is_smooth(self):\n",
" f = self.polynomial\n",
" if f.discriminant() == 0:\n",
" return 0\n",
" return 1\n",
" \n",
" def genus(self):\n",
" r = self.polynomial.degree()\n",
" m = self.exponent\n",
" delta = GCD(r, m)\n",
" return 1/2*((r-1)*(m-1) - delta + 1)\n",
" \n",
"class superelliptic_function:\n",
" def __init__(self, C, g):\n",
" p = C.characteristic\n",
" RxXy.<x, X, y> = PolynomialRing(QQ, 3)\n",
" FxXy = FractionField(RXy)\n",
" f = C.polynomial\n",
" r = f.degree()\n",
" m = C.exponent\n",
" \n",
" self.curve = C\n",
" self.function = g\n",
" \n",
" def __repr__(self):\n",
" return str(self.function)\n",
" \n",
" def jth_component(self, j):\n",
" g = self.function\n",
" C = self.curve\n",
" p = C.characteristic\n",
" RxX.<x,X> = PolynomialRing(QQ, 2)\n",
" FxX.<x> = FractionField(RxX)\n",
" FxXRy.<y> = PolynomialRing(FxX)\n",
" g = FxXRy(g)\n",
" return coff(g, j)\n",
" \n",
" def __add__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" return superelliptic_function(C, g1+g2)\n",
" \n",
" def __sub__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" return superelliptic_function(C, g1 - g2)\n",
" \n",
" def __mul__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" #g = reduction(C, g1 * g2)\n",
" return superelliptic_function(C, g1*g2)\n",
" \n",
" def __truediv__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" return superelliptic_function(C, g1 / g2)\n",
" \n",
"def diffn(self):\n",
" C = self.curve\n",
" f = C.polynomial\n",
" m = C.exponent\n",
" p = C.characteristic\n",
" g = self.function\n",
" RxXy.<x, X, y> = PolynomialRing(QQ, 3)\n",
" FxXy = FractionField(RXy)\n",
" g = RxXy(g)\n",
" A = g.derivative(X)*X^(-(p-1))/p\n",
" t = teichmuller(f)\n",
" B = g.derivative(y)*t.derivative()/(m*y^(m-1))*X^(-(p-1))/p\n",
" A1 = 0\n",
" return superelliptic_form(C, A+A1+B)\n",
" \n",
"class superelliptic_form:\n",
" def __init__(self, C, g):\n",
" p = C.characteristic\n",
" Rxy.<x, y> = PolynomialRing(QQ, 2)\n",
" Fxy = FractionField(Rxy)\n",
" g = Fxy(reduction_form(C, g))\n",
" self.form = g\n",
" self.curve = C \n",
" \n",
" def __add__(self, other):\n",
" C = self.curve\n",
" g1 = self.form\n",
" g2 = other.form\n",
" g = reduction(C, g1 + g2)\n",
" return superelliptic_form(C, g)\n",
" \n",
" def __sub__(self, other):\n",
" C = self.curve\n",
" g1 = self.form\n",
" g2 = other.form\n",
" g = reduction(C, g1 - g2)\n",
" return superelliptic_form(C, g)\n",
" \n",
" def __repr__(self):\n",
" g = self.form\n",
" if len(str(g)) == 1:\n",
" return str(g) + ' dx'\n",
" return '('+str(g) + ') dx'\n",
"\n",
" def __rmul__(self, constant):\n",
" C = self.curve\n",
" omega = self.form\n",
" return superelliptic_form(C, constant*omega) \n",
"\n",
" \n",
" def coordinates(self):\n",
" C = self.curve\n",
" p = C.characteristic\n",
" m = C.exponent\n",
" Rx.<x> = PolynomialRing(QQ)\n",
" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" g = C.genus()\n",
" degrees_holo = C.degree_holo\n",
" degrees_holo_inv = {b:a for a, b in degrees_holo.items()}\n",
" basis = C.basis_holomorphic_differentials\n",
" \n",
" for j in range(1, m):\n",
" omega_j = Fx(self.jth_component(j))\n",
" if omega_j != Fx(0):\n",
" d = degree_of_rational_fctn(omega_j, p)\n",
" index = degrees_holo_inv[(d, j)]\n",
" a = coeff_of_rational_fctn(omega_j, p)\n",
" a1 = coeff_of_rational_fctn(basis[index].jth_component(j), p)\n",
" elt = self - (a/a1)*basis[index]\n",
" return elt.coordinates() + a/a1*vector([QQ(i == index) for i in range(0, g)])\n",
" \n",
" return vector(g*[0])\n",
" \n",
" def jth_component(self, j):\n",
" g = self.form\n",
" C = self.curve\n",
" p = C.characteristic\n",
" Rx.<x> = PolynomialRing(QQ)\n",
" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" Fxy = FractionField(FxRy)\n",
" Ryinv.<y_inv> = PolynomialRing(Fx)\n",
" g = Fxy(g)\n",
" g = g(y = 1/y_inv)\n",
" g = Ryinv(g)\n",
" return coff(g, j)\n",
" \n",
" def is_regular_on_U0(self):\n",
" C = self.curve\n",
" p = C.characteristic\n",
" m = C.exponent\n",
" Rx.<x> = PolynomialRing(QQ)\n",
" for j in range(1, m):\n",
" if self.jth_component(j) not in Rx:\n",
" return 0\n",
" return 1\n",
" \n",
" def is_regular_on_Uinfty(self):\n",
" C = self.curve\n",
" p = C.characteristic\n",
" m = C.exponent\n",
" f = C.polynomial\n",
" r = f.degree()\n",
" delta = GCD(m, r)\n",
" M = m/delta\n",
" R = r/delta\n",
" \n",
" for j in range(1, m):\n",
" A = self.jth_component(j)\n",
" d = degree_of_rational_fctn(A, p)\n",
" if(-d*M + j*R -(M+1)<0):\n",
" return 0\n",
" return 1\n",
" \n",
" \n",
"class superelliptic_cech:\n",
" def __init__(self, C, omega, fct):\n",
" self.omega0 = omega\n",
" self.omega8 = omega - diffn(fct)\n",
" self.f = fct\n",
" self.curve = C\n",
" \n",
" def __add__(self, other):\n",
" C = self.curve\n",
" return superelliptic_cech(C, self.omega0 + other.omega0, self.f + other.f)\n",
" \n",
" def __sub__(self, other):\n",
" C = self.curve\n",
" return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)\n",
"\n",
" def __rmul__(self, constant):\n",
" C = self.curve\n",
" w1 = self.omega0.form\n",
" f1 = self.f.function\n",
" w2 = superelliptic_form(C, constant*w1)\n",
" f2 = superelliptic_function(C, constant*f1)\n",
" return superelliptic_cech(C, w2, f2) \n",
" \n",
" def __repr__(self):\n",
" return \"(\" + str(self.omega0) + \", \" + str(self.f) + \", \" + str(self.omega8) + \")\" \n",
"\n",
" def coordinates(self):\n",
" C = self.curve\n",
" p = C.characteristic\n",
" m = C.exponent\n",
" Rx.<x> = PolynomialRing(QQ)\n",
" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" g = C.genus()\n",
" degrees_holo = C.degree_holo\n",
" degrees_holo_inv = {b:a for a, b in degrees_holo.items()}\n",
" degrees0 = C.degree_de_rham0\n",
" degrees0_inv = {b:a for a, b in degrees0.items()}\n",
" degrees1 = C.degree_de_rham1\n",
" degrees1_inv = {b:a for a, b in degrees1.items()}\n",
" basis = C.basis_de_rham\n",
" \n",
" omega = self.omega0\n",
" fct = self.f\n",
" \n",
" if fct.function == Rx(0) and omega.form != Rx(0):\n",
" for j in range(1, m):\n",
" omega_j = Fx(omega.jth_component(j))\n",
" if omega_j != Fx(0):\n",
" d = degree_of_rational_fctn(omega_j, p)\n",
" index = degrees_holo_inv[(d, j)]\n",
" a = coeff_of_rational_fctn(omega_j, p)\n",
" a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), p)\n",
" elt = self - (a/a1)*basis[index]\n",
" return elt.coordinates() + a/a1*vector([QQ(i == index) for i in range(0, 2*g)])\n",
" \n",
" for j in range(1, m):\n",
" fct_j = Fx(fct.jth_component(j))\n",
" if (fct_j != Rx(0)):\n",
" d = degree_of_rational_fctn(fct_j, p)\n",
" \n",
" if (d, j) in degrees1.values():\n",
" index = degrees1_inv[(d, j)]\n",
" a = coeff_of_rational_fctn(fct_j, p)\n",
" elt = self - (a/m)*basis[index]\n",
" return elt.coordinates() + a/m*vector([QQ(i == index) for i in range(0, 2*g)])\n",
" \n",
" if d<0:\n",
" a = coeff_of_rational_fctn(fct_j, p)\n",
" h = superelliptic_function(C, FxRy(a*y^j*x^d))\n",
" elt = superelliptic_cech(C, self.omega0, self.f - h)\n",
" return elt.coordinates()\n",
" \n",
" if (fct_j != Rx(0)):\n",
" G = superelliptic_function(C, y^j*x^d)\n",
" a = coeff_of_rational_fctn(fct_j, p)\n",
" elt =self - a*superelliptic_cech(C, diffn(G), G)\n",
" return elt.coordinates()\n",
"\n",
" return vector(2*g*[0])\n",
" \n",
" def is_cocycle(self):\n",
" w0 = self.omega0\n",
" w8 = self.omega8\n",
" fct = self.f\n",
" if not w0.is_regular_on_U0() and not w8.is_regular_on_Uinfty():\n",
" return('w0 & w8')\n",
" if not w0.is_regular_on_U0():\n",
" return('w0')\n",
" if not w8.is_regular_on_Uinfty():\n",
" return('w8')\n",
" if w0.is_regular_on_U0() and w8.is_regular_on_Uinfty():\n",
" return 1\n",
" return 0\n",
" \n",
"def degree_of_rational_fctn(f, p):\n",
" Rx.<x> = PolynomialRing(QQ)\n",
" Fx = FractionField(Rx)\n",
" f = Fx(f)\n",
" f1 = f.numerator()\n",
" f2 = f.denominator()\n",
" d1 = f1.degree()\n",
" d2 = f2.degree()\n",
" return(d1 - d2)\n",
"\n",
"def coeff_of_rational_fctn(f, p):\n",
" Rx.<x> = PolynomialRing(QQ)\n",
" Fx = FractionField(Rx)\n",
" f = Fx(f)\n",
" if f == Rx(0):\n",
" return 0\n",
" f1 = f.numerator()\n",
" f2 = f.denominator()\n",
" d1 = f1.degree()\n",
" d2 = f2.degree()\n",
" a1 = f1.coefficients(sparse = false)[d1]\n",
" a2 = f2.coefficients(sparse = false)[d2]\n",
" return(a1/a2)\n",
"\n",
"def coff(f, d):\n",
" lista = f.coefficients(sparse = false)\n",
" if len(lista) <= d:\n",
" return 0\n",
" return lista[d]\n",
"\n",
"def cut(f, i):\n",
" R = f.parent()\n",
" coeff = f.coefficients(sparse = false)\n",
" return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))\n",
"\n",
"def polynomial_part(p, h):\n",
" Rx.<x> = PolynomialRing(QQ)\n",
" h = Rx(h)\n",
" result = Rx(0)\n",
" for i in range(0, h.degree()+1):\n",
" if (i%p) == p-1:\n",
" power = Integer((i-(p-1))/p)\n",
" result += Integer(h[i]) * x^(power) \n",
" return result"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"6*X^2 + x + 3*X + 8"
]
},
"execution_count": 4,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Rx.<x> = PolynomialRing(QQ)\n",
"f = Rx(x - 1)\n",
"teichmuller(f)"
]
},
{
"cell_type": "code",
"execution_count": 73,
"metadata": {
"collapsed": false
},
"outputs": [
{
"ename": "TypeError",
"evalue": "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'",
"output_type": "error",
"traceback": [
"\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
"\u001b[0;31mTypeError\u001b[0m Traceback (most recent call last)",
"\u001b[0;32m/tmp/ipykernel_1111/3447231159.py\u001b[0m in \u001b[0;36m<module>\u001b[0;34m\u001b[0m\n\u001b[1;32m 1\u001b[0m \u001b[0mRx\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mPolynomialRing\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mQQ\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mnames\u001b[0m\u001b[0;34m=\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m'x'\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m;\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mRx\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_first_ngens\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m----> 2\u001b[0;31m \u001b[0mC\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0msuperelliptic\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m2\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m3\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m",
"\u001b[0;32m/tmp/ipykernel_1111/3436947063.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, f, m, p)\u001b[0m\n\u001b[1;32m 68\u001b[0m \u001b[0mdr_basis_converted\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 69\u001b[0m \u001b[0;32mfor\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0ma\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;32min\u001b[0m \u001b[0mdr_basis\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m---> 70\u001b[0;31m \u001b[0mdr_basis_converted\u001b[0m \u001b[0;34m+=\u001b[0m \u001b[0;34m[\u001b[0m\u001b[0msuperelliptic_cech\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msuperelliptic_form\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0ma\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0msuperelliptic_function\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mb\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m]\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 71\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 72\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbasis_de_rham\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mdr_basis_converted\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/tmp/ipykernel_1111/3436947063.py\u001b[0m in \u001b[0;36m__init__\u001b[0;34m(self, C, omega, fct)\u001b[0m\n\u001b[1;32m 260\u001b[0m \u001b[0;32mdef\u001b[0m \u001b[0m__init__\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0momega\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mfct\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 261\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0momega0\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0momega\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 262\u001b[0;31m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0momega8\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0momega\u001b[0m \u001b[0;34m-\u001b[0m \u001b[0mdiffn\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mfct\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 263\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mf\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mfct\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 264\u001b[0m \u001b[0mself\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mcurve\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mC\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/tmp/ipykernel_1111/3436947063.py\u001b[0m in \u001b[0;36mdiffn\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m 153\u001b[0m \u001b[0mA\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mderivative\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mXp\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mXp\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 154\u001b[0m \u001b[0mt\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mteichmuller\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mf\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m--> 155\u001b[0;31m \u001b[0mB\u001b[0m \u001b[0;34m=\u001b[0m \u001b[0mg\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mderivative\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mt\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mderivative\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0my\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mm\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m*\u001b[0m\u001b[0mXp\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m-\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m1\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m/\u001b[0m\u001b[0mp\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 156\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0msuperelliptic_form\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mC\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mA\u001b[0m\u001b[0;34m+\u001b[0m\u001b[0mB\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 157\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/element.pyx\u001b[0m in \u001b[0;36msage.structure.element.Element.__mul__ (build/cythonized/sage/structure/element.c:12253)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1514\u001b[0m \u001b[0;32mreturn\u001b[0m \u001b[0;34m(\u001b[0m\u001b[0;34m<\u001b[0m\u001b[0mElement\u001b[0m\u001b[0;34m>\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0m_mul_\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mright\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1515\u001b[0m \u001b[0;32mif\u001b[0m \u001b[0mBOTH_ARE_ELEMENT\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mcl\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1516\u001b[0;31m \u001b[0;32mreturn\u001b[0m \u001b[0mcoercion_model\u001b[0m\u001b[0;34m.\u001b[0m\u001b[0mbin_op\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mleft\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mright\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mmul\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 1517\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1518\u001b[0m \u001b[0mcdef\u001b[0m \u001b[0mlong\u001b[0m \u001b[0mvalue\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;32m/ext/sage/9.5/local/var/lib/sage/venv-python3.9.9/lib/python3.9/site-packages/sage/structure/coerce.pyx\u001b[0m in \u001b[0;36msage.structure.coerce.CoercionModel.bin_op (build/cythonized/sage/structure/coerce.c:11751)\u001b[0;34m()\u001b[0m\n\u001b[1;32m 1246\u001b[0m \u001b[0;31m# We should really include the underlying error.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1247\u001b[0m \u001b[0;31m# This causes so much headache.\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0;32m-> 1248\u001b[0;31m \u001b[0;32mraise\u001b[0m \u001b[0mbin_op_exception\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mop\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n\u001b[0m\u001b[1;32m 1249\u001b[0m \u001b[0;34m\u001b[0m\u001b[0m\n\u001b[1;32m 1250\u001b[0m \u001b[0mcpdef\u001b[0m \u001b[0mcanonical_coercion\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0mself\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0mx\u001b[0m\u001b[0;34m,\u001b[0m \u001b[0my\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m:\u001b[0m\u001b[0;34m\u001b[0m\u001b[0;34m\u001b[0m\u001b[0m\n",
"\u001b[0;31mTypeError\u001b[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'"
]
}
],
"source": [
"Rx.<x> = PolynomialRing(QQ)\n",
"C = superelliptic(x^3 - x, 2, 3)"
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[((1/y) dx, 0, (1/y) dx), ((x/y) dx, 2/x*y, ((-1)/(x*y)) dx)]"
]
},
"execution_count": 58,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"C.basis_de_rham"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"g = basis_de_rham_degrees(x^3 - x, 2, 3)"
]
},
{
"cell_type": "code",
"execution_count": 54,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"RXpy.<Xp, y> = PolynomialRing(QQ, 2)\n",
"FXpy = FractionField(RXpy)"
]
},
{
"cell_type": "code",
"execution_count": 55,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"2*y/Xp^3"
]
},
"execution_count": 55,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"g(x = Xp^p, y = y)"
]
},
{
"cell_type": "code",
"execution_count": 68,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"t = teichmuller(x^3 - x)"
]
},
{
"cell_type": "code",
"execution_count": 71,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"6*Xp^6 + 6*Xp^4 + 6*Xp^2"
]
},
"execution_count": 71,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"t.derivative()"
]
},
{
"cell_type": "code",
"execution_count": 102,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"RxX.<x, X> = PolynomialRing(QQ)"
]
},
{
"cell_type": "code",
"execution_count": 103,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"g = RxX(x*X + 2*x*X^3)"
]
},
{
"cell_type": "code",
"execution_count": 82,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"3"
]
},
"execution_count": 82,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"g.monomials()[0].degree(X)"
]
},
{
"cell_type": "code",
"execution_count": 106,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"<class 'sage.rings.rational.Rational'>"
]
},
"execution_count": 106,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"type(g.monomial_coefficient(X))"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.5",
"language": "sagemath",
"metadata": {
"cocalc": {
"description": "Open-source mathematical software system",
"priority": 10,
"url": "https://www.sagemath.org/"
}
},
"name": "sage-9.5",
"resource_dir": "/ext/jupyter/kernels/sage-9.5"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.9.9"
}
},
"nbformat": 4,
"nbformat_minor": 4
}

559
deRhamComputation.ipynb
View File

@ -1,559 +0,0 @@
{
"cells": [
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"# Theory\n",
"Let $C : y^m = f(x)$. Then:\n",
"\n",
" - the basis of $H^0(C, \\Omega_{C/k})$ is given by:\n",
" $$x^{i-1} dx/y^j,$$\n",
" where $1 \\le i \\le r-1$, $1 \\le j \\le m-1$, $-mi + rj \\ge \\delta$ and $\\delta := GCD(m, r)$, $r := \\deg f$.\n",
" \n",
" - the above forms along with\n",
" $$\\lambda_{ij} = \\left[ \\left( \\frac{\\psi_{ij} \\, dx}{m x^{i+1} y^{m - j}},\n",
" \\frac{-\\phi_{ij} \\, dx}{m x^{i+1} y^{m - j}}, \\frac{y^j}{x^i} \\right) \\right]$$\n",
" (where $s_{ij} = jx f'(x) - mi f(x)$, \n",
" $\\psi_{ij}(x) = s_{ij}^{\\ge i+1}$,\n",
" $\\phi_{ij}(x) = s_{ij}^{< i+1}$)\n",
"form a basis of $H^1_{dR}(C/K)$."
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"# The program computes the basis of holomorphic differentials of y^m = f(x) in char p.\n",
"# The coefficient j means that we compute the j-th eigenpart, i.e.\n",
"# forms y^j * f(x) dx. Output is [f(x), 0]\n",
"\n",
"def baza_holo(m, f, j, p):\n",
" R.<x> = PolynomialRing(GF(p))\n",
" f = R(f)\n",
" r = f.degree()\n",
" delta = GCD(m, r)\n",
" baza = {}\n",
" k = 0\n",
" for i in range(1, r):\n",
" if (r*j - m*i >= delta):\n",
" baza[k] = [x^(i-1), R(0), j]\n",
" k = k+1\n",
" return baza"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"# The program computes the basis of de Rham cohomology of y^m = f(x) in char p.\n",
"# We treat them as pairs [omega, f], where omega is regular on the affine part\n",
"# and omega - df is regular on the second atlas.\n",
"# The coefficient j means that we compute the j-th eigenpart, i.e.\n",
"# [f(x) dx/y^j, y^(m-j)*g(x)]. Output is [f(x), g(x)]\n",
"\n",
"def baza_dr(m, f, j, p):\n",
" R.<x> = PolynomialRing(GF(p))\n",
" f = R(f) \n",
" r = f.degree()\n",
" delta = GCD(m, r)\n",
" baza = {}\n",
" holo = baza_holo(m, f, j, p)\n",
" for k in range(0, len(holo)):\n",
" baza[k] = holo[k]\n",
" \n",
" k = len(baza)\n",
" \n",
" for i in range(1, r):\n",
" if (r*(m-j) - m*i >= delta):\n",
" s = R(m-j)*R(x)*R(f.derivative()) - R(m)*R(i)*f\n",
" psi = R(obciecie(s, i, p))\n",
" baza[k] = [psi, R(m)/x^i, j]\n",
" k = k+1\n",
" return baza"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"#auxiliary programs\n",
"def stopnie_bazy_holo(m, f, j, p):\n",
" baza = baza_holo(m, f, j, p)\n",
" stopnie = {}\n",
" for k in range(0, len(baza)):\n",
" stopnie[k] = baza[k][0].degree()\n",
" return stopnie\n",
"\n",
"def stopnie_bazy_dr(m, f, j, p):\n",
" baza = baza_dr(m, f, j, p)\n",
" stopnie = {}\n",
" for k in range(0, len(baza)):\n",
" stopnie[k] = baza[k][0].degree()\n",
" return stopnie\n",
"\n",
"def stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p):\n",
" baza = baza_dr(m, f, j, p)\n",
" stopnie = {}\n",
" for k in range(0, len(baza)):\n",
" if baza[k][1] != 0:\n",
" stopnie[k] = baza[k][1].denominator().degree()\n",
" return stopnie\n",
"\n",
"def obciecie(f, i, p):\n",
" R.<x> = PolynomialRing(GF(p))\n",
" f = R(f)\n",
" coeff = f.coefficients(sparse = false)\n",
" return sum(x^(j-i-1) * coeff[j] for j in range(i+1, f.degree() + 1))\n",
"\n",
"\n",
"#Any element [f dx, g] is represented as a combination of the basis vectors.\n",
"\n",
"def zapis_w_bazie_dr(elt, m, f, p):\n",
" j = elt[2]\n",
" R.<x> = PolynomialRing(GF(p))\n",
" RR = FractionField(R)\n",
" f = R(f)\n",
" r = f.degree()\n",
" delta = GCD(m, r)\n",
" baza = baza_dr(m, f, j, p)\n",
" wymiar = len(baza)\n",
" zapis = vector([GF(p)(0) for i in baza])\n",
" stopnie = stopnie_bazy_dr(m, f, j, p)\n",
" inv_stopnie = {v: k for k, v in stopnie.items()}\n",
" stopnie_holo = stopnie_bazy_holo(m, f, j, p)\n",
" inv_stopnie_holo = {v: k for k, v in stopnie_holo.items()} \n",
" \n",
" ## zmiana\n",
" if elt[0]== 0 and elt[1] == 0:\n",
" return zapis\n",
" \n",
" if elt[1] == 0:\n",
" elt[0] = R(elt[0])\n",
" d = elt[0].degree()\n",
" a = elt[0].coefficients(sparse = false)[d]\n",
" k = inv_stopnie_holo[d] #ktory element bazy jest stopnia d? ten o indeksie k\n",
" \n",
" a1 = baza[k][0].coefficients(sparse = false)[d]\n",
" elt1 = [R(0),R(0),j]\n",
" elt1[0] = elt[0] - a/a1 * baza[k][0]\n",
" elt1[1] = R(0)\n",
" return zapis_w_bazie_dr(elt1, m, f, p) + vector([a/a1*GF(p)(i == k) for i in range(0, len(baza))])\n",
"\n",
" g = elt[1]\n",
" a = wspolczynnik_wiodacy(g)\n",
" d = -stopien_roznica(g)\n",
" Rr = r/delta\n",
" Mm = m/delta\n",
" \n",
" stopnie2 = stopnie_drugiej_wspolrzednej_bazy_dr(m, f, j, p)\n",
" inv_stopnie2 = {v: k for k, v in stopnie2.items()}\n",
" if (d not in stopnie2.values()):\n",
" if d> 0:\n",
" j1 = m-j\n",
" elt1 = [elt[0], RR(elt[1]) - a*1/R(x^d), j]\n",
" else:\n",
" d1 = -d\n",
" j1 = m-j\n",
" elt1 = [elt[0] - a*(j1*x^(d1) * f.derivative()/m + d1*f*x^(d1 - 1)), RR(elt[1]) - a*R(x^(d1)), j]\n",
" return zapis_w_bazie_dr(elt1, m, f, p)\n",
" \n",
" k = inv_stopnie2[d]\n",
" b = wspolczynnik_wiodacy(baza[k][1])\n",
" elt1 = [R(0), R(0), j]\n",
" elt1[0] = elt[0] - a/b*baza[k][0]\n",
" elt1[1] = elt[1] - a/b*baza[k][1]\n",
" return zapis_w_bazie_dr(elt1, m, f, p) + vector([a*GF(p)(i == k) for i in range(0, len(baza))])\n",
" \n",
" \n",
"def zapis_w_bazie_holo(elt, m, f, p):\n",
" j = elt[2]\n",
" R.<x> = PolynomialRing(GF(p))\n",
" f = R(f) \n",
" r = f.degree()\n",
" delta = GCD(m, r)\n",
" baza = baza_holo(m, f, j, p)\n",
" wymiar = len(baza)\n",
" zapis = vector([GF(p)(0) for i in baza])\n",
" stopnie = stopnie_bazy_holo(m, f, j, p)\n",
" inv_stopnie = {v: k for k, v in stopnie.items()}\n",
" \n",
" if elt[0] == 0:\n",
" return zapis\n",
" \n",
" d = elt[0].degree()\n",
" a = elt[0].coefficients(sparse = false)[d]\n",
" \n",
" k = inv_stopnie[d] #ktory element bazy jest stopnia d? ten o indeksie k\n",
" \n",
" a1 = baza[k][0].coefficients(sparse = false)[d]\n",
" elt1 = [R(0),R(0), j]\n",
" elt1[0] = elt[0] - a/a1 * baza[k][0]\n",
" \n",
" return zapis_w_bazie_holo(elt1, m, f, p) + vector([a/a1 * GF(p)(i == k) for i in range(0, len(baza))])\n"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"We have: $V(\\omega, f) = (C(\\omega), 0)$ and $F(\\omega, f) = (0, f^p)$, where C denotes the Cartier operator. Moreover:\n",
"\n",
"let $t = multord_m(p)$, $M := (p^t - 1)/m$. Then: $y^{p^t - 1} = f(x)^M$ and $1/y = f(x)^M/y^{p^t}$. Thus:\n",
"\n",
"\n",
"$$ C(P(x) \\, dx / y^j) = C(P(x) \\, f(x)^{M \\cdot j} \\, dx /y^{p^t \\cdot j}) = \\frac{1}{y^{p^{t - 1} \\cdot j}} C(P(x) \\, f(x)^{M \\cdot j} \\, dx) = \\frac{1}{y^{(p^{t - 1} \\cdot j) \\, mod \\, m}} \\cdot \\frac{1}{f(x)^{[p^{t - 1} \\cdot j/m]}} \\cdot C(P(x) \\, f(x)^{M \\cdot j} \\, dx)$$\n"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"def czesc_wielomianu(p, h):\n",
" R.<x> = PolynomialRing(GF(p))\n",
" h = R(h)\n",
" wynik = R(0)\n",
" for i in range(0, h.degree()+1):\n",
" if (i%p) == p-1:\n",
" potega = Integer((i-(p-1))/p)\n",
" wynik = wynik + Integer(h[i]) * x^(potega) \n",
" return wynik\n",
"\n",
"def cartier_dr(p, m, f, elt, j): #Cartier na y^m = f dla elt = [forma rozniczkowa, fkcja]\n",
" R.<x> = PolynomialRing(GF(p))\n",
" f = R(f)\n",
" r = f.degree()\n",
" delta = GCD(m, r)\n",
" rzad = Integers(m)(p).multiplicative_order()\n",
" M = Integer((p^(rzad)-1)/m)\n",
" W = R(elt[0])\n",
" h = R(W*f^(M*j))\n",
" B = floor(p^(rzad-1)*j/m)\n",
" g = czesc_wielomianu(p, h)/f^B\n",
" jj = (p^(rzad-1)*j)%m\n",
" #jj = Integers(m)(j/p)\n",
" return [g, 0, jj] #jest to w czesci indeksowanej jj\n",
"\n",
"def macierz_cartier_dr(p, m, f, j):\n",
" baza = baza_dr(m, f, j, p)\n",
" A = matrix(GF(p), len(baza), len(baza))\n",
" for k in range(0, len(baza)):\n",
" cart = cartier_dr(p, m, f, baza[k], j)\n",
" v = zapis_w_bazie_dr(cart, m, f, p)\n",
" A[k, :] = matrix(v)\n",
" return A.transpose()"
]
},
{
"cell_type": "markdown",
"metadata": {
"collapsed": false
},
"source": [
"$F((\\omega, P(x) \\cdot y^j)) = (0, P(x)^p \\cdot y^{p \\cdot j}) = (0, P(x)^p \\cdot f(x)^{[p \\cdot j/m]} \\cdot y^{(p \\cdot j) \\, mod \\, m})$"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"def frobenius_dr(p, m, f, elt, j): #Cartier na y^m = f dla elt = [forma rozniczkowa, fkcja]\n",
" R.<x> = PolynomialRing(GF(p))\n",
" RR = FractionField(R)\n",
" f = R(f)\n",
" j1 = m-j\n",
" M = floor(j1*p/m)\n",
" return [0, f^M * RR(elt[1])^p, (j1*p)%m] #eigenspace = j1*p mod m\n",
"\n",
"def macierz_frob_dr(p, m, f, j):\n",
" baza = baza_dr(m, f, j, p)\n",
" A = matrix(GF(p), len(baza), len(baza))\n",
" for k in range(0, len(baza)):\n",
" frob = frobenius_dr(p, m, f, baza[k], j)\n",
" v = zapis_w_bazie_dr(frob, m, f, p)\n",
" A[k, :] = matrix(v)\n",
" return A.transpose()\n",
"\n",
"def wspolczynnik_wiodacy(f):\n",
" R.<x> = PolynomialRing(GF(p))\n",
" RR = FractionField(R)\n",
" f = RR(f)\n",
" f1 = f.numerator()\n",
" f2 = f.denominator()\n",
" d1 = f1.degree()\n",
" d2 = f2.degree()\n",
" a1 = f1.coefficients(sparse = false)[d1]\n",
" a2 = f2.coefficients(sparse = false)[d2]\n",
" return(a1/a2)\n",
"\n",
"def stopien_roznica(f):\n",
" R.<x> = PolynomialRing(GF(p))\n",
" RR = FractionField(R)\n",
" f = RR(f)\n",
" f1 = f.numerator()\n",
" f2 = f.denominator()\n",
" d1 = f1.degree()\n",
" d2 = f2.degree()\n",
" return(d1 - d2)\n",
"\n",
"def czy_w_de_rhamie(elt, m, f, j, p):\n",
" j1 = m - j\n",
" R.<x> = PolynomialRing(GF(p))\n",
" RR = FractionField(R)\n",
" f = R(f)\n",
" elt = [RR(elt[0]), RR(elt[1])]\n",
" auxiliary = elt[0] - j1/m*elt[1]*f.derivative() - f*elt[1].derivative()\n",
" deg = stopien_roznica(auxiliary)\n",
" \n",
" r = f.degree()\n",
" delta = GCD(r, m)\n",
" Rr = r/delta\n",
" Mm = m/delta\n",
" return(j*Rr - deg*Mm >= 0)\n",
"\n",
"def full_cartier(m, f, p):\n",
" R.<x> = PolynomialRing(GF(p))\n",
" f = R(f)\n",
" r = f.degree()\n",
" delta = GCD(m, r)\n",
" g = 1/2*((m-1)*(r-1) - delta)\n",
" print(g)\n",
" \n",
" wymiary = [0]+[len(baza_holo(m, f, j, p)) for j in range(1, m)]\n",
" print(wymiary)\n",
" for j1 in range(1, m):\n",
" for j2 in range(1, m):\n",
" print(j1, j2)\n",
" print(matrix(GF(p), wymiary[j1], wymiary[j2]))\n",
" lista = [[matrix(GF(p), wymiary[j1], wymiary[j2]) for j1 in range(0, m)] for j2 in range(0, m)]\n",
" rzad = Integers(m)(p).multiplicative_order()\n",
" \n",
" for j in range(1, m):\n",
" jj = (p^(rzad-1)*j)%m\n",
" print(j, jj)\n",
" print('wymiary', macierz_cartier_dr(p, m, f, j).dimensions(), wymiary[j], wymiary[jj])\n",
" lista[j][jj] = macierz_cartier_dr(p, m, f, j)\n",
" return lista \n",
" return block_matrix(lista)"
]
},
{
"cell_type": "code",
"execution_count": 243,
"metadata": {
"collapsed": false
},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"5/2\n",
"[0, 0, 1, 2]\n",
"1 1\n",
"[]\n",
"1 2\n",
"[]\n",
"1 3\n",
"[]\n",
"2 1\n",
"[]\n",
"2 2\n",
"[0]\n",
"2 3\n",
"[0 0]\n",
"3 1\n",
"[]\n",
"3 2\n",
"[0]\n",
"[0]\n",
"3 3\n",
"[0 0]\n",
"[0 0]\n",
"1 1\n",
"wymiary (2, 2) 0 0\n",
"2 2\n",
"wymiary (2, 2) 1 1\n",
"3 3\n",
"wymiary (2, 2) 2 2\n"
]
}
],
"source": [
"m = 4\n",
"p = 5\n",
"f = x^3 + x+2\n",
"lista = full_cartier(m, f, p)"
]
},
{
"cell_type": "code",
"execution_count": 241,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[\n",
" [2 3] [0]\n",
"[], [], [0 0], [0]\n",
"]"
]
},
"execution_count": 241,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"lista[2]"
]
},
{
"cell_type": "code",
"execution_count": 247,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[0 0]\n",
"[0 0]"
]
},
"execution_count": 247,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"macierz_cartier_dr(p, m, f, 1)"
]
},
{
"cell_type": "code",
"execution_count": 249,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"{0: [3*x, 4/x, 0], 1: [4, 4/x^2, 0]}"
]
},
"execution_count": 249,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"baza_dr(m, f, 0, p)"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"{0: [1, 0, 3], 1: [0, 2/x, 3]}"
]
},
"execution_count": 8,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"p = 5\n",
"R.<x> = PolynomialRing(GF(p))\n",
"f = x^3 + x + 2\n",
"m = 7\n",
"baza_dr(m, f, 3, p)"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.5",
"language": "sagemath",
"metadata": {
"cocalc": {
"description": "Open-source mathematical software system",
"priority": 10,
"url": "https://www.sagemath.org/"
}
},
"name": "sage-9.5",
"resource_dir": "/ext/jupyter/kernels/sage-9.5"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.3"
}
},
"nbformat": 4,
"nbformat_minor": 4
}

2453
elementary_covers_of_superelliptic_curves.ipynb
View File

File diff suppressed because it is too large Load Diff

21
sage/.run.term-0.term
View File

@ -55303,4 +55303,23 @@ Untracked files:
no changes added to commit (use "git add" and/or "git commit -a") no changes added to commit (use "git add" and/or "git commit -a")
[?2004h]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ sage add git add C.x^33C.x^3C.x^33C.x^3 C.x^33C.x^3 sage/superelliptic/frobenius_kernel.sagesage/superelliptic/frobenius_kernel.sage [?2004h]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ sage add git add C.x^33C.x^3C.x^33C.x^3 C.x^33C.x^3 sage/superelliptic/frobenius_kernel.sagesage/superelliptic/frobenius_kernel.sage
[?2004l [?2004h]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git add -u [?2004l [?2004h]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git add -u
[?2004l [?2004h]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git commit -m "frobenius kerlnnel"
[?2004l [master 599aa94] frobenius kernel
8 files changed, 20509 insertions(+), 53 deletions(-)
rewrite sage/drafty/draft.sage (100%)
create mode 100644 sage/superelliptic/frobenius_kernel.sage
[?2004h]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ git push
[?2004l Username for 'https://git.wmi.amu.edu.pl': jgarnek
Password for 'https://jgarnek@git.wmi.amu.edu.pl':
Enumerating objects: 25, done.
Counting objects: 4% (1/25) Counting objects: 8% (2/25) Counting objects: 12% (3/25) Counting objects: 16% (4/25) Counting objects: 20% (5/25) Counting objects: 24% (6/25) Counting objects: 28% (7/25) Counting objects: 32% (8/25) Counting objects: 36% (9/25) Counting objects: 40% (10/25) Counting objects: 44% (11/25) Counting objects: 48% (12/25) Counting objects: 52% (13/25) Counting objects: 56% (14/25) Counting objects: 60% (15/25) Counting objects: 64% (16/25) Counting objects: 68% (17/25) Counting objects: 72% (18/25) Counting objects: 76% (19/25) Counting objects: 80% (20/25) Counting objects: 84% (21/25) Counting objects: 88% (22/25) Counting objects: 92% (23/25) Counting objects: 96% (24/25) Counting objects: 100% (25/25) Counting objects: 100% (25/25), done.
Delta compression using up to 4 threads
Compressing objects: 7% (1/14) Compressing objects: 14% (2/14) Compressing objects: 21% (3/14) Compressing objects: 28% (4/14) Compressing objects: 35% (5/14) Compressing objects: 42% (6/14) Compressing objects: 50% (7/14) Compressing objects: 57% (8/14) Compressing objects: 64% (9/14) Compressing objects: 71% (10/14) Compressing objects: 78% (11/14) Compressing objects: 85% (12/14) Compressing objects: 92% (13/14) Compressing objects: 100% (14/14) Compressing objects: 100% (14/14), done.
Writing objects: 7% (1/14) Writing objects: 14% (2/14) Writing objects: 21% (3/14) Writing objects: 28% (4/14) Writing objects: 35% (5/14) Writing objects: 42% (6/14) Writing objects: 50% (7/14) Writing objects: 57% (8/14) Writing objects: 64% (9/14) Writing objects: 71% (10/14) Writing objects: 78% (11/14) Writing objects: 85% (12/14) Writing objects: 92% (13/14) Writing objects: 100% (14/14) Writing objects: 100% (14/14), 306.88 KiB | 1.59 MiB/s, done.
Total 14 (delta 10), reused 0 (delta 0), pack-reused 0
remote: . Processing 1 references
remote: Processed 1 references in total
To https://git.wmi.amu.edu.pl/jgarnek/DeRhamComputation.git
5e738d4..599aa94 master -> master
[?2004h]0;~/Research/2021 De Rham/DeRhamComputation~/Research/2021 De Rham/DeRhamComputation$ g

966
superelliptic.ipynb
View File

@ -1,966 +0,0 @@
{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"class superelliptic:\n",
" \n",
" def __init__(self, f, m):\n",
" Rx = f.parent()\n",
" x = Rx.gen()\n",
" F = Rx.base() \n",
" Rx.<x> = PolynomialRing(F)\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
" Fxy = FractionField(Rxy)\n",
" self.polynomial = Rx(f)\n",
" self.exponent = m\n",
" self.base_ring = F\n",
" self.characteristic = F.characteristic()\n",
" \n",
" r = Rx(f).degree()\n",
" delta = GCD(r, m)\n",
" \n",
" def __repr__(self):\n",
" f = self.polynomial\n",
" m = self.exponent\n",
" F = self.base_ring\n",
" return 'Superelliptic curve with the equation y^' + str(m) + ' = ' + str(f)+' over ' + str(F)\n",
"\n",
" \n",
" def basis_holomorphic_differentials_degree(self):\n",
" f = self.polynomial\n",
" m = self.exponent\n",
" r = f.degree()\n",
" delta = GCD(r, m)\n",
" F = self.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
" Fxy = FractionField(Rxy)\n",
" #########basis of holomorphic differentials and de Rham\n",
"\n",
" basis_holo = []\n",
" degrees0 = {}\n",
" k = 0\n",
"\n",
" for j in range(1, m):\n",
" for i in range(1, r):\n",
" if (r*j - m*i >= delta):\n",
" basis_holo += [superelliptic_form(self, Fxy(x^(i-1)/y^j))]\n",
" degrees0[k] = (i-1, j)\n",
" k = k+1\n",
"\n",
" return(basis_holo, degrees0)\n",
"\n",
" def holomorphic_differentials_basis(self):\n",
" basis_holo, degrees0 = self.basis_holomorphic_differentials_degree()\n",
" return basis_holo\n",
" \n",
" def degrees_holomorphic_differentials(self):\n",
" basis_holo, degrees0 = self.basis_holomorphic_differentials_degree()\n",
" return degrees0\n",
"\n",
" def basis_de_rham_degrees(self):\n",
" f = self.polynomial\n",
" m = self.exponent\n",
" r = f.degree()\n",
" delta = GCD(r, m)\n",
" F = self.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
" Fxy = FractionField(Rxy)\n",
" basis_holo = self.holomorphic_differentials_basis()\n",
" basis = []\n",
" for k in range(0, len(basis_holo)):\n",
" basis += [superelliptic_cech(self, basis_holo[k], superelliptic_function(self, 0))]\n",
"\n",
" ## non-holomorphic elts of H^1_dR\n",
" t = len(basis)\n",
" degrees0 = {}\n",
" degrees1 = {}\n",
" for j in range(1, m):\n",
" for i in range(1, r):\n",
" if (r*(m-j) - m*i >= delta): \n",
" s = Rx(m-j)*Rx(x)*Rx(f.derivative()) - Rx(m)*Rx(i)*f\n",
" psi = Rx(cut(s, i))\n",
" basis += [superelliptic_cech(self, superelliptic_form(self, Fxy(psi/y^j)), superelliptic_function(self, Fxy(m*y^(m-j)/x^i)))]\n",
" degrees0[t] = (psi.degree(), j)\n",
" degrees1[t] = (-i, m-j)\n",
" t += 1\n",
" return basis, degrees0, degrees1\n",
"\n",
" def de_rham_basis(self):\n",
" basis, degrees0, degrees1 = self.basis_de_rham_degrees()\n",
" return basis\n",
"\n",
" def degrees_de_rham0(self):\n",
" basis, degrees0, degrees1 = self.basis_de_rham_degrees()\n",
" return degrees0\n",
"\n",
" def degrees_de_rham1(self):\n",
" basis, degrees0, degrees1 = self.basis_de_rham_degrees()\n",
" return degrees1 \n",
" \n",
" def is_smooth(self):\n",
" f = self.polynomial\n",
" if f.discriminant() == 0:\n",
" return 0\n",
" return 1\n",
" \n",
" def genus(self):\n",
" r = self.polynomial.degree()\n",
" m = self.exponent\n",
" delta = GCD(r, m)\n",
" return 1/2*((r-1)*(m-1) - delta + 1)\n",
" \n",
" def verschiebung_matrix(self):\n",
" basis = self.de_rham_basis()\n",
" g = self.genus()\n",
" p = self.characteristic\n",
" F = self.base_ring\n",
" M = matrix(F, 2*g, 2*g)\n",
" for i in range(0, len(basis)):\n",
" w = basis[i]\n",
" v = w.verschiebung().coordinates()\n",
" M[i, :] = v\n",
" return M\n",
" \n",
" def frobenius_matrix(self):\n",
" basis = self.de_rham_basis()\n",
" g = self.genus()\n",
" p = self.characteristic\n",
" F = self.base_ring\n",
" M = matrix(F, 2*g, 2*g)\n",
" \n",
" for i in range(0, len(basis)):\n",
" w = basis[i]\n",
" v = w.frobenius().coordinates()\n",
" M[i, :] = v\n",
" return M\n",
"\n",
" def cartier_matrix(self):\n",
" basis = self.holomorphic_differentials_basis()\n",
" g = self.genus()\n",
" p = self.characteristic\n",
" F = self.base_ring\n",
" M = matrix(F, g, g)\n",
" for i in range(0, len(basis)):\n",
" w = basis[i]\n",
" v = w.cartier().coordinates()\n",
" M[i, :] = v\n",
" return M \n",
"\n",
"# def p_rank(self):\n",
"# return self.cartier_matrix().rank()\n",
" \n",
" def a_number(self):\n",
" g = C.genus()\n",
" return g - self.cartier_matrix().rank()\n",
" \n",
" def final_type(self, test = 0):\n",
" Fr = self.frobenius_matrix()\n",
" V = self.verschiebung_matrix()\n",
" p = self.characteristic\n",
" return flag(Fr, V, p, test)\n",
" \n",
"def reduction(C, g):\n",
" p = C.characteristic\n",
" F = C.base_ring\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
" Fxy = FractionField(Rxy)\n",
" f = C.polynomial\n",
" r = f.degree()\n",
" m = C.exponent\n",
" g = Fxy(g)\n",
" g1 = g.numerator()\n",
" g2 = g.denominator()\n",
" \n",
" Rx.<x> = PolynomialRing(F)\n",
" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx) \n",
" (A, B, C) = xgcd(FxRy(g2), FxRy(y^m - f))\n",
" g = FxRy(g1*B/A)\n",
" \n",
" while(g.degree(Rxy(y)) >= m):\n",
" d = g.degree(Rxy(y))\n",
" G = coff(g, d)\n",
" i = floor(d/m)\n",
" g = g - G*y^d + f^i * y^(d%m) *G\n",
" \n",
" return(FxRy(g))\n",
"\n",
"def reduction_form(C, g):\n",
" F = C.base_ring\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
" Fxy = FractionField(Rxy)\n",
" f = C.polynomial\n",
" r = f.degree()\n",
" m = C.exponent\n",
" g = reduction(C, g)\n",
"\n",
" g1 = Rxy(0)\n",
" Rx.<x> = PolynomialRing(F)\n",
" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" \n",
" g = FxRy(g)\n",
" for j in range(0, m):\n",
" if j==0:\n",
" G = coff(g, 0)\n",
" g1 += FxRy(G)\n",
" else:\n",
" G = coff(g, j)\n",
" g1 += Fxy(y^(j-m)*f*G)\n",
" return(g1)\n",
" \n",
"class superelliptic_function:\n",
" def __init__(self, C, g):\n",
" F = C.base_ring\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
" Fxy = FractionField(Rxy)\n",
" f = C.polynomial\n",
" r = f.degree()\n",
" m = C.exponent\n",
" \n",
" self.curve = C\n",
" g = reduction(C, g)\n",
" self.function = g\n",
" \n",
" def __repr__(self):\n",
" return str(self.function)\n",
" \n",
" def jth_component(self, j):\n",
" g = self.function\n",
" C = self.curve\n",
" F = C.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" Fx.<x> = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" g = FxRy(g)\n",
" return coff(g, j)\n",
" \n",
" def __add__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" g = reduction(C, g1 + g2)\n",
" return superelliptic_function(C, g)\n",
" \n",
" def __sub__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" g = reduction(C, g1 - g2)\n",
" return superelliptic_function(C, g)\n",
" \n",
" def __mul__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" g = reduction(C, g1 * g2)\n",
" return superelliptic_function(C, g)\n",
" \n",
" def __truediv__(self, other):\n",
" C = self.curve\n",
" g1 = self.function\n",
" g2 = other.function\n",
" g = reduction(C, g1 / g2)\n",
" return superelliptic_function(C, g)\n",
"\n",
" def diffn(self):\n",
" C = self.curve\n",
" f = C.polynomial\n",
" m = C.exponent\n",
" F = C.base_ring\n",
" g = self.function\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
" Fxy = FractionField(Rxy)\n",
" g = Fxy(g)\n",
" A = g.derivative(x)\n",
" B = g.derivative(y)*f.derivative(x)/(m*y^(m-1))\n",
" return superelliptic_form(C, A+B)\n",
"\n",
" \n",
" def expansion_at_infty(self, i = 0, prec=10):\n",
" C = self.curve\n",
" f = C.polynomial\n",
" m = C.exponent\n",
" F = C.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" f = Rx(f)\n",
" Rt.<t> = LaurentSeriesRing(F, default_prec=prec)\n",
" RptW.<W> = PolynomialRing(Rt)\n",
" RptWQ = FractionField(RptW)\n",
" Rxy.<x, y> = PolynomialRing(F)\n",
" RxyQ = FractionField(Rxy)\n",
" fct = self.function\n",
" fct = RxyQ(fct)\n",
" r = f.degree()\n",
" delta, a, b = xgcd(m, r)\n",
" b = -b\n",
" M = m/delta\n",
" R = r/delta\n",
" while a<0:\n",
" a += R\n",
" b += M\n",
" \n",
" g = (x^r*f(x = 1/x))\n",
" gW = RptWQ(g(x = t^M * W^b)) - W^(delta)\n",
" ww = naive_hensel(gW, F, start = root_of_unity(F, delta), prec = prec)\n",
" xx = Rt(1/(t^M*ww^b))\n",
" yy = 1/(t^R*ww^a)\n",
" return Rt(fct(x = Rt(xx), y = Rt(yy)))\n",
" \n",
"def naive_hensel(fct, F, start = 1, prec=10):\n",
" Rt.<t> = LaurentSeriesRing(F, default_prec=prec)\n",
" RtQ = FractionField(Rt)\n",
" RptW.<W> = PolynomialRing(RtQ)\n",
" fct = RptW(fct)\n",
" alpha = (fct.derivative())(W = start)\n",
" w0 = Rt(start)\n",
" i = 1\n",
" while(i < prec):\n",
" w0 = w0 - fct(W = w0)/alpha + O(t^(prec))\n",
" i += 1\n",
" return w0\n",
"\n",
"class superelliptic_form:\n",
" def __init__(self, C, g):\n",
" F = C.base_ring\n",
" Rxy.<x, y> = PolynomialRing(F, 2)\n",
" Fxy = FractionField(Rxy)\n",
" g = Fxy(reduction_form(C, g))\n",
" self.form = g\n",
" self.curve = C \n",
" \n",
" def __add__(self, other):\n",
" C = self.curve\n",
" g1 = self.form\n",
" g2 = other.form\n",
" g = reduction(C, g1 + g2)\n",
" return superelliptic_form(C, g)\n",
" \n",
" def __sub__(self, other):\n",
" C = self.curve\n",
" g1 = self.form\n",
" g2 = other.form\n",
" g = reduction(C, g1 - g2)\n",
" return superelliptic_form(C, g)\n",
" \n",
" def __repr__(self):\n",
" g = self.form\n",
" if len(str(g)) == 1:\n",
" return str(g) + ' dx'\n",
" return '('+str(g) + ') dx'\n",
"\n",
" def __rmul__(self, constant):\n",
" C = self.curve\n",
" omega = self.form\n",
" return superelliptic_form(C, constant*omega) \n",
" \n",
" def cartier(self):\n",
" C = self.curve\n",
" m = C.exponent\n",
" p = C.characteristic\n",
" f = C.polynomial\n",
" F = C.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" Fxy = FractionField(FxRy)\n",
" result = superelliptic_form(C, FxRy(0))\n",
" mult_order = Integers(m)(p).multiplicative_order()\n",
" M = Integer((p^(mult_order)-1)/m)\n",
" \n",
" for j in range(1, m):\n",
" fct_j = self.jth_component(j)\n",
" h = Rx(fct_j*f^(M*j))\n",
" j1 = (p^(mult_order-1)*j)%m\n",
" B = floor(p^(mult_order-1)*j/m)\n",
" result += superelliptic_form(C, polynomial_part(p, h)/(f^B*y^(j1)))\n",
" return result \n",
" \n",
" def coordinates(self):\n",
" C = self.curve\n",
" F = C.base_ring\n",
" m = C.exponent\n",
" Rx.<x> = PolynomialRing(F)\n",
" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" g = C.genus()\n",
" degrees_holo = C.degrees_holomorphic_differentials()\n",
" degrees_holo_inv = {b:a for a, b in degrees_holo.items()}\n",
" basis = C.holomorphic_differentials_basis()\n",
" \n",
" for j in range(1, m):\n",
" omega_j = Fx(self.jth_component(j))\n",
" if omega_j != Fx(0):\n",
" d = degree_of_rational_fctn(omega_j, F)\n",
" index = degrees_holo_inv[(d, j)]\n",
" a = coeff_of_rational_fctn(omega_j, F)\n",
" a1 = coeff_of_rational_fctn(basis[index].jth_component(j), F)\n",
" elt = self - (a/a1)*basis[index]\n",
" return elt.coordinates() + a/a1*vector([F(i == index) for i in range(0, g)])\n",
" \n",
" return vector(g*[0])\n",
" \n",
" def jth_component(self, j):\n",
" g = self.form\n",
" C = self.curve\n",
" F = C.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" Fxy = FractionField(FxRy)\n",
" Ryinv.<y_inv> = PolynomialRing(Fx)\n",
" g = Fxy(g)\n",
" g = g(y = 1/y_inv)\n",
" g = Ryinv(g)\n",
" return coff(g, j)\n",
" \n",
" def is_regular_on_U0(self):\n",
" C = self.curve\n",
" F = C.base_ring\n",
" m = C.exponent\n",
" Rx.<x> = PolynomialRing(F)\n",
" for j in range(1, m):\n",
" if self.jth_component(j) not in Rx:\n",
" return 0\n",
" return 1\n",
" \n",
" def is_regular_on_Uinfty(self):\n",
" C = self.curve\n",
" F = C.base_ring\n",
" m = C.exponent\n",
" f = C.polynomial\n",
" r = f.degree()\n",
" delta = GCD(m, r)\n",
" M = m/delta\n",
" R = r/delta\n",
" \n",
" for j in range(1, m):\n",
" A = self.jth_component(j)\n",
" d = degree_of_rational_fctn(A, F)\n",
" if(-d*M + j*R -(M+1)<0):\n",
" return 0\n",
" return 1\n",
"\n",
" def expansion_at_infty(self, i = 0, prec=10):\n",
" g = self.form\n",
" C = self.curve\n",
" g = superelliptic_function(C, g)\n",
" g = g.expansion_at_infty(i = i, prec=prec)\n",
" x_series = superelliptic_function(C, x).expansion_at_infty(i = i, prec=prec)\n",
" dx_series = x_series.derivative()\n",
" return g*dx_series\n",
" \n",
"class superelliptic_cech:\n",
" def __init__(self, C, omega, fct):\n",
" self.omega0 = omega\n",
" self.omega8 = omega - fct.diffn()\n",
" self.f = fct\n",
" self.curve = C\n",
" \n",
" def __add__(self, other):\n",
" C = self.curve\n",
" return superelliptic_cech(C, self.omega0 + other.omega0, self.f + other.f)\n",
" \n",
" def __sub__(self, other):\n",
" C = self.curve\n",
" return superelliptic_cech(C, self.omega0 - other.omega0, self.f - other.f)\n",
"\n",
" def __rmul__(self, constant):\n",
" C = self.curve\n",
" w1 = self.omega0.form\n",
" f1 = self.f.function\n",
" w2 = superelliptic_form(C, constant*w1)\n",
" f2 = superelliptic_function(C, constant*f1)\n",
" return superelliptic_cech(C, w2, f2) \n",
" \n",
" def __repr__(self):\n",
" return \"(\" + str(self.omega0) + \", \" + str(self.f) + \", \" + str(self.omega8) + \")\" \n",
" \n",
" def verschiebung(self):\n",
" C = self.curve\n",
" omega = self.omega0\n",
" F = C.base_ring\n",
" Rx.<x> = PolynomialRing(F)\n",
" return superelliptic_cech(C, omega.cartier(), superelliptic_function(C, Rx(0)))\n",
" \n",
" def frobenius(self):\n",
" C = self.curve\n",
" fct = self.f.function\n",
" p = C.characteristic\n",
" Rx.<x> = PolynomialRing(F)\n",
" return superelliptic_cech(C, superelliptic_form(C, Rx(0)), superelliptic_function(C, fct^p))\n",
"\n",
" def coordinates(self):\n",
" C = self.curve\n",
" F = C.base_ring\n",
" m = C.exponent\n",
" Rx.<x> = PolynomialRing(F)\n",
" Fx = FractionField(Rx)\n",
" FxRy.<y> = PolynomialRing(Fx)\n",
" g = C.genus()\n",
" degrees_holo = C.degrees_holomorphic_differentials()\n",
" degrees_holo_inv = {b:a for a, b in degrees_holo.items()}\n",
" degrees0 = C.degrees_de_rham0()\n",
" degrees0_inv = {b:a for a, b in degrees0.items()}\n",
" degrees1 = C.degrees_de_rham1()\n",
" degrees1_inv = {b:a for a, b in degrees1.items()}\n",
" basis = C.de_rham_basis()\n",
" \n",
" omega = self.omega0\n",
" fct = self.f\n",
" \n",
" if fct.function == Rx(0) and omega.form != Rx(0):\n",
" for j in range(1, m):\n",
" omega_j = Fx(omega.jth_component(j))\n",
" if omega_j != Fx(0):\n",
" d = degree_of_rational_fctn(omega_j, F)\n",
" index = degrees_holo_inv[(d, j)]\n",
" a = coeff_of_rational_fctn(omega_j, F)\n",
" a1 = coeff_of_rational_fctn(basis[index].omega0.jth_component(j), F)\n",
" elt = self - (a/a1)*basis[index]\n",
" return elt.coordinates() + a/a1*vector([F(i == index) for i in range(0, 2*g)])\n",
" \n",
" for j in range(1, m):\n",
" fct_j = Fx(fct.jth_component(j))\n",
" if (fct_j != Rx(0)):\n",
" d = degree_of_rational_fctn(fct_j, p)\n",
" \n",
" if (d, j) in degrees1.values():\n",
" index = degrees1_inv[(d, j)]\n",
" a = coeff_of_rational_fctn(fct_j, F)\n",
" elt = self - (a/m)*basis[index]\n",
" return elt.coordinates() + a/m*vector([F(i == index) for i in range(0, 2*g)])\n",
" \n",
" if d<0:\n",
" a = coeff_of_rational_fctn(fct_j, F)\n",
" h = superelliptic_function(C, FxRy(a*y^j*x^d))\n",
" elt = superelliptic_cech(C, self.omega0, self.f - h)\n",
" return elt.coordinates()\n",
" \n",
" if (fct_j != Rx(0)):\n",
" G = superelliptic_function(C, y^j*x^d)\n",
" a = coeff_of_rational_fctn(fct_j, F)\n",
" elt =self - a*superelliptic_cech(C, diffn(G), G)\n",
" return elt.coordinates()\n",
"\n",
" return vector(2*g*[0])\n",
" \n",
" def is_cocycle(self):\n",
" w0 = self.omega0\n",
" w8 = self.omega8\n",
" fct = self.f\n",
" if not w0.is_regular_on_U0() and not w8.is_regular_on_Uinfty():\n",
" return('w0 & w8')\n",
" if not w0.is_regular_on_U0():\n",
" return('w0')\n",
" if not w8.is_regular_on_Uinfty():\n",
" return('w8')\n",
" if w0.is_regular_on_U0() and w8.is_regular_on_Uinfty():\n",
" return 1\n",
" return 0\n",
" \n",
"def degree_of_rational_fctn(f, F):\n",
" Rx.<x> = PolynomialRing(F)\n",
" Fx = FractionField(Rx)\n",
" f = Fx(f)\n",
" f1 = f.numerator()\n",
" f2 = f.denominator()\n",
" d1 = f1.degree()\n",
" d2 = f2.degree()\n",
" return(d1 - d2)\n",
"\n",
"def coeff_of_rational_fctn(f, F):\n",
" Rx.<x> = PolynomialRing(F)\n",
" Fx = FractionField(Rx)\n",
" f = Fx(f)\n",
" if f == Rx(0):\n",
" return 0\n",
" f1 = f.numerator()\n",
" f2 = f.denominator()\n",
" d1 = f1.degree()\n",
" d2 = f2.degree()\n",
" a1 = f1.coefficients(sparse = false)[d1]\n",
" a2 = f2.coefficients(sparse = false)[d2]\n",
" return(a1/a2)\n",
"\n",
"def coff(f, d):\n",
" lista = f.coefficients(sparse = false)\n",
" if len(lista) <= d:\n",
" return 0\n",
" return lista[d]\n",
"\n",
"def cut(f, i):\n",
" R = f.parent()\n",
" coeff = f.coefficients(sparse = false)\n",
" return sum(R(x^(j-i-1)) * coeff[j] for j in range(i+1, f.degree() + 1))\n",
"\n",
"def polynomial_part(p, h):\n",
" F = GF(p)\n",
" Rx.<x> = PolynomialRing(F)\n",
" h = Rx(h)\n",
" result = Rx(0)\n",
" for i in range(0, h.degree()+1):\n",
" if (i%p) == p-1:\n",
" power = Integer((i-(p-1))/p)\n",
" result += Integer(h[i]) * x^(power) \n",
" return result\n",
"\n",
"#Find delta-th root of unity in field F\n",
"def root_of_unity(F, delta):\n",
" Rx.<x> = PolynomialRing(F)\n",
" cyclotomic = x^(delta) - 1\n",
" for root, a in cyclotomic.roots():\n",
" powers = [root^d for d in delta.divisors() if d!= delta]\n",
" if 1 not in powers:\n",
" return root\n",
" \n",
"def preimage(U, V, M): #preimage of subspace U under M\n",
" basis_preimage = M.right_kernel().basis()\n",
" imageU = U.intersection(M.transpose().image())\n",
" basis = imageU.basis()\n",
" for v in basis:\n",
" w = M.solve_right(v)\n",
" basis_preimage = basis_preimage + [w]\n",
" return V.subspace(basis_preimage)\n",
"\n",
"def image(U, V, M):\n",
" basis = U.basis()\n",
" basis_image = []\n",
" for v in basis:\n",
" basis_image += [M*v]\n",
" return V.subspace(basis_image)\n",
"\n",
"def flag(F, V, p, test = 0):\n",
" dim = F.dimensions()[0]\n",
" space = VectorSpace(GF(p), dim)\n",
" flag_subspaces = (dim+1)*[0]\n",
" flag_used = (dim+1)*[0]\n",
" final_type = (dim+1)*['?']\n",
" \n",
" flag_subspaces[dim] = space\n",
" flag_used[dim] = 1\n",
" \n",
" \n",
" while 1 in flag_used:\n",
" index = flag_used.index(1)\n",
" flag_used[index] = 0\n",
" U = flag_subspaces[index]\n",
" U_im = image(U, space, V)\n",
" d_im = U_im.dimension()\n",
" final_type[index] = d_im\n",
" U_pre = preimage(U, space, F)\n",
" d_pre = U_pre.dimension()\n",
" \n",
" if flag_subspaces[d_im] == 0:\n",
" flag_subspaces[d_im] = U_im\n",
" flag_used[d_im] = 1\n",
" \n",
" if flag_subspaces[d_pre] == 0:\n",
" flag_subspaces[d_pre] = U_pre\n",
" flag_used[d_pre] = 1\n",
" \n",
" if test == 1:\n",
" print('(', final_type, ')')\n",
" \n",
" for i in range(0, dim+1):\n",
" if final_type[i] == '?' and final_type[dim - i] != '?':\n",
" i1 = dim - i\n",
" final_type[i] = final_type[i1] - i1 + dim/2\n",
" \n",
" final_type[0] = 0\n",
" \n",
" for i in range(1, dim+1):\n",
" if final_type[i] == '?':\n",
" prev = final_type[i-1]\n",
" if prev != '?' and prev in final_type[i+1:]:\n",
" final_type[i] = prev\n",
" \n",
" for i in range(1, dim+1):\n",
" if final_type[i] == '?':\n",
" final_type[i] = min(final_type[i-1] + 1, dim/2)\n",
" \n",
" if is_final(final_type, dim/2):\n",
" return final_type[1:dim/2 + 1]\n",
" print('error:', final_type[1:dim/2 + 1])\n",
" \n",
"def is_final(final_type, dim):\n",
" n = len(final_type)\n",
" if final_type[0] != 0:\n",
" return 0\n",
" \n",
" if final_type[n-1] != dim:\n",
" return 0\n",
" \n",
" for i in range(1, n):\n",
" if final_type[i] != final_type[i - 1] and final_type[i] != final_type[i - 1] + 1:\n",
" return 0\n",
" return 1"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "code",
"execution_count": 58,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "code",
"execution_count": 51,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"3*t^2 + 3*t^14 + t^16 + 3*t^26 + 3*t^28 + 3*t^30 + 3*t^38 + t^40 + 3*t^50 + 3*t^62 + t^72 + t^74 + 2*t^76 + 4*t^84 + 2*t^86 + 4*t^88 + 4*t^90 + 2*t^96 + 4*t^98 + t^100 + O(t^114)"
]
},
"execution_count": 51,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"omega.omega8().exp"
]
},
{
"cell_type": "code",
"execution_count": 30,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
]
},
{
"cell_type": "code",
"execution_count": 33,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"F = QQ\n",
"Rt.<t> = LaurentSeriesRing(F, default_prec=20)\n",
"RtQ = FractionField(Rt)\n",
"RptW.<W> = PolynomialRing(RtQ)\n",
"fct = W^2 - (1+t)"
]
},
{
"cell_type": "code",
"execution_count": 35,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"A = naive_hensel(fct, F, start = 1, prec=10)"
]
},
{
"cell_type": "code",
"execution_count": 36,
"metadata": {
"collapsed": false,
"scrolled": true
},
"outputs": [
{
"name": "stderr",
"output_type": "stream",
"text": [
"WARNING: Some output was deleted.\n"
]
}
],
"source": [
"A^2 - (1+t)"
]
},
{
"cell_type": "code",
"execution_count": 37,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"FF = GF(5^6)"
]
},
{
"cell_type": "code",
"execution_count": 56,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"[(4, 1),\n",
" (1, 1),\n",
" (3*z6^5 + 4*z6^4 + 3*z6^2 + 2*z6 + 2, 1),\n",
" (3*z6^5 + 4*z6^4 + 3*z6^2 + 2*z6 + 1, 1),\n",
" (2*z6^5 + z6^4 + 2*z6^2 + 3*z6 + 4, 1),\n",
" (2*z6^5 + z6^4 + 2*z6^2 + 3*z6 + 3, 1)]"
]
},
"execution_count": 56,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Rx.<x> = PolynomialRing(FF)\n",
"(x^6 - 1).roots()"
]
},
{
"cell_type": "code",
"execution_count": 57,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"3*z2 + 3"
]
},
"execution_count": 57,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"root_of_unity(GF(5^2), 3)"
]
},
{
"cell_type": "code",
"execution_count": 52,
"metadata": {
"collapsed": false
},
"outputs": [
{
"data": {
"text/plain": [
"x^2 + x + 1"
]
},
"execution_count": 52,
"metadata": {
},
"output_type": "execute_result"
}
],
"source": [
"Rx.<x> = PolynomialRing(GF(5^3))\n",
"Rx(x^2 + x + 1).factor()"
]
},
{
"cell_type": "code",
"execution_count": 53,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"a = GF(5^3).random_element()"
]
},
{
"cell_type": "code",
"execution_count": 0,
"metadata": {
"collapsed": false
},
"outputs": [
],
"source": [
"a"
]
}
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.5",
"language": "sagemath",
"metadata": {
"cocalc": {
"description": "Open-source mathematical software system",
"priority": 1,
"url": "https://www.sagemath.org/"
}
},
"name": "sage-9.5",
"resource_dir": "/ext/jupyter/kernels/sage-9.5"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.9.9"
}
},
"nbformat": 4,
"nbformat_minor": 4
}

1189
superelliptic_alpha.ipynb
View File

File diff suppressed because it is too large Load Diff