{ "cells": [ { "cell_type": "markdown", "metadata": {}, "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", " - " ] }, { "cell_type": "code", "execution_count": 3, "metadata": {}, "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. = 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)]\n", " k = k+1\n", " return baza" ] }, { "cell_type": "code", "execution_count": 4, "metadata": {}, "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", "# [y^j * f(x) dx, g(x)/y^j]. Output is [f(x), g(x)]\n", "\n", "def baza_dr(m, f, j, p):\n", " R. = 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]\n", " k = k+1\n", " return baza" ] }, { "cell_type": "code", "execution_count": 155, "metadata": {}, "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. = 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, j, p):\n", " #print(elt)\n", " R. = 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", " print('p1')\n", " return zapis\n", " \n", " if elt[1] == 0:\n", " print('p2')\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)]\n", " elt1[0] = elt[0] - a/a1 * baza[k][0]\n", " elt1[1] = R(0)\n", " return zapis_w_bazie_dr(elt1, m, f, j, p) + vector([a/a1*GF(p)(i == k) for i in range(0, len(baza))])\n", "\n", " g = elt[1]\n", " g1 = R(elt[1].numerator())\n", " g2 = R(elt[1].denominator())\n", " d1 = g1.degree()\n", " d2 = g2.degree()\n", " a1 = g1.coefficients(sparse = false)[d1]\n", " a2 = g2.coefficients(sparse = false)[d2]\n", " a = a1/a2\n", " d = d2 - d1\n", " Rr = r/delta\n", " M = 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", " \n", " if (d not in inv_stopnie2):\n", " print('p3')\n", " if d>= 0:\n", " elt1 = [elt[0], RR(elt[1]) - a1/a2*1/R(x^d)]\n", " else:\n", " elt1 = [elt[0], RR(elt[1]) - a1/a2*R(x^(-d))]\n", " return zapis_w_bazie_dr(elt1, m, f, j, p)\n", " \n", " print('p4')\n", " k = inv_stopnie2[d]\n", " elt1 = [R(0), R(0)]\n", " elt1[0] = elt[0] - a*baza[k][0]\n", " elt1[1] = elt[1] - a*baza[k][1]\n", " return zapis_w_bazie_dr(elt1, m, f, j, 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, j, p):\n", " R. = 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)]\n", " elt1[0] = elt[0] - a/a1 * baza[k][0]\n", " \n", " return zapis_w_bazie_holo(elt1, m, f, j, p) + vector([a/a1 * GF(p)(i == k) for i in range(0, len(baza))])\n" ] }, { "cell_type": "markdown", "metadata": {}, "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": 156, "metadata": {}, "outputs": [], "source": [ "def czesc_wielomianu(p, h):\n", " R. = 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. = 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] #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, j, p)\n", " A[k, :] = matrix(v)\n", " return A.transpose()" ] }, { "cell_type": "markdown", "metadata": {}, "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": 157, "metadata": {}, "outputs": [], "source": [ "def frobenius_dr(p, m, f, elt, j): #Cartier na y^m = f dla elt = [forma rozniczkowa, fkcja]\n", " R. = PolynomialRing(GF(p))\n", " RR = FractionField(R)\n", " f = R(f)\n", " M = floor(j*p/m)\n", " return [0, f^M * RR(elt[1])^p] #eigenspace = j*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, j, p)\n", " A[k, :] = matrix(v)\n", " return A.transpose()" ] }, { "cell_type": "code", "execution_count": 158, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "{0: [1, 0], 1: [x, 2/x]}" ] }, "execution_count": 158, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = 2\n", "j = 1\n", "p = 5\n", "f = x^3 + x+3\n", "baza_dr(m, f, j, p)\n", "#macierz_frob_dr(p, m, f, j)" ] }, { "cell_type": "code", "execution_count": 159, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0, (2*x^6 + 4*x^4 + 2*x^3 + 2*x^2 + 2*x + 3)/x^5]" ] }, "execution_count": 159, "metadata": {}, "output_type": "execute_result" } ], "source": [ "frobenius_dr(p, m, f, [x, 2/x], j)" ] }, { 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"p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n", "p4\n" ] }, { "ename": "RecursionError", "evalue": "maximum recursion depth exceeded while calling a Python object", "output_type": "error", "traceback": [ "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m", "\u001b[0;31mRecursionError\u001b[0m Traceback (most recent call last)", "\u001b[0;32m\u001b[0m in \u001b[0;36m\u001b[0;34m()\u001b[0m\n\u001b[0;32m----> 1\u001b[0;31m \u001b[0mzapis_w_bazie_dr\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;34m[\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m0\u001b[0m\u001b[0;34m)\u001b[0m\u001b[0;34m,\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[0mx\u001b[0m\u001b[0;34m**\u001b[0m\u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m6\u001b[0m\u001b[0;34m)\u001b[0m \u001b[0;34m+\u001b[0m \u001b[0mInteger\u001b[0m\u001b[0;34m(\u001b[0m\u001b[0;36m4\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;36m4\u001b[0m\u001b[0;34m)\u001b[0m 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range(0, p^2))\n", "czesc_wielomianu(p, h)" ] }, { "cell_type": "code", "execution_count": 60, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "{0: [x^2, 0], 1: [x, 0], 2: [1, 0]}" ] }, "execution_count": 60, "metadata": {}, "output_type": "execute_result" } ], "source": [ "m = 5\n", "j = 1\n", "p = 5\n", "f = x^4 + x+2\n", "baza_dr(m, f, j, p)" ] }, { "cell_type": "code", "execution_count": 61, "metadata": {}, "outputs": [ { "data": { "text/plain": [ "[0 0 0]\n", "[0 0 0]\n", "[0 0 0]" ] }, "execution_count": 61, "metadata": {}, "output_type": "execute_result" } ], "source": [ "macierz_frob_dr(p, m, f, j)" ] }, { "cell_type": "code", "execution_count": null, "metadata": {}, "outputs": [], "source": [] } ], "metadata": { "kernelspec": { "display_name": "SageMath 9.1", "language": "sage", "name": "sagemath" }, "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": 2 }