{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A demo of the vector_bundle package."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "from vector_bundle import *\n",
    "F.<x> = FunctionField(GF(101))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constructing an indecomposable bundle on an elliptic curve\n",
    "We construct an indecomposable vector bundle of rank 5 and degree 3 on a function field of genus 1.  \n",
    "This construction may be done automatically using the ```atiyah_bundle``` function but we break it down step by step."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
      "[(1)]\n",
      "[((x/(x^2 + 1))*y)]\n"
     ]
    }
   ],
   "source": [
    "R.<y> = F[]\n",
    "K.<y> = F.extension(y^2 - x^3 - x)\n",
    "deg_1_bundle = VectorBundle(K, K.places_infinite()[0].divisor())\n",
    "E = deg_1_bundle\n",
    "print(E.coefficient_ideals())\n",
    "print(E.basis_finite())\n",
    "print(E.basis_infinite())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
      "[(1, 0), (0, 1)]\n",
      "[(1, 0), (100*x^3/(x^2 + 1), (x/(x^2 + 1))*y)]\n"
     ]
    }
   ],
   "source": [
    "E = E.extension_by_global_sections()\n",
    "print(E.coefficient_ideals())\n",
    "print(E.basis_finite())\n",
    "print(E.basis_infinite())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
      "[(1, 0), (0, 1)]\n",
      "[((x/(x^2 + 1))*y, 0), ((100*x^4/(x^4 + 2*x^2 + 1))*y, x^3/(x^2 + 1))]\n"
     ]
    }
   ],
   "source": [
    "E = E.tensor_product(deg_1_bundle)\n",
    "print(E.coefficient_ideals())\n",
    "print(E.basis_finite())\n",
    "print(E.basis_infinite())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (x^2/(x^2 + 3)) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
      "[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (0, 0, 0, 1, 0), (0, 0, 0, 0, 1)]\n",
      "[(1, 0, 0, 0, 0), (0, 1, 0, 0, 0), (0, 0, 1, 0, 0), (100*x^3/(x^2 + 1), 0, 100/x^2*y, (x/(x^2 + 1))*y, 0), (x^6/(x^4 + 2*x^2 + 1), (100*x^4/(x^4 + 2*x^2 + 1))*y, (x/(x^2 + 1))*y, (100*x^4/(x^4 + 2*x^2 + 1))*y, x^3/(x^2 + 1))]\n"
     ]
    }
   ],
   "source": [
    "E = E.extension_by_global_sections()\n",
    "print(E.coefficient_ideals())\n",
    "print(E.basis_finite())\n",
    "print(E.basis_infinite())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "5\n",
      "3\n"
     ]
    }
   ],
   "source": [
    "print(E.rank())\n",
    "print(E.degree())"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We check the algebra of global endomorphisms of E to ensure that it is indecomposable."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[\n",
       "[1 0 0 0 0]\n",
       "[0 1 0 0 0]\n",
       "[0 0 1 0 0]\n",
       "[0 0 0 1 0]\n",
       "[0 0 0 0 1]\n",
       "]"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "E.end().h0()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Constructing a weakly stable vector bundle following Savin's method\n",
    "We construct a weakly stable vector bundle of rank 3 and degree 5 by successive extensions by line bundles.  \n",
    "See [Sav07] in the references for details.  \n",
    "This, again, may be achieved directly using the ```savin_bundle``` function."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3\n",
      "5\n",
      "[Ideal (1, 1/x*y) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x, Ideal (1) of Maximal order of Function field in y defined by y^2 + 100*x^3 + 100*x]\n",
      "[(1, 0, 0), (0, 1, 0), (0, 0, 1)]\n",
      "[(1, 0, 0), ((100*x/(x^2 + 1))*y, x^3/(x^2 + 1), 0), (0, (100*x^4/(x^4 + 2*x^2 + 1))*y, x^3/(x^2 + 1))]\n"
     ]
    }
   ],
   "source": [
    "F = VectorBundle(K, 2*K.places_infinite()[0].divisor())\n",
    "F1 = VectorBundle(K, K.places_finite()[0].divisor())\n",
    "E1 = F1\n",
    "E2 = F.non_trivial_extension(E1)\n",
    "E3 = F.non_trivial_extension(E2)\n",
    "print(E3.rank())\n",
    "print(E3.degree())\n",
    "print(E3.coefficient_ideals())\n",
    "print(E3.basis_finite())\n",
    "print(E3.basis_infinite())"
   ]
  }
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