SpacesAndLattices.ipynb

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   "source": [
    "# Programming in julia/Hecke/Oscar - Session 4\n",
    "## Working with $\\mathbf Z$-lattices"
   ]
  },
  {
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   "source": [
    "### Quadratic spaces\n",
    "Before lattices, there are quadratic spaces, which are represented using a Gram matrix."
   ]
  },
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     "text": [
      "┌ Info: Precompiling Hecke [3e1990a7-5d81-5526-99ce-9ba3ff248f21]\n",
      "└ @ Base loading.jl:1278\n"
     ]
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     "text": [
      "\n",
      "Welcome to \n",
      "\n",
      "\u001b[31m    _    _           _\u001b[39m\n",
      "\u001b[31m   | |  | |         | |\u001b[39m\n",
      "\u001b[31m   | |__| | ___  ___| | _____\u001b[39m\n",
      "\u001b[31m   |  __  |/ _ \\/ __| |/ / _ \\\u001b[39m\n",
      "\u001b[31m   | |  | |  __/ (__|   <  __/\u001b[39m\n",
      "\u001b[31m   |_|  |_|\\___|\\___|_|\\_\\___|\u001b[39m\n",
      "\u001b[31m    \u001b[39m\n",
      "Version\u001b[32m 0.9.1-dev \u001b[39m... \n",
      " ... which comes with absolutely no warranty whatsoever\n",
      "(c) 2015-2020 by Claus Fieker, Tommy Hofmann and Carlo Sircana\n",
      "\n"
     ]
    }
   ],
   "source": [
    "using Hecke;"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "Quadratic space over\n",
       "Rational Field\n",
       "with Gram matrix\n",
       "[1 2; 2 1]"
      ]
     },
     "execution_count": 3,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "V = quadratic_space(QQ, QQ[1 2; 2 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1  2]\n",
       "[2  1]"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gram_matrix(V)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "To get the inner product we can do the following:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "3"
      ],
      "text/plain": [
       "3"
      ]
     },
     "execution_count": 5,
     "metadata": {},
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    }
   ],
   "source": [
    "inner_product(V, [1, 1], [0, 1])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "fragment"
    }
   },
   "source": [
    "Some (incomplete) documentation can be found here: http://www.thofma.com/Hecke.jl/dev/quad_forms/basics/. Here are two examples"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2-element Array{fmpq,1}:\n",
       " 1\n",
       " -3"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "diagonal(V)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "hasse_invariant(V, 2)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Lattices\n",
    "Once we have a quadratic space, we can define a lattice by specifying a basis matrix. Thus, we represent lattices using an ambient space and a basis matrix of the underlying $\\mathbf Z$-module."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Quadratic lattice of rank 1 and degree 2 over the rationals"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L = lattice(V, ZZ[1 0;]) "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {
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     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1  0]"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "basis_matrix(L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1]"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gram_matrix(L)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
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   },
   "source": [
    "Note that there is an alternative syntax which allows one to skip the creation of the quadratic space:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Quadratic lattice of rank 1 and degree 2 over the rationals"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "L = Zlattice(ZZ[1 0]; gram = QQ[1 2; 2 1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[1  0]"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "basis_matrix(L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {
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     "slide_type": "fragment"
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    {
     "data": {
      "text/plain": [
       "[1]"
      ]
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "gram_matrix(L)"
   ]
  },
  {
   "cell_type": "markdown",
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     "slide_type": "fragment"
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   "source": [
    "Note that neither the Gram matrix nor the basis matrix have to be integral."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
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   },
   "source": [
    "One can also recover the ambient quadratic space:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "Quadratic space over\n",
       "Rational Field\n",
       "with Gram matrix\n",
       "[1 2; 2 1]"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ambient_space(L)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "We can also compute $L \\otimes \\mathbf Q$, which has Gram matrix equal to the Gram matrix of $L$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "Quadratic space over\n",
       "Rational Field\n",
       "with Gram matrix\n",
       "[1]"
      ]
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "rational_span(L)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "At the moment there are not that many functions for $\\mathbf Z$-lattices. Most functionality exists for lattices over number fields and needs to be ported over (with simplifications). Here are some functions that exist:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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   "outputs": [
    {
     "data": {
      "text/latex": [
       "1"
      ],
      "text/plain": [
       "1"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "discriminant(L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 30,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
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    {
     "data": {
      "text/latex": [
       "2"
      ],
      "text/plain": [
       "2"
      ]
     },
     "execution_count": 30,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "automorphism_group_order(L)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 31,
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "1-element Array{fmpq_mat,1}:\n",
       " [-1 0; -4 1]"
      ]
     },
     "execution_count": 31,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "automorphism_group_generators(L)"
   ]
  }
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