{
  "aliases": null,
  "backlinks": [
    {
      "field": "uses",
      "target": "3NZBZXYWAJ"
    },
    {
      "field": "uses",
      "target": "68SJQEC11F"
    },
    {
      "field": "uses",
      "target": "CG17WSS64W"
    },
    {
      "field": "uses",
      "target": "CS1NVJBZMH"
    },
    {
      "field": "uses",
      "target": "NDYRB4MQZT"
    },
    {
      "field": "uses",
      "target": "RWXXNZNQYJ"
    },
    {
      "field": "uses",
      "target": "SV5EEBF861"
    },
    {
      "field": "uses",
      "target": "X9PBTW2J8D"
    },
    {
      "field": "specializes",
      "target": "A50Y012RBP"
    },
    {
      "field": "specializes",
      "target": "EYREVY31BN"
    },
    {
      "field": "about",
      "target": "CS1NVJBZMH"
    },
    {
      "field": "about",
      "target": "RWXXNZNQYJ"
    },
    {
      "field": "defines",
      "target": "M8NA9X8BTV"
    }
  ],
  "body": "\n## Summary\n\nA modular form of weight $k$ is a [[holomorphic function|0NAHXR1FRH]] on the [[upper half-plane|X6XGGH5FPW]].\nIt transforms in a fixed way under the [[modular group|VEKV1EM77B]] and stays bounded at infinity.\nIts Fourier expansion $\\sum_{n \\ge 0} a_n q^n$ in $q = e^{2 \\pi i \\tau}$ often has arithmetic coefficients.\nThe spaces of modular forms of a given weight are finite-dimensional, so identities between them can be proved by comparing a few coefficients.\nModular forms are central in modern number theory, for example in the proof of Fermat's Last Theorem through the [[modularity theorem|CS1NVJBZMH]].\n\n## Explanations\n\n### L1 human\n\nA modular form is a function with an enormous amount of symmetry.\nIt is periodic, $f(\\tau + 1) = f(\\tau)$, and it also changes in a simple way under $\\tau \\mapsto -1/\\tau$.\nThese symmetries are so strong that, for each weight, the modular forms form a space of small dimension.\nThe [[Eisenstein series|EYREVY31BN]] $E_4(\\tau) = 1 + 240 \\sum_{n \\ge 1} \\sigma_3(n) q^n$ is an example of weight $4$.\nCoefficients of modular forms count things, such as the number of ways to write $n$ as a sum of squares.\n",
  "commit": "547b6eddb8f2feb8d579f2d633d22a27b7e132fc",
  "front": {
    "kind": "concept",
    "msc": [
      "11F11",
      "11F03"
    ],
    "name": "Modular form",
    "same_as": {
      "wikidata": "Q870797",
      "wikipedia": "Modular form"
    },
    "sources": [
      {
        "url": "https://en.wikipedia.org/wiki/Modular_form"
      }
    ],
    "specializes": [
      "3C73HND6XA"
    ],
    "tag": "GWKN5VG78A"
  },
  "kind": "concept",
  "link": "https://math2.tamnd.com/node/GWKN5VG78A/",
  "links": [
    {
      "field": "specializes",
      "target": "3C73HND6XA"
    }
  ],
  "markdown": "https://math2.tamnd.com/node/GWKN5VG78A.md",
  "msc": [
    "11F11",
    "11F03"
  ],
  "name": "Modular form",
  "prov": null,
  "role": "",
  "source": "https://github.com/tamnd/math2/blob/main/nodes/GWK/GWKN5VG78A.md",
  "status": null,
  "tag": "GWKN5VG78A",
  "uri": "m2://GWKN5VG78A",
  "url": "https://math2.tamnd.com/wiki/modular-form/",
  "writing": {
    "tag": "GWKN5VG78A",
    "kind": "concept",
    "level": "W0",
    "next": [
      {
        "what": "a source with a locator",
        "have": 0,
        "want": 1
      }
    ]
  }
}