{
  "id": "2001.07270",
  "version": "v1",
  "published": "2020-01-20T22:43:15.000Z",
  "updated": "2020-01-20T22:43:15.000Z",
  "title": "Computing actions on cusp forms",
  "authors": [
    "David Zywina"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For positive integers $k$ and $N$, we describe how to compute the natural action of $SL_2(\\mathbb{Z})$ on the space of cusp forms $S_k(\\Gamma(N))$, where a cusp form is given by sufficiently many terms of its $q$-expansion. This will reduce to computing the action of the Atkin--Lehner operator on $S_k(\\Gamma)$ for a congruence subgroup $\\Gamma_1(N)\\subseteq \\Gamma \\subseteq \\Gamma_0(N)$. Our motivating application of such fundamental computations is to compute explicit models of some modular curves $X_G$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-20T22:43:15.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F11",
      "11G18"
    ],
    "keywords": [
      "cusp form",
      "computing actions",
      "natural action",
      "atkin-lehner operator",
      "congruence subgroup"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}