{
  "id": "2112.09481",
  "version": "v2",
  "published": "2021-12-17T12:46:13.000Z",
  "updated": "2022-07-18T22:15:55.000Z",
  "title": "Congruences like Atkin's for the partition function",
  "authors": [
    "Scott Ahlgren",
    "Patrick B. Allen",
    "Shiang Tang"
  ],
  "comment": "Minor revisions to improve exposition",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p(n)$ be the ordinary partition function. In the 1960s Atkin found a number of examples of congruences of the form $p( Q^3 \\ell n+\\beta)\\equiv0\\pmod\\ell$ where $\\ell$ and $Q$ are prime and $5\\leq \\ell\\leq 31$; these lie in two natural families distinguished by the square class of $1-24\\beta\\pmod\\ell$. In recent decades much work has been done to understand congruences of the form $p(Q^m\\ell n+\\beta)\\equiv 0\\pmod\\ell$. It is now known that there are many such congruences when $m\\geq 4$, that such congruences are scarce (if they exist at all) when $m=1, 2$, and that for $m=0$ such congruences exist only when $\\ell=5, 7, 11$. For congruences like Atkin's (when $m=3$), more examples have been found for $5\\leq \\ell\\leq 31$ but little else seems to be known. Here we use the theory of modular Galois representations to prove that for every prime $\\ell\\geq 5$, there are infinitely many congruences like Atkin's in the first natural family which he discovered and that for at least $17/24$ of the primes $\\ell$ there are infinitely many congruences in the second family.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2022-07-18T22:15:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "modular galois representations",
      "ordinary partition function",
      "understand congruences",
      "square class",
      "1960s atkin"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}