{
  "id": "2311.03259",
  "version": "v1",
  "published": "2023-11-06T16:44:44.000Z",
  "updated": "2023-11-06T16:44:44.000Z",
  "title": "$p$-Adic hypergeometric functions and the trace of Frobenius of elliptic curves",
  "authors": [
    "Sulakashna",
    "Rupam Barman"
  ],
  "comment": "27 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $p$ be an odd prime and $q=p^r$, $r\\geq 1$. For positive integers $n$, let ${_n}G_n[\\cdots]_q$ denote McCarthy's $p$-adic hypergeometric functions. In this article, we prove an identity expressing a ${_4}G_4[\\cdots]_q$ hypergeometric function as a sum of two ${_2}G_2[\\cdots]_q$ hypergeometric functions. This identity generalizes some known identities satisfied by the finite field hypergeometric functions. We also prove a transfomation that relates ${_{n+2}}G_{n+2}[\\cdots]_q$ and ${_n}G_n[\\cdots]_q$ hypergeometric functions. Next, we express the trace of Frobenius of elliptic curves in terms of special values of ${_4}G_4[\\cdots]_q$ and ${_6}G_6[\\cdots]_q$ hypergeometric functions. Our results extend the recent works of Tripathi and Meher on the finite field hypergeometric functions to wider classes of primes.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-11-06T16:44:44.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "adic hypergeometric functions",
      "elliptic curves",
      "finite field hypergeometric functions",
      "denote mccarthys",
      "identity generalizes"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 27,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}