{
  "id": "2009.11578",
  "version": "v1",
  "published": "2020-09-24T10:02:04.000Z",
  "updated": "2020-09-24T10:02:04.000Z",
  "title": "Orders occurring as endomorphism ring of a Drinfeld module in some isogeny classes of Drinfeld modules of higher ranks",
  "authors": [
    "Sedric Nkotto Nkung Assong"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "The question we propose to answer throughout this paper is the following: Given an isogeny class of Drinfeld modules over a finite field, what are the orders of the corresponding endomorphism algebra (which is an isogeny invariant) that occur as endomorphism ring of a Drinfeld module in that isogeny class? It is worth mentioning that this question is different from the ones investigated by the authors Kuhn, Pink in [6] and Garai, Papikian in [3]. The former authors rather provided an answer to the question, given a Drinfeld module {\\phi}, how does one efficiently compute the endomorphism ring of {\\phi}? The importance of our question resides in the fact that it might be very helpful to better understand isogeny graphs of Drinfeld modules of higher rank (r > 2) and may be reopen the debate concerning the application to isogeny-based cryptography. We answer that question for the case whereby the endomorphism algebra is a field by providing a necessary and sufficient condition for a given order to be the endomorphism ring of a Drinfeld module. We apply our result to rank r = 3 Drinfeld modules and explicitly compute those orders occurring as endomorphism rings of rank 3 Drinfeld modules over a finite field.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-09-24T10:02:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "drinfeld module",
      "endomorphism ring",
      "isogeny class",
      "higher rank",
      "orders occurring"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}