{
  "id": "2106.15677",
  "version": "v1",
  "published": "2021-06-29T18:47:37.000Z",
  "updated": "2021-06-29T18:47:37.000Z",
  "title": "Supersingular Loci from Traces of Hecke Operators",
  "authors": [
    "Kevin Gomez",
    "Kaya Lakein",
    "Anne Larsen"
  ],
  "comment": "14 pages, 3 tables",
  "categories": [
    "math.NT"
  ],
  "abstract": "A classical observation of Deligne shows that, for any prime $p \\geq 5$, the divisor polynomial of the Eisenstein series $E_{p-1}(z)$ mod $p$ is closely related to the supersingular polynomial at $p$, $$S_p(x) := \\prod_{E/\\overline{\\mathbb{F}}_p \\text{ supersingular}}(x-j(E)) \\in \\mathbb{F}_p[x].$$ Deuring, Hasse, and Kaneko and Zagier found other families of modular forms which also give the supersingular polynomial at $p$. In a new approach, we prove an analogue of Deligne's result for the Hecke trace forms $T_k(z)$ defined by the Hecke action on the space of cusp forms $S_k$. We use the Eichler-Selberg trace formula to identify congruences between trace forms of different weights mod $p$, and then relate their divisor polynomials to $S_p(x)$ using Deligne's observation.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-06-29T18:47:37.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F33",
      "11G07"
    ],
    "keywords": [
      "hecke operators",
      "supersingular loci",
      "supersingular polynomial",
      "divisor polynomial",
      "hecke trace forms"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 14,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}