{
  "id": "2012.12534",
  "version": "v1",
  "published": "2020-12-23T08:11:46.000Z",
  "updated": "2020-12-23T08:11:46.000Z",
  "title": "Chebotarev Density Theorem and Extremal Primes for non-CM elliptic curves",
  "authors": [
    "Amita Malik",
    "Neha Prabhu"
  ],
  "categories": [
    "math.NT"
  ],
  "abstract": "For a fixed non-CM elliptic curve $E$ over $\\mathbb{Q}$ and a prime $\\ell$, we prove an asymptotic formula on the number of primes $p \\leq x$ for which the Frobenius trace $a_p(E)$ satisfies the congruence $a_p(E)\\equiv [2\\sqrt{p}] \\pmod \\ell$. In order to achieve this, we establish a joint distribution concerning the fractional part of $\\alpha p^\\theta$ for $\\theta \\in [0,1], \\alpha>0$, and primes $p$ satisfying the Chebotarev condition. As a corollary, we also obtain upper bounds for the number of extremal primes. The results rely on GRH for Dedekind zeta functions for Galois extensions of number fields",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-23T08:11:46.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11G05",
      "11N05"
    ],
    "keywords": [
      "chebotarev density theorem",
      "extremal primes",
      "fixed non-cm elliptic curve",
      "dedekind zeta functions",
      "galois extensions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}