{
  "id": "1506.09170",
  "version": "v1",
  "published": "2015-06-30T17:37:02.000Z",
  "updated": "2015-06-30T17:37:02.000Z",
  "title": "Linnik's Theorem for Sato-Tate Laws on Elliptic Curves with Complex Multiplication",
  "authors": [
    "Evan Chen",
    "Peter S. Park",
    "Ashvin A. Swaminathan"
  ],
  "comment": "11 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "Let $E/\\mathbb{Q}$ be an elliptic curve with complex multiplication (CM), and for each prime $p$ of good reduction, let $a_E(p) = p + 1 - \\#E(\\mathbb{F}_p)$ denote the trace of Frobenius. By the Hasse bound, $a_E(p) = 2\\sqrt{p} \\cos \\theta_p$ for a unique $\\theta_p \\in [0, \\pi]$. In this paper, we prove that the least prime $p$ such that $\\theta_p \\in [\\alpha, \\beta] \\subset [0, \\pi]$ satisfies \\[ p \\ll \\left(\\frac{N_E}{\\beta - \\alpha}\\right)^A, \\] where $N_E$ is the conductor of $E$ and the implied constant and exponent $A > 2$ are absolute and effectively computable. Our result is an analogue for CM elliptic curves of Linnik's Theorem for arithmetic progressions, which states that the least prime $p \\equiv a \\pmod q$ for $(a,q)=1$ satisfies $p \\ll q^L$ for an absolute constant $L > 0$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-30T17:37:02.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11F11",
      "14H52",
      "11M41",
      "11N05"
    ],
    "keywords": [
      "complex multiplication",
      "linniks theorem",
      "sato-tate laws",
      "cm elliptic curves",
      "arithmetic progressions"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 11,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150609170C"
    }
  }
}