{
  "id": "2206.08282",
  "version": "v1",
  "published": "2022-06-16T16:23:40.000Z",
  "updated": "2022-06-16T16:23:40.000Z",
  "title": "Hyperbolic angles from Heegner points",
  "authors": [
    "Giacomo Cherubini",
    "Alessandro Fazzari"
  ],
  "comment": "21 pages",
  "categories": [
    "math.NT"
  ],
  "abstract": "We study lattice points on hyperbolic circles centred at Heegner points of class number one. Our main result is that, on a density one subset of radii tending to infinity, the angles of such points equidistribute on the unit circle. To prove this, we establish a connection between lattice points and algebraic integers in the associated field having norm of a special form and satisfying a congruence condition. As a by-product of this, we obtain an explicit formulation of the classical hyperbolic circle problem as a shifted convolution sum for the function that counts the number of algebraic integers with given norm. Along the way, we also prove a lower bound for shifted B-numbers, which is done by sieve methods.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-06-16T16:23:40.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "11E25",
      "11N36",
      "11N37",
      "11R11",
      "11F99",
      "11L99"
    ],
    "keywords": [
      "heegner points",
      "hyperbolic angles",
      "algebraic integers",
      "study lattice points",
      "classical hyperbolic circle problem"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}