{
  "id": "2401.03982",
  "version": "v1",
  "published": "2024-01-08T16:02:25.000Z",
  "updated": "2024-01-08T16:02:25.000Z",
  "title": "Sharp bounds for the number of rational points on algebraic curves and dimension growth, over all global fields",
  "authors": [
    "Gal Binyamini",
    "Raf Cluckers",
    "Fumiharu Kato"
  ],
  "comment": "16 pages",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "Let $C\\subset{\\mathbb P}_K^2$ be an algebraic curve over a number field $K$, and denote by $d_K$ the degree of $K$ over ${\\mathbb Q}$. We prove that the number of $K$-rational points of height at most $H$ in $C$ is bounded by $c d^{2}H^{2d_K/d}(\\log H)^\\kappa$ where $c,\\kappa$ are absolute constants. We also prove analogous results for global fields in positive characteristic, and, for higher dimensional varieties. The quadratic dependence on $d$ in the bound as well as the exponent of $H$ are optimal; the novel aspect is the quadratic dependence on $d$ which answers a question raised by Salberger. We derive new results on Heath-Brown's and Serre's dimension growth conjecture for global fields, which generalize in particular the results by the first two authors and Novikov from the case $K={\\mathbb Q}$. The proofs however are of a completely different nature, replacing the real analytic approach previously used by the $p$-adic determinant method. The optimal dependence on $d$ is achieved using a technical improvement in the treatment of high multiplicity points on mod $p$ reductions of algebraic curves.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2024-01-08T16:02:25.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "global fields",
      "algebraic curve",
      "rational points",
      "sharp bounds",
      "quadratic dependence"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}