{
  "id": "2101.11705",
  "version": "v1",
  "published": "2021-01-27T22:00:22.000Z",
  "updated": "2021-01-27T22:00:22.000Z",
  "title": "Diophantine triples and K3 surfaces",
  "authors": [
    "Matija Kazalicki",
    "Bartosz Naskręcki"
  ],
  "comment": "21 pages, 1 figure",
  "categories": [
    "math.NT",
    "math.AG"
  ],
  "abstract": "A Diophantine $m$-tuple over $K$ is a set of $m$ non-zero (distinct) elements of $K$ with the property that the product of any two distinct elements is one less than a square in $K$. Let $X: (x^2-1)(y^2-1)(z^2-1)=k^2,$ be a threefold. Its $K$-rational points parametrize Diophantine triples over $K$ such that the product of the elements of the triple that corresponds to the point $(x,y,z,k)\\in X(K)$ is equal to $k$. We denote by $\\overline{X}$ the projective closure of $X$ and for a fixed $k$ by $X_k$ a fibre over $k$. First, we prove that the variety $\\overline{X}$ is rational which leads us to a new rational parametrization of the set of Diophantine triples. Next, specializing to finite fields, we find a Shioda-Inose structure of the K3 surface $X_k$ for a given $k\\in\\mathbb{F}_{p}^{\\times}$ in the prime field $\\mathbb{F}_{p}$ of odd characteristic, determined by an abelian surface which is a square $E_k\\times E_k$ of an explicit elliptic curve. We derive an explicit formula for $N(p,k)$, the number of Diophantine triples over $\\mathbb{F}_{p}$ with the product of elements equal to $k$. Moreover, we show that the variety $\\overline{X}$ admits a fibration by rational elliptic surfaces and from it we derive the formula for the number of points on $\\overline{X}$ over an arbitrary finite field $\\mathbb{F}_{q}$. We reprove the formula for the number of Diophantine triples over $\\mathbb{F}_{q}$ from Dujella-Kazalicki(2021). From the interplay of the two (K3 and rational) fibrations of $\\overline{X}$, we derive the formula for the second moment of the elliptic surface $E_k$ (and thus confirming Steven J. Miller's Bias conjecture in this particular case) which we describe in terms of Fourier coefficients of a rational newform generating $S_4(\\Gamma_{0}(8))$. Finally, in the Appendix, Luka Lasi\\'c defines circular Diophantine $m$-tuples and describes the parametrization of these sets.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2021-01-27T22:00:22.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H52",
      "14J28",
      "11D09",
      "11G05"
    ],
    "keywords": [
      "k3 surface",
      "luka lasic defines circular diophantine",
      "rational points parametrize diophantine triples",
      "elliptic surface",
      "finite field"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 21,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}