Rational points in geometric progressions on certain hyperelliptic curves

Andrew Bremner, Maciej Ulas

Research output: Contribution to journalArticlepeer-review

9 Scopus citations

Abstract

We pose a simple Diophantine problem which may be expressed in the language of geometry. Let C be a hyperelliptic curve given by the equation y2 = f(x), where f ∈ ℤ[x] is without multiple roots. We say that points Pi = (xi; yi) ∈ C(ℚ) for i = 1, 2, . ., k, are in geometric progression if the numbers xi for i = 1, 2, . . ., k, are in geometric progression. Let n ≥ 3 be a given integer. In this paper we show that there exist polynomials a, b 2 ℤ[t] such that on the curve y2 = a(t)xn + b(t) (defined over the field ℚ(t)) we can find four points in geometric progression. In particular this result generalizes earlier results of Berczes and Ziegler concerning the existence of geometric progressions on Pell type quadrics y2 = ax2 + b. We also investigate for fixed b ∈ ℤ, when there can exist rationals yi, i = 1, . . ., 4, with -y2 i - b} forming a geometric progression, with particular attention to the case b = 1. Finally, we show that there exist infinitely many parabolas y2 = ax + b which contain five points in geometric progression.

Original languageEnglish (US)
Pages (from-to)669-683
Number of pages15
JournalPublicationes Mathematicae
Volume82
Issue number3-4
DOIs
StatePublished - Jun 24 2013

Keywords

  • Geometric progressions
  • Hyperelliptic curves
  • Rational points

ASJC Scopus subject areas

  • Mathematics(all)

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