Pell surfaces

  title={Pell surfaces},
  author={J'anos Koll'ar},
  journal={Acta Mathematica Hungarica},
  • J. Koll'ar
  • Published 20 June 2019
  • Mathematics
  • Acta Mathematica Hungarica
In 1826 Abel started the study of the polynomial Pell equation x 2 − g ( u ) y 2 = 1. Its solvability in polynomials x ( u ), y ( u ) depends on a certain torsion point on the Jacobian of the hyperelliptic curve v 2 = g ( u ). In this paper we study the affine surfaces defined by the Pell equations in 3-space with coordinates x, y, u , and aim to describe all affine lines on it. These are polynomial solutions of the equation x ( t ) 2 − g ( u ( t )) y ( t ) 2 = 1. Our results are rather… 
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