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Doklady Mathematics

, Volume 100, Issue 2, pp 440–444 | Cite as

On the Finiteness of the Number of Elliptic Fields with Given Degrees of \(S\) -Units and Periodic Expansion of \(\sqrt f \)

  • V. P. PlatonovEmail author
  • M. M. PetruninEmail author
  • Yu. N. ShteinikovEmail author
MATHEMATICS
  • 9 Downloads

Abstract

For a field k of characteristic 0, up to a natural equivalence relation, it is proved that the number of nontrivial elliptic fields \(k(x)(\sqrt f )\) with a periodic continued fraction expansion of \(\sqrt f \in k((x))\) for which the corresponding elliptic curve contains a k-point of even order at most 18 or a k-point of odd order at most 11 is finite. In the case when k is a quadratic extension of \(\mathbb{Q}\), all such fields are found.

Notes

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Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.Scientific Research Institute for System Analysis, Russian Academy of SciencesMoscowRussia
  2. 2.Steklov Mathematical Institute, Russian Academy of SciencesMoscowRussia

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