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manuscripta mathematica

, Volume 100, Issue 4, pp 519–533 | Cite as

Finding rational points on bielliptic genus 2 curves

  • E. Victor Flynn
  • Joseph L. Wetherell

Abstract:

We discuss a technique for trying to find all rational points on curves of the form Y 2=f 3 X 6+f 2 X 4+f 1 X 2+f 0, where the sextic has nonzero discriminant. This is a bielliptic curve of genus 2. When the rank of the Jacobian is 0 or 1, Chabauty's Theorem may be applied. However, we shall concentrate on the situation when the rank is at least 2. In this case, we shall derive an associated family of elliptic curves, defined over a number field ℚα. If each of these elliptic curves has rank less than the degree of ℚα : ℚ, then we shall describe a Chabauty-like technique which may be applied to try to find all the points (x,y) defined over ℚα) on the elliptic curves, for which x∈ℚ. This in turn allows us to find all ℚ-rational points on the original genus 2 curve. We apply this to give a solution to a problem of Diophantus (where the sextic in X is irreducible over ℚ), which simplifies the recent solution of Wetherell. We also present two examples where the sextic in X is reducible over ℚ.

Mathematics Subject Classification (1991):11G30, 14H40 

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Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • E. Victor Flynn
    • 1
  • Joseph L. Wetherell
    • 2
  1. 1.Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, United Kingdom. E-mail: evflynn@liv.ac.ukUK
  2. 2.Department of Mathematics, University of Southern California, 1042 West 36th Place, Los Angeles, CA 90089-1113, USA. E-mail: jlwether@alum.mit.eduUS

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