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Poonen's question concerning isogenies
between Smart's genus 2 curves


Author: Paul van Wamelen
Journal: Math. Comp. 69 (2000), 1685-1697
MSC (1991): Primary 14-04; Secondary 14K02
Published electronically: August 18, 1999
MathSciNet review: 1677415
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Abstract | References | Similar Articles | Additional Information

Abstract: We describe a method for proving that two explicitly given genus two curves have isogenous jacobians. We apply the method to the list of genus 2 curves with good reduction away from 2 given by Smart. This answers a question of Poonen.


References [Enhancements On Off] (What's this?)

  • 1. Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206
  • 2. Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523
  • 3. David Mumford, Tata lectures on theta. II, Progress in Mathematics, vol. 43, Birkhäuser Boston, Inc., Boston, MA, 1984. Jacobian theta functions and differential equations; With the collaboration of C. Musili, M. Nori, E. Previato, M. Stillman and H. Umemura. MR 742776
  • 4. Bjorn Poonen, Computational aspects of curves of genus at least 2, Algorithmic number theory (Talence, 1996) Lecture Notes in Comput. Sci., vol. 1122, Springer, Berlin, 1996, pp. 283–306. MR 1446520, 10.1007/3-540-61581-4_63
  • 5. N. P. Smart, 𝑆-unit equations, binary forms and curves of genus 2, Proc. London Math. Soc. (3) 75 (1997), no. 2, 271–307. MR 1455857, 10.1112/S002461159700035X
  • 6. P. van Wamelen. Proving that a genus 2 curve has complex multiplication. Math. Comp. 68 (1999), 1663-1677.

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Additional Information

Paul van Wamelen
Affiliation: Department of Mathematics, University of South Africa, P. O. Box 392, Pretoria, 0003, South Africa
Address at time of publication: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918
Email: wamelen@math.lsu.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01179-5
Keywords: Isogenies, genus 2 curves, good reduction
Received by editor(s): June 9, 1998
Received by editor(s) in revised form: December 7, 1998
Published electronically: August 18, 1999
Article copyright: © Copyright 2000 American Mathematical Society