Mathematics of Computation

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

Author(s): Paul van Wamelen.
Journal: Math. Comp. 69 (2000), 1685-1697.
MSC (1991): Primary 14-04; Secondary 14K02
Posted: August 18, 1999
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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:

1.: H. Cohen. A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics 138. Springer-Verlag, 1995. MR 94i:11105
2.: P. Griffiths and J. Harris Principles of Algebraic Geometry. John Wiley & Sons, Inc., New York, 1994. MR 95d:14001
3.: D. Mumford. Tata Lectures on Theta II, volume 43 of Progr. Math. Birkhäuser, 1984. MR 86b:14017
4.: B. Poonen. Computational aspects of curves of genus at least $2$ . Algorithmic Number Theory. (Talence, 1996), Lecture Notes in Comput. Sci., 1122, Springer, Berlin, 1996, 283-306. MR 98c:11059
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 98d:11072
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: 10.1090/S0025-5718-99-01179-5
PII: S 0025-5718(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
Posted: August 18, 1999
Copyright of article: Copyright 2000, American Mathematical Society


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