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

DOI:
https://doi.org/10.1090/S0025-5718-99-01179-5

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.

**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 .*Algorithmic Number Theory.*(Talence, 1996), Lecture Notes in Comput. Sci., 1122, Springer, Berlin, 1996, 283-306. MR**98c:11059****5.**N. P. Smart. -unit equations, binary forms and curves of genus .*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:
https://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