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

van Wamelen, Paul

doi:10.1090/S0025-5718-99-01179-5

Poonen’s question concerning isogenies between Smart’s genus 2 curves
HTML articles powered by AMS MathViewer

by Paul van Wamelen PDF

Math. Comp. 69 (2000), 1685-1697 Request permission

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

Henri Cohen, A course in computational algebraic number theory, Graduate Texts in Mathematics, vol. 138, Springer-Verlag, Berlin, 1993. MR 1228206, DOI 10.1007/978-3-662-02945-9
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, DOI 10.1002/9781118032527
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, DOI 10.1007/978-0-8176-4578-6
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, DOI 10.1007/3-540-61581-4_{6}3
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, DOI 10.1112/S002461159700035X
P. van Wamelen. Proving that a genus 2 curve has complex multiplication. Math. Comp. 68 (1999), 1663–1677.