Abstract
We solve two genus 2 curve problems using Magma. First we give examples of how Magma can be used to find the equation of a genus 2 curve whose Jacobian has prescribed Complex Multiplication. We treat 2 fields, one easy and one harder. Secondly we show how Magma can be used to find, and ultimately prove existence of, rational isogenies between the Jacobians of two genus 2 curves.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
1. Wieb Bosma, John Cannon, Catherine Playoust, The Magma algebra system I: The user language, J. Symbolic Comput. 24 (1997), 235–265. See also the Magma home page at http://magma.maths.usyd.edu.au/magma/.
2. H. M. Farkas, I. Kra, Riemann surfaces, Graduate Texts in Mathematics 71, New York: Springer-Verlag, 1992 (second edition).
3. William Fulton, Algebraic curves. An introduction to algebraic geometry, New York-Amsterdam: W. A. Benjamin, Inc., 1969. Notes written with the collaboration of Richard Weiss, Mathematics Lecture Notes Series.
4. Phillip Gri.ths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, New York: John Wiley & Sons Inc., 1994. Reprint of the 1978 original.
5. S. Lang, Introduction to Algebraic and Abelian Functions, Springer-Verlag, 1982.
6. S. Lang, Complex Multiplication, Springer-Verlag, 1983.
7. H. Lange, Ch. Birkenhake, Complex Abelian Varieties, Springer-Verlag, 1992.
8. J. S. Milne, Jacobian varieties, in: G. Cornell, J. H. Silverman (editors), Arithmetic Geometry, Springer-Verlag, 1986.
9. D. Mumford, Tata Lectures on Theta I, Progr. Math. 28, Birkhäuser, 1983.
10. D. Mumford, Tata Lectures on Theta II, Progr. Math. 43, Birkhäuser, 1984.
11. Naoki Murabayashi, Atsuki Umegaki, Determination of all Q-rational Cmpoints in the moduli space of principally polarized abelian surfaces, J. Algebra 235-1 (2001), 267–274.
12. Michael Rosen, Abelian varieties over C, pp. 79–101 in: Arithmetic geometry (Storrs, Conn., 1984), New York: Springer, 1986.
13. N.P. Smart, S-unit equations, binary forms and curves of genus 2, Proc. London Math. Soc. (3) 75-2 (1997), 271–307.
14. Paul vanWamelen, Examples of genus two CM curves de.ned over the rationals, Math. Comp. 68-225 (1999), 307–320.
15. Paul van Wamelen, Proving that a genus 2 curve has complex multiplication, Math. Comp. 68-228 (1999), 1663–1677.
16. Paul van Wamelen, Poonen's question concerning isogenies between Smart's genus 2 curves, Math. Comp. 69-232 (2000), 1685–1697.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2006 Springer
About this chapter
Cite this chapter
van Wamelen, P.B. (2006). Computing with the analytic Jacobian of a genus 2 curve. In: Bosma, W., Cannon, J. (eds) Discovering Mathematics with Magma. Algorithms and Computation in Mathematics, vol 19. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-37634-7_5
Download citation
DOI: https://doi.org/10.1007/978-3-540-37634-7_5
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-37632-3
Online ISBN: 978-3-540-37634-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)