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Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Examples of genus two CM curves
defined over the rationals


Author: Paul van Wamelen
Journal: Math. Comp. 68 (1999), 307-320
MSC (1991): Primary 14-04; Secondary 14K22
DOI: https://doi.org/10.1090/S0025-5718-99-01020-0
MathSciNet review: 1609658
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Abstract: We present the results of a systematic numerical search for genus two curves defined over the rationals such that their Jacobians are simple and have endomorphism ring equal to the ring of integers of a quartic CM field. Including the well-known example $y^2 = x^5 - 1$ we find 19 non-isomorphic such curves. We believe that these are the only such curves.


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

  • 1. E. Gottschling. Explizite bestimmung der randflächen des fundamentalbereiches der modulgruppe zweiten grades. Math. Annalen, 138:103-124, 1959. MR 21:5748
  • 2. K. Hardy, R. H. Hudson, D. Richman, and K. S. Williams. The determination of all imaginary cyclic quartic fields with class number 2. Trans. Amer. Math. Soc., 311(1):1-55, 1989. MR 89f:11148
  • 3. J. I. Igusa. Arithmetic variety of moduli for genus two. Ann. of Math., 72(3):612-649, 1960. MR 22:5637
  • 4. S. Lang. Complex Multiplication. Springer-Verlag, 1983. MR 85f:10042
  • 5. H. Lange and C. Birkenhake. Complex Abelian Varieties. Springer-Verlag, 1992. MR 94j:14001
  • 6. S. Louboutin and R. Okazaki. Determination of all non-normal quartic CM-fields and of all non-abelian normal octic CM-fields with class number one. Acta Arith., 67(1):47-62, 1994. MR 95g:11107
  • 7. J.-F. Mestre. Construction de courbes de genre 2 à partir de leurs modules. In Effective Methods in Algebraic Geometry (Castiglioncello, 1990), volume 94 of Progr. Math. Birkhäuser, 1991, pp. 313-334. MR 92g:14022
  • 8. D. Mumford. Tata Lectures on Theta II, volume 43 of Progr. Math. Birkhäuser, 1984. MR 86b:14017
  • 9. J.-P. Serre. A Course in Arithmetic. Springer-Verlag, 1973. MR 49:8956
  • 10. B. Setzer. The determination of all imaginary, quartic, abelian number fields with class number 1. Math. Comp., 35(152):1383-1386, 1980. MR 81k:12005
  • 11. G. Shimura and Y. Taniyama. Complex Multiplication of Abelian Varieties. The Mathematical Society of Japan, 1961. MR 23:A2419
  • 12. A.-M. Spallek. Kurven vom geschlecht 2 und ihre anwendung in public-key-kryptosystemen. Preprint 18, Universität GH Essen, Ellernstraße 29, 45326 Essen, Germany, 1994.
  • 13. P. B. van Wamelen. Equations for the algebraic Jacobian of a hyperelliptic curve. Submitted to Trans. Amer. Math. Soc.
  • 14. L. C. Washington. Introduction to Cyclotomic Fields. Springer-Verlag, 1982. MR 85g:11001

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Additional Information

Paul van Wamelen
Affiliation: Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803-4918
Email: wamelen@math.lsu.edu

DOI: https://doi.org/10.1090/S0025-5718-99-01020-0
Received by editor(s): June 13, 1996
Additional Notes: This work was partially supported by grant LEQSF(1995-97)-RD-A-09 from the Louisiana Educational Quality Support Fund
Article copyright: © Copyright 1999 American Mathematical Society

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