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Examples of genus two CM curves defined over the rationals
Author(s):
Paul
van Wamelen.
Journal:
Math. Comp.
68
(1999),
307-320.
MSC (1991):
Primary 14-04;
Secondary 14K22
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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  we find 19 non-isomorphic such curves. We believe that these are the only such curves.
References:
- 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:
10.1090/S0025-5718-99-01020-0
PII:
S 0025-5718(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
Copyright of article:
Copyright
1999,
American Mathematical Society
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