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Proving that a genus 2 curve has complex multiplication
Author(s):
Paul
van Wamelen.
Journal:
Math. Comp.
68
(1999),
1663-1677.
MSC (1991):
Primary 14-04;
Secondary 14K22
Posted:
May 17, 1999
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Abstract:
Recently examples of genus 2 curves defined over the rationals were found which, conjecturally, should have complex multiplication. We prove this conjecture. This involves computing an explicit representation of a rational map defining complex multiplication.
References:
- 1.
- H. Cohen. A Course in Computational Algebraic Number Theory. Graduate Texts in Mathematics 138. Springer-Verlag, 1993. MR 94i:11105
- 2.
- E. Gottschling. Explizite bestimmung der randflächen des fundamentalbereiches der modulgruppe zweiten grades. Math. Annalen, 138:103-124, 1959. MR 21:5748
- 3.
- J. Milne. Jacobian varieties. In G. Cornell and J. Silverman, editors, Arithmetic Geometry. Springer-Verlag, 1986, pp. 167-212. MR 89b:14029
- 4.
- D. Mumford. Tata Lectures on Theta II, Progr. Math. 43, Birkhäuser, 1984. MR 86b:14017
- 5.
- P. van Wamelen. Examples of genus two CM curves defined over the raionals, Math. Comp. 68 (1999), 307-320. CMP 99:03
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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:
10.1090/S0025-5718-99-01101-1
PII:
S 0025-5718(99)01101-1
Keywords:
CM-curves,
complex multiplication,
genus 2 curves
Received by editor(s):
December 16, 1997
Posted:
May 17, 1999
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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