Hostname: page-component-7f64f4797f-l842n Total loading time: 0 Render date: 2025-11-10T03:01:31.574Z Has data issue: false hasContentIssue false
  • English
  • Français

Trace of Frobenius endomorphism of an elliptic curve with complex multiplication

Published online by Cambridge University Press:  17 April 2009

Noburo Ishii
Noburo Ishii
Affiliation:
Department of Mathematics and Information Science, Osaka Prefecture University, 1–1 Gakuen-cho, Sakai, Osaka, 599–8531, Japan e-mail: ishii@mi.cias.osakafu-u.ac.jp
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let E be an elliptic curve with complex multiplication by R, where R is an order of discriminant D < −4 of an imaginary quadratic field K. If a prime number p is decomposed completely in the ring class field associated with R, then E has good reduction at a prime ideal p of K dividing p and there exist positive integers u and υ such that 4p = u2 – Du;2. It is well known that the absolute value of the trace ap of the Frobenius endomorphism of the reduction of E modulo p is equal to u. We determine whether ap = u or ap = −u in the case where the class number of R is 2 or 3 and D is divisible by 3, 4 or 5.

Information

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 70 , Issue 1 , August 2004 , pp. 125 - 142
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Brewer, B.W., ‘On certain character sums’, Trans. Amer. Math. Soc. 99 (1961), 241–245.Google Scholar
[2]Brewer, B.W., ‘On primes of the form u 2 + 5u 2’, Proc. Amer. Math. Soc. 17 (1966), 502–509.Google Scholar
[3]Cox, D.A., Primes of the form x2 + ny2 (John Wiley & Sons Inc., New York, 1989).Google Scholar
[4]Hiramatsu, T. and Ishii, N., ‘Quartic residuacity and cusp forms of weight one’, Comment Math. Univ. St. Paul. 34 (1985), 91–103.Google Scholar
[5]Joux, A. and Morain, F., ‘Sur les sommes de caractères liées aux courbes elliptiques à multiplication complexe’, J. Number Theory 55 (1995), 108–128.Google Scholar
[6]Lang, S., Elliptic functions (Addison-Wesley Publishing Co. Ins., 1973).Google Scholar
[7]Leprévost, F. and Morain, F., ‘Revêtements de courbes elliptiques à multiplication complexe par des courbes hyperelliptiques et sommes de caractères’, J. Number Theory 64 (1997), 165–182.Google Scholar
[8]Lenstra, H.W. Jr, ‘Complex multiplication structure of elliptic curves’, J. Number Theory 56 (1996), 227–241.CrossRefGoogle Scholar
[9]Serre, J-P., Local fields, Graduate Texts in Mathematics 67 (Springer-Verlag, New York, 1979).Google Scholar
[10]Silverman, J.H., The arithmetic of elliptic curves, Graduate Texts in Mathematics 106 (Springer-Verlag, New York, 1986).Google Scholar
[11]Silverman, J.H., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151 (Springer-Verlag, New York, 1994).CrossRefGoogle Scholar