Hostname: page-component-8448b6f56d-xtgtn Total loading time: 0 Render date: 2024-04-20T02:11:39.871Z Has data issue: false hasContentIssue false
  • English
  • Français

On the units generated by Weierstrass forms

Part of: Algebraic number theory: global fields Computational number theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  01 August 2014

Ömer Küçüksakallı
Ömer Küçüksakallı*
Affiliation:
Department of Mathematics , Middle East Technical University, 06800 Ankara, Turkey email komer@metu.edu.tr

Abstract

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.

There is an algorithm of Schoof for finding divisors of class numbers of real cyclotomic fields of prime conductor. In this paper we introduce an improvement of the elliptic analogue of this algorithm by using a subgroup of elliptic units given by Weierstrass forms. These elliptic units which can be expressed in terms of $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}x$-coordinates of points on elliptic curves enable us to use the fast arithmetic of elliptic curves over finite fields.

MSC classification

Secondary: 11G16: Elliptic and modular units 11R29: Class numbers, class groups, discriminants 11Y40: Algebraic number theory computations
Type
Research Article
Information
LMS Journal of Computation and Mathematics , Volume 17 , Special Issue A: Algorithmic Number Theory Symposium XI , 2014 , pp. 303 - 313
Copyright
© The Author 2014 

References

Buhler, J. P. and Harvey, D., ‘Irregular primes to 163 million’, Math. Comp. 80 (2011) no. 276, 2435–2444.CrossRefGoogle Scholar
Cox, D. A., Primes of the form x 2 + n y 2 : Fermat, class field theory, and complex multiplication (Wiley, New York, 1989).Google Scholar
Gee, A., ‘Class fields by Shimura reciprocity’, PhD Thesis, Universiteit Leiden, 2001.Google Scholar
Kubert, D. and Lang, S., Modular units, Grundlehren der mathematischen Wissenschaften 244 (Spinger, Berlin, 1981).CrossRefGoogle Scholar
Kucuksakalli, O., ‘Class numbers of ray class fields of imaginary quadratic fields’, Math. Comp. 80 (2011) no. 274, 1099–1122.Google Scholar
Lang, S., Elliptic functions, 2nd edn,Graduate Texts in Mathematics 112 (Springer, New York, 1987).CrossRefGoogle Scholar
Oukhaba, H., ‘Index formulas for ramified elliptic units’, Compos. Math. 137 (2003) no. 1, 1–22.CrossRefGoogle Scholar
Robert, G., ‘Nombres de Hurwitz et unités elliptiques. Un critère de régularité pour les extensions abéliennes d’un corps quadratique imaginaire’, Ann. Sci. Éc. Norm. Supér. (4) 11 (1978) no. 3, 297–389.CrossRefGoogle Scholar
Schoof, R., ‘Class numbers of real cyclotomic fields of prime conductor’, Math. Comp. 72 (2003) no. 242, 913–937.CrossRefGoogle Scholar
Silverman, J. S., Advanced topics in the arithmetic of elliptic curves, Graduate Texts in Mathematics 151 (Springer, New York, 1994).Google Scholar
Sinnott, W., ‘On the Stickelberger ideal and the circular units of a cyclotomic field’, Ann. of Math. (2) 108 (1978) no. 1, 107–134.CrossRefGoogle Scholar
Stark, H. M., ‘L-functions at s = 1. IV. First derivatives at s = 0’, Adv. Math. 35 (1980) no. 3, 197–235.CrossRefGoogle Scholar
Washington, L., Elliptic curves. Number theory and cryptography, 2nd edn,Discrete Mathematics and its Applications (Chapman & Hall/CRC, Boca Raton, FL, 2008).Google Scholar
PARI/GP, version 2.3.5, http://pari.math.u-bordeaux.fr/, Bordeaux, 2010.Google Scholar