We introduce generalized class invariants as quotients of Thetanullwerte, which realize the computation of class polynomials more efficiently than as quotients of values of the Dedekind h-function. Furthermore, we prove that these invariants are units in the corresponding class field as most of their classical counterparts.
We introduce generalized class invariants as quotients of Thetanullwerte, which realize the computation of class polynomials more efficiently than as quotients of values of the Dedekind h-function. Furthermore, we prove that these invariants are units in the corresponding class field as most of their classical counterparts.
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Belding, J., Br¨ oker, R., Enge, A. and Lauter, K.: Computing Hilbert Class Polynomials, Springer ANTS-VIII, vol. 5011 of Lect. Notes Comp. Sci., 282–295 (2008).
Brauer, R.: On the Zeta-Function of Algebraic Number Fields, American Journal of Mathematics 69, 243–250 (1947). Deuring, M.: Die Klassenk¨ orper der komplexen Multiplikation, Enzykl. d. math. Wiss., 2. Auflage, Heft 10, Stuttgart (1958).
Dupont, R.: Moyenne arithm´ etico-g´ eom´ etrique, suites de Borchardt et applications, Phd Thesis, ´ Ecole Polytechnique (2006).
Dupont, R.: Fast Evaluation of Modular Functions Using Newton Iterations and the AGM, Math. Comp. 80, 1823– 1847 (2011).
Enge, A. and Morain, F.: Generalised Weber Functions I, preprint, 2009 http://hal.inria.fr/inria-00385608/.
Gee, A.: Class Fields by Shimura Reciprocity, Phd Thesis, Universiteit Leiden (2001).
Hart, W. B.: Schl¨ afli Modular Equations for Generalized Weber Functions, Ramanujan Journal, 15, 435–468 (2008). Hilbert, D.: Ein neuer Beweis des Kroneckerschen Fundamentalsatzes ¨ uber Abelsche K¨ orper, Nach. K. Ges. Wiss. G¨ ottingen, 29–39 (1896) (Ges. Abh., 53–62).
Lang, S.: Elliptic Functions, Addison-Wesley 1973.
Lepr´ evost, F., Pohst, M. and Uzunkol, O.: On the computation of class polynomials with “Thetanullwerte” and its applications to the unit group computation, Experimental Mathematics, 20(3), 271–281 (2011).
MAGMA: Computer Algebra Software package, Computational Algebra System: http://magma.maths.usyd.edu.au/magma/. Rauch, H. E. and Farkas, H. M.: Theta Functions with Applications to Riemann Surfaces, The Williams-Wilkins Company 1974.
Schertz, R.: Weber’s Class Invariants Revisited, Journal de th´ eorie des nombres de Bordeaux 14, 325–343 (2002). Schertz, R.: Complex Multiplication, Cambridge University Press, Cambridge 2010.
Silverman, J. H.: Advanced Topics in the Arithmetic of Elliptic Curves, Springer Verlag, Chapter II, 1994.
Uzunkol, O.: Atkin’s ECPP Algorithm, M. Sc. Thesis TU-Kaiserslautern 2004.
Uzunkol, O.: ¨ Uber die Konstruktion algebraischer Kurven mittels komplexer Multiplikation, Phd Thesis, Technische Universit¨ at Berlin (2010).
Weber, H.: Lehrbuch der Algebra, Bd. 3, 2. Aufl. Braunschweig (1908).