Generalized class invariants with `Thetanullwerte'

Generalized class invariants with `Thetanullwerte'

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 $eta$-function. Furthermore, we prove that these invariants are units in the corresponding class field as most of their classical counterparts.

___

  • [1] Belding, J., Bröker, R., Enge, A. and Lauter, K.: Computing Hilbert Class Polynomials, Springer ANTS-VIII, vol. 5011 of Lect. Notes Comp. Sci., 282–295 (2008).
  • [2] Brauer, R.: On the Zeta-Function of Algebraic Number Fields, American Journal of Mathematics 69, 243–250 (1947).
  • [3] Deuring, M.: Die Klassenk¨orper der komplexen Multiplikation, Enzykl. d. math. Wiss., 2. Auflage, Heft 10, Stuttgart (1958).
  • [4] Dupont, R.: Moyenne arithmetico-geometrique, suites de Borchardt et applications, Phd Thesis, ´Ecole Polytechnique (2006).
  • [5] Dupont, R.: Fast Evaluation of Modular Functions Using Newton Iterations and the AGM, Math. Comp. 80, 1823– 1847 (2011).
  • [6] Enge, A. and Morain, F.: Generalised Weber Functions I, preprint, 2009 http://hal.inria.fr/inria-00385608/.
  • [7] Gee, A.: Class Fields by Shimura Reciprocity, Phd Thesis, Universiteit Leiden (2001).
  • [8] Hart, W. B.: Schl¨afli Modular Equations for GeneralizedWeber Functions, Ramanujan Journal, 15, 435–468 (2008).
  • [9] 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).
  • [10] Lang, S.: Elliptic Functions, Addison-Wesley 1973.
  • [11] 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).
  • [12] MAGMA: Computer Algebra Software package, Computational Algebra System: http://magma.maths.usyd.edu.au/magma/.
  • [13] Rauch, H. E. and Farkas, H. M.: Theta Functions with Applications to Riemann Surfaces, The Williams-Wilkins Company 1974.
  • [14] Schertz, R.: Weber’s Class Invariants Revisited, Journal de th´eorie des nombres de Bordeaux 14, 325–343 (2002).
  • [15] Schertz, R.: Complex Multiplication, Cambridge University Press, Cambridge 2010.
  • [16] Silverman, J. H.: Advanced Topics in the Arithmetic of Elliptic Curves, Springer Verlag, Chapter II, 1994.
  • [17] Uzunkol, O.: Atkin’s ECPP Algorithm, M. Sc. Thesis TU-Kaiserslautern 2004.
  • [18] Uzunkol, O.: Uber die Konstruktion algebraischer Kurven mittels komplexer Multiplikation, Phd Thesis, Technische Universit¨at Berlin (2010).
  • [19] Weber, H.: Lehrbuch der Algebra, Bd. 3, 2. Aufl. Braunschweig (1908).