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On the singular values
of Weber modular functions

Authors: Noriko Yui and Don Zagier
Journal: Math. Comp. 66 (1997), 1645-1662
MSC (1991): Primary 11G15; Secondary 11R37, 11F03, 11G16
MathSciNet review: 1415803
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Abstract | References | Similar Articles | Additional Information

Abstract: The minimal polynomials of the singular values of the classical Weber modular functions give far simpler defining polynomials for the class fields of imaginary quadratic fields than the minimal polynomials of singular moduli of level 1. We describe computations of these polynomials and give conjectural formulas describing the prime decomposition of their resultants and discriminants, extending the formulas of Gross-Zagier for the level 1 case.

References [Enhancements On Off] (What's this?)

  • 1. Atkin, O. and Morain, F., Elliptic curves and primality proving, Math. Comp. 61 (1993), 29-68. MR 93m:11136
  • 2. Berwick, W.E.H., Modular invariants expressible in terms of quadratic and cubic irrationalities, Proc. London Math. Soc. 28 (1928), 53-69.
  • 3. Birch, B., Weber's class invariants, Mathematika 16 (1969), 283-294. MR 41:6816
  • 4. Cohn, H., Introduction to the Construction of Class Fields, Cambridge studies in advanced mathematics 6, Cambridge University Press, 1985. MR 87i:11165
  • 5. Cox, D., Primes of the form $x^{2}+ny^{2}$, John Wiley & Sons, 1989. MR 90m:11016
  • 6. Deuring, M., Teilbarkeitseigenschaften der singulären Moduln der elliptischen Funktionen und die Diskriminante der Klassengleichung, Comm. Math. Helvetici 19 (1946), 74-82. MR 8:318d
  • 7. Gross, B. and Zagier, D., On singular moduli, J. Reine Angew. Math. 355 (1985), 191-220. MR 86j:11041
  • 8. Kaltofen, E., and Yui, N., Explicit construction of the Hilbert class fields of imaginary quadratic fields by integer lattice reduction, Number Theory, New York Seminars 1989-90, Springer-Verlag, 1991, pp. 149-202. MR 92j:11133
  • 9. Schertz, R., Die singulären Werte der Weberschen Funktionen $\mathfrak {f}$, $\mathfrak {\mathfrak f}_{1}$, $\mathfrak {\mathfrak f}_{2}$, $\gamma _{2}$, $\gamma _{3}$, J. Reine Angew. Math. 286/287 (1976), 46-74. MR 54:10205
  • 10. Watson, G.N., Singular moduli (3), Proc. London Math. Soc. 40 (1936), 83-142.
  • 11. Weber, H., Lehrbuch der Algebra, Bd. III, Braunschweig, 1908.
  • 12. Hajir, F. and Rodriguez Villegas, F., Explicit elliptic units I, preprint, 1995.

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Additional Information

Noriko Yui
Affiliation: Department of Mathematics, Queen’s University, Kingston, Ontario, Canada K7L 3N6

Don Zagier
Affiliation: Max-Planck-Institut für Mathematik, Gottfried-Claren-Straße 26, 53225 Bonn, Germany

Received by editor(s): June 8, 1994
Received by editor(s) in revised form: June 19, 1996
Article copyright: © Copyright 1997 American Mathematical Society

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