Class invariants by Shimura's reciprocity law

Alice Gee

Journal de théorie des nombres de Bordeaux (1999)

  • Volume: 11, Issue: 1, page 45-72
  • ISSN: 1246-7405

Abstract

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We apply the Shimura reciprocity law to determine when values of modular functions of higher level can be used to generate the Hilbert class field of an imaginary quadratic field. In addition, we show how to find the corresponding polynomial in these cases. This yields a proof for conjectural formulas of Morain and Zagier concerning such polynomials.

How to cite

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Gee, Alice. "Class invariants by Shimura's reciprocity law." Journal de théorie des nombres de Bordeaux 11.1 (1999): 45-72. <http://eudml.org/doc/248345>.

@article{Gee1999,
abstract = {We apply the Shimura reciprocity law to determine when values of modular functions of higher level can be used to generate the Hilbert class field of an imaginary quadratic field. In addition, we show how to find the corresponding polynomial in these cases. This yields a proof for conjectural formulas of Morain and Zagier concerning such polynomials.},
author = {Gee, Alice},
journal = {Journal de théorie des nombres de Bordeaux},
keywords = {minimal polynomial; Hilbert class field; class invariant; Shimura reciprocity law; Weber modular functions},
language = {eng},
number = {1},
pages = {45-72},
publisher = {Université Bordeaux I},
title = {Class invariants by Shimura's reciprocity law},
url = {http://eudml.org/doc/248345},
volume = {11},
year = {1999},
}

TY - JOUR
AU - Gee, Alice
TI - Class invariants by Shimura's reciprocity law
JO - Journal de théorie des nombres de Bordeaux
PY - 1999
PB - Université Bordeaux I
VL - 11
IS - 1
SP - 45
EP - 72
AB - We apply the Shimura reciprocity law to determine when values of modular functions of higher level can be used to generate the Hilbert class field of an imaginary quadratic field. In addition, we show how to find the corresponding polynomial in these cases. This yields a proof for conjectural formulas of Morain and Zagier concerning such polynomials.
LA - eng
KW - minimal polynomial; Hilbert class field; class invariant; Shimura reciprocity law; Weber modular functions
UR - http://eudml.org/doc/248345
ER -

References

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  1. [1] B. Birch, Weber's class invariants. Mathematika16 (1969), pp. 283-294. Zbl0226.12005MR262206
  2. [2] S. Lang, Elliptic functions. 2nd edition, Springer GTM112, 1987. Zbl0615.14018MR890960
  3. [3] F. Morain, Primality Proving Using Elliptic Curves: An Update. Algorithmic Number Theory, Springer LNCS 1423 (1998), pp. 111-130. Zbl0908.11061MR1726064
  4. [4] R. Schertz, Die singulären Werte der Weberschen Funktionen f, f1, f2, γ2, γ3. J. Reine Angew. Math.286/287 (1976), pp. 46-74. Zbl0335.12018
  5. [5] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami Shoten and Princeton University Press, 1971. Zbl0221.10029MR314766
  6. [6] G. Shimura, Complex Multiplication, Modular functions of One Variable I. Springer LNM 320 (1973), pp. 39-56. Zbl0268.10015MR498404
  7. [7] H. Weber, Lehrbuch der Algebra. Band III: Elliptische Funktionen und algebraische Zahlen. 2nd edition, Braunschweig, 1908. (Reprint by Chelsea, New York, 1961.) 
  8. [8] N. Yui and D. Zagier, On the singular values of Weber modular functions. Math. Comp.66 (1997), no 220, pp. 1645-1662. Zbl0892.11022MR1415803

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