no. 1
Class invariants by Shimura's reciprocity law
Journal de théorie des nombres de Bordeaux, Tome 11 (1999) no. 1, pp. 45-72.

On applique la loi de réciprocité de Shimura pour décider quand les valeurs des fonctions modulaires de haut niveau peuvent être utilisées pour engendrer le corps de classes de Hilbert d'un corps quadratique imaginaire. Lorsque c'est le cas, nous montrons aussi comment trouver le polynôme correspondant. Cela donne une preuve de certaines formules conjecturales de Morain et Zagier relatives à ces polynômes.

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.

@article{JTNB_1999__11_1_45_0,
     author = {Gee, Alice},
     title = {Class invariants by {Shimura's} reciprocity law},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {45--72},
     publisher = {Universit\'e Bordeaux I},
     volume = {11},
     number = {1},
     year = {1999},
     mrnumber = {1730432},
     zbl = {0957.11048},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_1999__11_1_45_0/}
}
TY  - JOUR
AU  - Gee, Alice
TI  - Class invariants by Shimura's reciprocity law
JO  - Journal de théorie des nombres de Bordeaux
PY  - 1999
SP  - 45
EP  - 72
VL  - 11
IS  - 1
PB  - Université Bordeaux I
UR  - http://www.numdam.org/item/JTNB_1999__11_1_45_0/
LA  - en
ID  - JTNB_1999__11_1_45_0
ER  - 
%0 Journal Article
%A Gee, Alice
%T Class invariants by Shimura's reciprocity law
%J Journal de théorie des nombres de Bordeaux
%D 1999
%P 45-72
%V 11
%N 1
%I Université Bordeaux I
%U http://www.numdam.org/item/JTNB_1999__11_1_45_0/
%G en
%F JTNB_1999__11_1_45_0
Gee, Alice. Class invariants by Shimura's reciprocity law. Journal de théorie des nombres de Bordeaux, Tome 11 (1999) no. 1, pp. 45-72. http://www.numdam.org/item/JTNB_1999__11_1_45_0/

[1] B. Birch, Weber's class invariants. Mathematika 16 (1969), pp. 283-294. | MR | Zbl

[2] S. Lang, Elliptic functions. 2nd edition, Springer GTM 112, 1987. | MR | Zbl

[3] F. Morain, Primality Proving Using Elliptic Curves: An Update. Algorithmic Number Theory, Springer LNCS 1423 (1998), pp. 111-130. | MR | Zbl

[4] R. Schertz, Die singulären Werte der Weberschen Funktionen f, f1, f2, γ2, γ3. J. Reine Angew. Math. 286/287 (1976), pp. 46-74. | Zbl

[5] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami Shoten and Princeton University Press, 1971. | MR | Zbl

[6] G. Shimura, Complex Multiplication, Modular functions of One Variable I. Springer LNM 320 (1973), pp. 39-56. | MR | Zbl

[7] H. Weber, Lehrbuch der Algebra. Band III: Elliptische Funktionen und algebraische Zahlen. 2nd edition, Braunschweig, 1908. (Reprint by Chelsea, New York, 1961.)

[8] N. Yui and D. Zagier, On the singular values of Weber modular functions. Math. Comp. 66 (1997), no 220, pp. 1645-1662. | MR | Zbl