# Weber's class invariants revisited

Journal de théorie des nombres de Bordeaux (2002)

- Volume: 14, Issue: 1, page 325-343
- ISSN: 1246-7405

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topSchertz, Reinhard. "Weber's class invariants revisited." Journal de théorie des nombres de Bordeaux 14.1 (2002): 325-343. <http://eudml.org/doc/248893>.

@article{Schertz2002,

abstract = {Let $K$ be a quadratic imaginary number field of discriminant $d$. For $t \in \mathbb \{N\}$ let $\mathfrak \{O\}_t$ denote the order of conductor $t$ in $K$ and $j(\mathfrak \{O\}_t)$ its modular invariant which is known to generate the ring class field modulo $t$ over $K$. The coefficients of the minimal equation of $j(\mathfrak \{O\}_t)$ being quite large Weber considered in [We] the functions $f,f_1, f_2, \gamma _2, \gamma _3$ defined below and thereby obtained simpler generators of the ring class fields. Later on the singular values of these functions played a crucial role in Heegner’s solution [He] of the class number one problem for quadratic imaginary number fields [He,Me2,St]. Actually these numbers are used in cryptography to find elliptic curves over finite fields with nice properties. It is the aim of this paper i) to enunciate some known results of [We,Bi,Me2,Sch1] cf. Theorem 1, 2 and 3, concerning singular values of the functions $f,f_1, f_2, \gamma _2, \gamma _3$, and ii) to give a short and easy proof of these results. That method also applies to other functions, such as those in the table preceding Theorem 4. The proofs of Theorems 1 to 4 are given at the end of our article. Our proofs rely on the reciprocity law of Shimura (cf. Theorem 5, and also Theorem 6 and 7), and on the knowledge of the $24$-th root of unity that acquires $\eta = \@root 24 \of \{\Delta \}$ by unimodular substitution (cf. Proposition 2, and [Mel] p.162); they also give via Proposition 3 explicit formulas for the conjugates of the singular values (of the above functions), that are quite useful for numerical calculations. Examples of such calculations are to be found immediately before the references.},

author = {Schertz, Reinhard},

journal = {Journal de théorie des nombres de Bordeaux},

keywords = {complex multiplication; ring class field; modular invariant; class invariant; Schläfli functions},

language = {eng},

number = {1},

pages = {325-343},

publisher = {Université Bordeaux I},

title = {Weber's class invariants revisited},

url = {http://eudml.org/doc/248893},

volume = {14},

year = {2002},

}

TY - JOUR

AU - Schertz, Reinhard

TI - Weber's class invariants revisited

JO - Journal de théorie des nombres de Bordeaux

PY - 2002

PB - Université Bordeaux I

VL - 14

IS - 1

SP - 325

EP - 343

AB - Let $K$ be a quadratic imaginary number field of discriminant $d$. For $t \in \mathbb {N}$ let $\mathfrak {O}_t$ denote the order of conductor $t$ in $K$ and $j(\mathfrak {O}_t)$ its modular invariant which is known to generate the ring class field modulo $t$ over $K$. The coefficients of the minimal equation of $j(\mathfrak {O}_t)$ being quite large Weber considered in [We] the functions $f,f_1, f_2, \gamma _2, \gamma _3$ defined below and thereby obtained simpler generators of the ring class fields. Later on the singular values of these functions played a crucial role in Heegner’s solution [He] of the class number one problem for quadratic imaginary number fields [He,Me2,St]. Actually these numbers are used in cryptography to find elliptic curves over finite fields with nice properties. It is the aim of this paper i) to enunciate some known results of [We,Bi,Me2,Sch1] cf. Theorem 1, 2 and 3, concerning singular values of the functions $f,f_1, f_2, \gamma _2, \gamma _3$, and ii) to give a short and easy proof of these results. That method also applies to other functions, such as those in the table preceding Theorem 4. The proofs of Theorems 1 to 4 are given at the end of our article. Our proofs rely on the reciprocity law of Shimura (cf. Theorem 5, and also Theorem 6 and 7), and on the knowledge of the $24$-th root of unity that acquires $\eta = \@root 24 \of {\Delta }$ by unimodular substitution (cf. Proposition 2, and [Mel] p.162); they also give via Proposition 3 explicit formulas for the conjugates of the singular values (of the above functions), that are quite useful for numerical calculations. Examples of such calculations are to be found immediately before the references.

LA - eng

KW - complex multiplication; ring class field; modular invariant; class invariant; Schläfli functions

UR - http://eudml.org/doc/248893

ER -

## References

top- [Bi] B.J. Birch, Weber's Class Invariants. Mathematika16 (1969), 283-294. Zbl0226.12005MR262206
- [De] M. Deuring, Die Klassenkörper der komplexen Multiplikation. Enzykl. d. math. Wiss.I/2, 2.Auflage, Heft 10, Stuttgart, 1958. Zbl0123.04001MR167481
- [Ha-Vi] F. Hajir, F.R. Villegas, Explicit elliptic units, I. Duke Math. J.90 (1997), 495-521. Zbl0898.11025MR1480544
- [He] K. Heegner, Diophantische Analysis und Modulfunktionen. Math. Zeitschrift56 (1952), 227-253. Zbl0049.16202MR53135
- [La] S. Lang, Elliptic functions. Addison Wesley, 1973. Zbl0316.14001MR409362
- [Me1] C. Meyer, Über einige Anwendungen Dedekindscher Summen. J. Reine Angew. Math.198 (1957), 143-203. Zbl0079.10303MR104643
- [Me2] C. Meyer, Bemerkungen zum Satz von Heegner-Stark über die imaginär-quadratischen Zahlkörper mit der Klassenzahl Eins. J. Reine Angew. Math.242 (1970), 179-214. Zbl0218.12007MR266896
- [Mo] F. Morain, Modular Curves, Class Invariants and Applications, preprint.
- [Sch1] R. Schertz, Die singulären Werte der Weberschen Funktionen f, f1, f2, γ2, γ3. J. Reine Angew. Math.286/287 (1976), 46-74. Zbl0335.12018
- [Sch2] R. Schertz, Zur Theorie der Ringklassenkörper über imaginär-quadratischen Zahlkörpern. J. Number Theory10 (1978), 70-82. Zbl0372.12013MR480431
- [Sch3] R. Schertz, Zur expliziten Berechnung von Ganzheitsbasen in Strahiklassenkörpern über einem imaginär-quadratischen Zahlkörper. J. Number Theory34 (1990), 41-53. Zbl0701.11059MR1039766
- [Sch4] R. Schertz, Construction of Ray Class Fields by Elliptic Units. J. Théor. Nombres Bordeaux9 (1997), 383-394. Zbl0902.11047MR1617405
- [Sh] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Iwanami Shoten and Princeton University Press, 1971. Zbl0221.10029MR314766
- [St] H. Stark, On the "Gap" in a Theorem of Heegner. J. Number Theory1 (1969),16-27. Zbl0198.37702MR241384
- [We] H. Weber, Lehrbuch der Algebra, Bd 3, 2. Aufl. Braunschweig, 1908; Neudruck, New York, 1962. Zbl39.0227.04

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