Weber's class invariants revisited

@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 -