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Class invariants by Shimura’s reciprocity law. (English) Zbl 0957.11048
Let \(K\) be an imaginary quadratic number field of discriminant \(d<-4\) with ring of integers \({\mathcal O}=\mathbb{Z}[\theta]\) and minimal polynomial of \(\theta\), \(f(X)=X^2 +BX+C\). According to the theory of complex multiplication, the absolute invariant \(j({\mathcal O})=j(\theta)\) generates the Hilbert class field \(H\) of \(K\). For a modular function \(h\) of level \(N\), the value \(h(\theta)\) is called a class invariant whenever \(h(\theta)\) and \(j(\theta)\) generate the same field over \(K\). Class invariants are useful because their minimal polynomial has, in general much smaller coefficients than the minimal polynomial of \(j({\mathcal O})\).
The author of the paper under review had the nice idea to use the Shimura reciprocity law in order to produce the following criterion:
If \(h\) is a modular function of level \(N\) and \(\mathbb{Q}(j) \subseteq \mathbb{Q} (h)\), then \(h(\theta)\) is a class invariant if and only if the set of matrices \[ W_{N,\theta}= \left\{\left(\begin{matrix} t-Bs & -Cs\\ s & t\end{matrix} \right)\in GL_2 (\mathbb{Z}/N \mathbb{Z})\mid t,s\in \mathbb{Z}/N \mathbb{Z}\right\} \] acts trivially on \(h\).
By using this result she gives, in an easy way results proven before by H. Weber, B. Birch or R. Schertz, concerning class invariants, of the Weber modular functions \(\gamma_2,\gamma_3,f,f_1,f_2\) as well as of Weber’s resolvents of \(\omega_0\) and \(\omega_3\) of degree 5.
After that she gives a modification of Shimura’s reciprocity law which has been used to give the action of the Galois group \(\text{Gal}(H/K)\) on class invariants and proves a conjectural formula of F. Morain [Algorithmic Number Theory, Lect. Notes Comput. Sci. 1423, 111-130 (1998; Zbl 0908.11061)] and another one of N. Yui and D. Zagier [Math. Comput. 66, 1645-1662 (1997; Zbl 0892.11022)] concerning conjugates of class invariants arising from some classical functions.
An extensive list containing the polynomials for the class invariants arising from the functions considered in this paper has been constructed but is not appended in this paper.
A really nice work!.

MSC:
11R37 Class field theory
11F03 Modular and automorphic functions
Software:
ECPP
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References:
[1] Birch, B., Weber’s class invariants. Mathematika16 (1969), pp. 283-294. · Zbl 0226.12005
[2] Lang, S., Elliptic functions. 2nd edition, Springer GTM112, 1987. · Zbl 0615.14018
[3] Morain, F., Primality Proving Using Elliptic Curves: An Update. Algorithmic Number Theory, (1998), pp. 111-130. · Zbl 0908.11061
[4] Schertz, R., Die singulären Werte der Weberschen Funktionen f, f1, f2, γ2, γ3. J. Reine Angew. Math.286/287 (1976), pp. 46-74. · Zbl 0335.12018
[5] Shimura, G., Introduction to the Arithmetic Theory of Automorphic Functions. Iwanami Shoten and Princeton University Press, 1971. · Zbl 0221.10029
[6] Shimura, G., Complex Multiplication, Modular functions of One Variable I. (1973), pp. 39-56. · Zbl 0268.10015
[7] Weber, H., Lehrbuch der Algebra. Band III: Elliptische Funktionen und algebraische Zahlen. 2nd edition, Braunschweig, 1908. (Reprint by Chelsea, New York, 1961.)
[8] Yui, N. and Zagier, D., On the singular values of Weber modular functions. Math. Comp.66 (1997), no 220, pp. 1645-1662. · Zbl 0892.11022
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