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!.


11R37 Class field theory
11F03 Modular and automorphic functions


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