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Legendre model of an elliptic curve with complex multiplication and monogeneity of rings of integers. II. (Modèle de Legendre d’une courbe elliptique à multiplication complexe et monogénéité d’anneaux d’entiers. II.) (French) Zbl 0708.11061

[For part I, cf. the review Zbl 0708.11060 above.]
Let \(K\) be a number field, \({\mathbb{Z}}_ k\) its ring of integers. If \(L/K\) is a finite algebraic extension of number fields, \({\mathbb{Z}}_ L\) is called \({\mathbb{Z}}_ k\)-monogene if \({\mathbb{Z}}_ L={\mathbb{Z}}_ k[\theta]\). Following recent results, one is tempted to conjecture:
Let \(K\) be an imaginary quadratic field, \(H_ K\) its field of Hilbert classes, \(F\) an ideal of \({\mathbb{Z}}_ K\). Then the integral ring of \(K^{(F)}\) is monogenic over the integral ring of \(H_ K.\)
All partial results involve the study of the elliptic functions evaluated in divisors on the elliptic curve \(E=C/{\mathbb{Z}}_ K\). The conjecture is proven e.g. for \(F\) prime to 2, 2 decomposable in \(K/{\mathbb{Q}}\) or for \(K={\mathbb{Q}}(\sqrt{-d})\) \((d\equiv 2 \bmod 4)\). Two more results are proven in this paper:
Theorem 1: Let \(K\) be an imaginary quadratic field in which 2 ramifies. Let \(F\) be an ideal of \({\mathbb{Z}}_ K\), prime to 2. Then the integral ring of \(K^{(F)}\) is monogenic over the integral ring of \(H_ K.\)
Theorem 2: Let \(K\) be an imaginary quadratic field in which 2 is inert. Let \(F\) be an ideal in \({\mathbb{Z}}_ k\), prime to 2, not being a power of a prime ideal. Then the integral ring of \(k^{(F)}\) is monogenic over the integral ring of \(H_ K.\)
There is good evidence that the cases for which the conjecture remains hard to prove are the cases where the ideals 2 and 3 are inert in \(K/{\mathbb{Q}}\) and the conductor is a prime power of a prime ideal.

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R20 Other abelian and metabelian extensions
11G15 Complex multiplication and moduli of abelian varieties
11R37 Class field theory

Citations:

Zbl 0708.11060
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