## 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

### Keywords:

elliptic curves; monogenic ring; Legendre function

Zbl 0708.11060
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