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**Unités modulaires et monogénéité d’anneaux d’entiers. (Modular units and monogenity of rings of integers).**
*(French)*
Zbl 0714.11078

Sémin. Théor. Nombres, Paris 1986-87, Prog. Math. 75, 35-64 (1988).

[For the entire collection see Zbl 0653.00005.]

It is well known that there is a cyclic extension N of \({\mathbb{Q}}\) of prime degree \(\ell \geq 5\) of which the ring of integers is monogene over \({\mathbb{Z}}\) iff \(p=2\ell +1\) is prime and \(N={\mathbb{Q}}(\cos 2\pi /p)\). This result is generalised by Cougnard to the case where the base field is a quadratic imaginary field K. His result leads to the following conjecture: Let K be a quadratic imaginary field and F an integral ideal of \({\mathcal O}_ K\) (the ring of integers of K). Let H be the field of Hilbert classes of K and N the field of “classes de rayon”, with conductor F. Then \({\mathcal O}_ N\) is monogene over \({\mathcal O}_ H\). - This is known to be true if one of the following conditions is fulfilled. (1) 2 is decomposed in K and F is odd. (2) 3 is decomposed in K and F is prime to 6. (3) \(K={\mathbb{Q}}(\sqrt{-1})\) and F is odd. (4) \(K={\mathbb{Q}}(\sqrt{-3})\) and F is prime to 6.

In this paper, the authors consider the technique of modular functions, instead of elliptic functions, which they employed in their previous work. This leads to a new proof for the cases (1) and (2) and to a new class of field satisfying the monogenity condition, i.e. the fields in which 2 and 3 are ramified.

It is well known that there is a cyclic extension N of \({\mathbb{Q}}\) of prime degree \(\ell \geq 5\) of which the ring of integers is monogene over \({\mathbb{Z}}\) iff \(p=2\ell +1\) is prime and \(N={\mathbb{Q}}(\cos 2\pi /p)\). This result is generalised by Cougnard to the case where the base field is a quadratic imaginary field K. His result leads to the following conjecture: Let K be a quadratic imaginary field and F an integral ideal of \({\mathcal O}_ K\) (the ring of integers of K). Let H be the field of Hilbert classes of K and N the field of “classes de rayon”, with conductor F. Then \({\mathcal O}_ N\) is monogene over \({\mathcal O}_ H\). - This is known to be true if one of the following conditions is fulfilled. (1) 2 is decomposed in K and F is odd. (2) 3 is decomposed in K and F is prime to 6. (3) \(K={\mathbb{Q}}(\sqrt{-1})\) and F is odd. (4) \(K={\mathbb{Q}}(\sqrt{-3})\) and F is prime to 6.

In this paper, the authors consider the technique of modular functions, instead of elliptic functions, which they employed in their previous work. This leads to a new proof for the cases (1) and (2) and to a new class of field satisfying the monogenity condition, i.e. the fields in which 2 and 3 are ramified.

Reviewer: G.Molenberghs

### MSC:

11R37 | Class field theory |

11R20 | Other abelian and metabelian extensions |

11R33 | Integral representations related to algebraic numbers; Galois module structure of rings of integers |

11F03 | Modular and automorphic functions |