Non-totally real number fields and toroidal groups

Dioguardi Burgio, Alessandro; Falcone, Giovanni; Galici, Mario

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 1, pp. 339-359

In this paper we study the relationship between non-totally real number fields $K$ and toroidal groups $𝒯$, as well as meromorphic periodic functions, exploiting a representation of $𝒯$ as the generalized Jacobian ${𝔍}_L\left(𝒞\right)$ of a suitable elliptic curve $𝒞$. We consider in detail the cubic and quartic cases.

In these cases, we write down the relations between the minimal polynomial of a suitable primitive element of $K$ and the parameters defining the generalized Jacobian ${𝔍}_L\left(𝒞\right)$ corresponding to the toroidal group associated with the ring of integers. Furthermore, for such a toroidal group we explicitly show the analytic and rational representations of its ring of endomorphisms, the former giving in turn a new (complex) representation of the ring of integers of $K$.

Moreover, for the cubic case, we give an explicit description of the $m$-torsion of $𝒯$ in the geometric correspondence of $𝒯$ with ${𝔍}_L\left(𝒞\right)$, as image of a fractional ideal of $K$.

Dans cet article, nous étudions la relation entre les corps de nombres non totalement réels $K$ et les groupes toroïdaux $𝒯$, ainsi que les fonctions périodiques méromorphes, en exploitant une représentation de $𝒯$ en termes de la jacobienne généralisée ${𝔍}_L\left(𝒞\right)$ d’une courbe elliptique appropriée $𝒞$. Nous considérons en détail les cas cubique et quartique.

Dans ces cas, nous écrivons les relations entre le polynôme minimal d’un élément primitif convenable de $K$ et les paramètres définissant la jacobienne généralisée ${𝔍}_L\left(𝒞\right)$ correspondant au groupe toroïdal associé à l’anneau des entiers. En outre, pour un tel groupe toroïdal, nous décrivons explicitement les représentations analytique et rationnelle de son anneau d’endomorphismes, le premier donnant une nouvelle représentation (complexe) de l’anneau des entiers de $K$.

De plus, dans le cas cubique, nous donnons une description explicite de la $m$-torsion de $𝒯$ en termes de la correspondance géométrique entre $𝒯$ et ${𝔍}_L\left(𝒞\right)$ comme l’image d’un idéal fractionnaire de $K$.

Classification : 11R16, 32G20, 11G15, 32M05, 22E10, 57T15
Keywords: Toroidal groups, non-totally real number fields, generalized Jacobians

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