Pour une d\'efinition commune des courbes elliptiques et modules de Drinfeld

TL;DR

This paper aims to unify the definitions of elliptic curves, Drinfeld modules, and related algebraic structures over Dedekind rings by introducing and classifying a new class of algebraic modules called 'modules élémentaires'.

Contribution

It introduces 'modules élémentaires', a new general class of algebraic A-modules that unify Drinfeld modules, elliptic curves, and multiplicative groups, and provides their classification.

Findings

Introduction of 'modules élémentaires' as a unifying concept.

Generalization of Drinfeld modules, elliptic curves, and multiplicative groups.

Classification results for these modules.

Abstract

It is often stated that the Carlitz module is to the ring of univariate polynomials over a finite field what the multiplicative group is to the ring of integers. This analogy extends to the "rank 2" case, where Drinfeld modules play a role similar to that of elliptic curves. This work grew out with the will of finding a common definition for these objects, depending only on the ring of coefficients, and thus elevating this analogy to a common theory. To that end, we introduce a class of algebraic $A$-modules for a finitely generated Dedekind ring $A$, called "modules \'el\'ementaires", which naturally generalize Drinfeld modules, forms of the multiplicative group, and elliptic curves over a field (when $A$ has the corresponding form). The objective of this text is the classification of these "modules \'el\'ementaires".