Pour une définition commune des courbes elliptiques et modules de Drinfeld

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 élémentaires", 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 élémentaires".