English

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

Number Theory 2023-06-26 v1 Algebraic Geometry

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 AA-modules for a finitely generated Dedekind ring AA, called "modules \'el\'ementaires", which naturally generalize Drinfeld modules, forms of the multiplicative group, and elliptic curves over a field (when AA has the corresponding form). The objective of this text is the classification of these "modules \'el\'ementaires".

Keywords

Cite

@article{arxiv.2306.13160,
  title  = {Pour une d\'efinition commune des courbes elliptiques et modules de Drinfeld},
  author = {Quentin Gazda and Damien Junger},
  journal= {arXiv preprint arXiv:2306.13160},
  year   = {2023}
}

Comments

21 pages, in French