Explicit description of isogeny and isomorphism classes of Drinfeld modules over finite field

TL;DR

This paper provides an explicit classification of isogeny and isomorphism classes of higher-rank Drinfeld modules over finite fields, extending known results for rank 2 and introducing new invariants.

Contribution

It explicitly describes Weil polynomials for higher-rank Drinfeld modules and introduces fine isomorphy invariants that classify these modules over finite fields.

Findings

Explicit Weil polynomial descriptions for rank r > 2 Drinfeld modules.

Introduction of fine isomorphy invariants for classification.

Complete determination of isomorphism classes using invariants and J-invariants.

Abstract

When travelling from the number fields theory to the function fields theory, one cannot miss the deep analogy between rank 1 Drinfeld modules and the group of root of unity and the analogy between rank 2 Drinfeld modules and elliptic curves. But so far, there is no known structure in number fields theory that is analogous to the Drinfeld modules of higher rank r > 2. In this paper we investigate the classes of those Drinfeld modules of higher rank r > 2. We describe explicitly the Weil polynomials defining the isogeny classes of rank r Drinfeld modules for any rank r > 2. our explicit description of the Weil polynomials depends heavily on Yu's classification of isogeny classes (analogue of Honda-Tate at abelian varieties). Actually Yu has also explicitly did that work for r = 2. To complete the classification, we define the new notion of fine isomorphy invariants for any rank r Drinfeld module and we prove that the fine isomorphy invariants together with J-invariants completely determine the L-isomorphism classes of rank r Drinfeld modules defined over the finite field L.