On Some Subrings of Ore Polynomials Connected with Finite Drinfeld Modules

Abstract

It is well known that there exists a deep analogy between global function fields and algebraic number fields. In that analogy, Drinfeld modules play the role of elliptic curves or abelian varieties, producing arithmetical Ž objects such as modular schemes, some kind of field extensions, rings of . endomorphisms, and so on of positive characteristic, similar to that of Ž characteristic zero and note that, even if it is defined over a field of positive characteristic, the objects produced by abelian varieties are of . characteristic zero . In this paper, we study arithmetical properties of endomorphism rings of Drinfeld modules defined over finite fields, following in that way E.-U. w x Gekeler GE . Our main results are the following: we prove that the ring of endomorphisms of such a Drinfeld module, which is not necessarily a Ž Ž .. maximal order see 4.7 , is maximal at some place which plays an important role in our situation, and we give a simple algebraic characterization of the rings of endomorphisms of finite Drinfeld modules. More precisely, let K be a global function field, let ` be a place of K, let A be the ring of elements of K which are regular outside of `, and let F be the q 4 field of constants of K. Let L be a finite extension of F and let L t be q Ž Ž .. the ring of Ore polynomials associated to the extension LrF see 1.1 . q Then a Drinfeld A-module F over L is a non-trivial embedding of A into 4 Ž . L t see Definition 1.4 . It follows that the ring of L-endomorphisms of F 4 Ž . is a subring of L t . We prove Theorem 3.4 that endomorphism rings of finite Drinfeld modules, and their centers too, have a local property of maximality. With the help of this result, we are able to prove the following