On finite Drinfeld modules

Abstract

In [5], Deuring determined the possible isomorphism types of endomorphism rings of elliptic curves, notably for those curves that are defined over a finite field. His results were later generalized to abelian varieties of higher rank by Tate [17] and Honda [13]. Now in the fundamental paper [6], Drinfeld transports the modular theory of elliptic curves to the function field case. He found the kind of diophantine objects (called by him “elliptic modules”) that over global function fields play the role of elliptic curves in number theory. By his theory, he was able to prove analogues of the theorem of KroneckerWeber, the main theorem of complex multiplication, and parts of the Langlands conjectures for GL(2) over function fields. Actually, in the course of the last few years, the theory of Drinfeld modules has shown to be the key tool in the arithmetic of function fields over finite fields. This comes from the fact that Drinfeld modules lead to moduli problems that are related to GL(r) (r arbitrary), and to Galois representations in local fields of positive characteristic, which one needs in order to describe the absolute Galois group of a global function field. In this paper, we treat Deuring’s problem of endomorphism rings in the Drinfeld module setting, i.e., we study Drinfeld modules that are defined over a finite field, and their endomorphism rings. Let (K, co) be a pair consisting of a function field K in one variable over a finite field, and a place co of K. Let further A be the ring of elements of K with poles at most at co, and p a prime of A with finite residue field IF, = A/p. As for elliptic curves, the classification up to isogeny of Drinfeld modules 4 over extensions of IF, is given by the isomorphism type of End(d)@,, K (Theorem 3.5). This ring turns out to be a certain division algebra central over the subfield E generated over K by the Frobenius endomorphism F of 4 (Theorem 2.9). (These two results are stated in [7, Prop. 2.11 in a somewhat disguised