Isogenies of Drinfeld modules over finite fields
Let \(A\) be the affine function ring of a smooth connected algebraic curve defined over a finite field, and with one point at infinity. As is well known, Drinfeld \(A\)-modules (notably those with rank \(d = 2)\) share many properties (moduli schemes, structure of torsion points, of endomorphism rings) with elliptic curves. Let now the finite field \(\mathbb{F}_q\) with \(q\) elements be equipped with a structure as an \(A\)-algebra. The main result of the paper is theorem 3, which states a bijection of the set of ``Weil numbers of rank \(d\)'' with the set of isogeny classes of Drinfeld \(A\)-modules of rank \(d\) over \(\mathbb{F}_q\). (N.b.: Theorem 3 had been stated in [\textit{V. G. Drinfeld}, Math. USSR, Sb. 31, 159-170 (1977); translation from Mat. Sb., Nov. Ser. 102(144), 182-194 (1977; Zbl 0363.20038)], but not rigorously proved.) Some more precise results (propositions 4-7 on the set of Weil numbers and the size of corresponding isogeny classes) are given in the case where the rank \(d\) equals two and \(A\) is a polynomial ring.
scientific article; zbMATH DE number 809114
scientific article