Local Isogeny Theorem for Drinfeld Modules with Nonintegral Invariants

Abstract

An isogeny theorem for the Drinfeld modules of rank 2 over a local field analogous to that of elliptic curves is proved. 0. INTRODUCTION Let k be a global function field over a finite constant field Fq. Drinfeld introduced the notion of elliptic modules, which are now known as Drinfeld modules, on k in analogy with classical elliptic curves. Hayes also studied this independently to generate certain class fields of k. Drinfeld modules of rank 2 have many interesting properties analogous to those of elliptic curves. We fix k to be the rational function field Fq(T). In [1] we introduced the Tate parametrization of Drinfeld modules of rank 2 with nonintegral invariants over a complete field. In this article we use the description of division points of Tate-Drinfeld modules and the methods in [6, 7] to get an isomorphism theorem for Drinfeld modules over a field with some restrictions on t and t'. In other words, there exist a and b in A = Fqq[T] such that Pa (t1) Pb (t1) is integral. This restriction does not appear in the classical case because a/fl is a unit if the valuations of a and fi are equal. From now on Drinfeld modules always mean Drinfeld modules of rank 2 defined on A = Fq[T]. 1. TATE-DRINFELD MODULES In this section we give a quick review of Tate-Drinfeld modules, which are the function field analogues of Tate elliptic curves [1]. Let k = Fq(T) and koo = Fq((T)), and let C be the completion of the algebraic closure of kok. Let IT be an element of C associated to the Carlitz module PT = TX+ Xq. Any rank 2 Drinfeld module q over C on A = Fq[T] is completely determined by (_T = TX + rl-qgXq + jt1-q2'AXq. Received by the editors April 29, 1991 and, in revised form, January 16, 1992. 1991 Mathematics Subject Classification. Primary 1 1G09, I 1R58. Partially supported by KOSEF Research Grant 91-08-00-07.