Rank-One Drinfeld Modules on Elliptic Curves

Abstract

The sgn-normalized rank-one Drinfeld modules 0 associated with all elliptic curves E over Fq for 4 < q < 13 are computed in explicit form. (Such 0 for q < 4 were computed previously.) These computations verify a conjecture of Dorman on the norm of j(q) = aq+l and also suggest some interesting new properties of 0 . We prove Dorman's conjecture in the ramified case. We also prove the formula deg N(a) = q(hk 1 + q), where N(a) is the norm of a and hk is the class number of k = Fq (E). We describe a remarkable conjectural property of the trace of a in even characteristic that holds in all the examples. In his recent paper [1] on the factorization of norms of j-invariants of ranktwo Drinfeld modules with complex multiplications, D. Dorman conjectured that such norms are monic elements of Fq[x]. Dorman computed the jinvariants in several examples and found his conjecture valid in all of them. For this purpose, he used the rank-one examples from [6]. The first extensive computational test of the conjecture was carried out by one of us, Dummit, who computed the rank-one Drinfeld modules associated with all elliptic curves over Fq with q < 13 (see ?3 below). The results of these computations may be found on the microfiche card included at the end of this issue. Inspired by these computations, one of us, Hayes, proved Dorman's conjecture when the infinite place is ramified (see ?2 below). In ?3, we describe an algorithm for computing the rank-one Drinfeld modules associated with any hyperelliptic curve over Fqq, and we prove some basic attributes of the algorithm. In ?4, we prove formulas for the degrees of the norm and trace of the j-invariants of Drinfeld modules associated with elliptic curves. These formulas were first observed computationally. They suggest a number of interesting questions for elliptic curves with complex multiplications in characteristic zero. In ?5, we state some conjectures about the remarkable form of the trace term in characteristic two. These conjectures are supported by all our computations.