The A-module structure induced by a Drinfeld A-module of rank 2 over a finite field

Abstract

Let F q be a finite field and let L / F q be a finite extension. Let F be the Frobenius of L ( F : x ↦ x # L ) and let ( P ) be the F [ T ] -characteristic of F. Let m be the degree of the extension L / F q [ T ] / ( P ) . There exists then c ∈ F q [ T ] and μ ∈ F q such that the characteristic polynomial P F of F is equal to P F ( X ) = X 2 − c X + μ P m . Our main result is an analogue of Deuring's Theorem on elliptic curves: let M = F q [ T ] ( i 1 ) ⊕ F q [ T ] ( i 2 ) , where i 1 and i 2 are two polynomials of F q [ T ] such that i 2 | i 1 and i 2 | ( c − 2 ) , there exists an ordinary Drinfeld F q [ T ] -module Φ of rank 2 over L such that the structure of the finite F q [ T ] -module L Φ induced by Φ over L is isomorphic to M. To cite this article: M.-S. Mohamed-Ahmed, C. R. Acad. Sci. Paris, Ser. I 346 (2008).