EXPLICIT SURJECTIVITY RESULTS FOR DRINFELD MODULES OF RANK 2

Abstract

Let K = F-q (T) and A = F-q [T]. Suppose that phi is a Drinfeld A module of rank 2 over K which does not have complex multiplication. We obtain an explicit upper bound (dependent on phi) on the degree of primes} of K such that the image of the Galois representation on the} - torsion points of phi is not surjective, in the case of q odd. Our results are a Drinfeld module analogue of Serre's explicit large image results for the Galois representations on p - torsion points of elliptic curves (Serre, Proprietes galoisiennes des points d'ordre fi ni des courbes elliptiques, Invent. Math. 15 (1972), 259{331; Serre, Quelques applications du theoreme de densite de Chebotarev, Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 323{401.) and are unconditional because the generalized Riemann hypothesis for function fi elds holds. An explicit isogeny theorem for Drinfeld A - modules of rank 2 over K is also proven.