Let K = F q (T) and A = F q [T]. Suppose that φ is a Drinfeld A-module of rank 2 over K which does not have complex multiplication. We obtain an explicit upper bound (dependent on φ) on the degree of primes p of K such that the image of the Galois representation on the p-torsion points of φ 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, Propriétés galoisiennes des points d'ordre fini des courbes elliptiques, Invent. Math. 15 (1972), 259-331; Serre, Quelques applications du théorème de densité de Chebotarev, Inst. Hautes Etudes Sci. Publ. Math. 54 (1981), 323-401.) and are unconditional because the generalized Riemann hypothesis for function fields holds. An explicit isogeny theorem for Drinfeld A-modules of rank 2 over K is also proven.
|Number of pages||29|
|Journal||Nagoya Mathematical Journal|
|State||Published - 1 Jun 2019|