Explicit isogeny theorems for drinfeld modules

Abstract

Let F = Fq (T) and A = Fq(T) Given two nonisogenous rank-r Drinfeld A-modules φ and φ' over K, where K is a finite extension of F, we obtain a partially explicit upper bound (dependent only on φ and φ') on the degree of primes p of K such that Ppφ ≠ Ppφ', where Pp(*) denotes the characteristic polynomial of Frobenius at p on a Tate module of *. The bounds are completely explicit in terms of the defining coefficients of φ and φ', except for one term, which can be made explicit in the case of r = 2. An ingredient in the proof of the partially explicit isogeny theorem for general rank is an explicit bound for the different divisor of torsion fields of Drinfeld modules, which detects primes of potentially good reduction. Our results are a Drinfeld module analogue of Serre's work (1981), but the results we obtain are unconditional because the generalized Riemann hypothesis holds for function fields. © 2013 Mathematical Sciences.