Explicit isogeny theorems for Drinfeld modules

Imin Chen and Yoonjin Lee

Vol. 263 (2013), No. 1, 87–116

DOI: 10.2140/pjm.2013.263.87

Abstract

Let F = 𝔽_q(T) and A = 𝔽_q[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 ℘ of K such that P_℘(ϕ)≠P_℘(ϕ′), where P_℘(∗) denotes the characteristic polynomial of Frobenius at ℘ 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.

Keywords

Drinfeld modules, Galois representations, Chebotarev density theorem, isogeny theorem, different, ramification

Mathematical Subject Classification 2010

Primary: 11G09

Secondary: 11R58

Milestones

Received: 20 December 2011

Accepted: 31 July 2012

Published: 14 May 2013