A lower bound for the canonical height associated to a Drinfeld module

TL;DR

This paper establishes a lower bound for the canonical height of points on Drinfeld modules over function fields, linking it to invariants like the j-invariant and minimal discriminant, with implications for specializations.

Contribution

It proves a lower bound for the canonical height on Drinfeld modules depending only on bad reduction places and invariants, extending height theory in function fields.

Findings

Existence of constants psilon>0 and C depending on bad reduction places.

Lower bound for canonical height in terms of j-invariant and discriminant.

Implications for specializations of Drinfeld modules.

Abstract

Denis associated to each Drinfeld module M over a global function function field L a canonical height function, which plays a role analogous to that of the Neron-Tate height in the context of elliptic curves. We prove that there exist constants \epsilon>0 and C, depending only on the number of places at which M has bad reduction, such that either x in M is a torsion point of bounded order, or else the canonical height of x is bound below by \epsilon max{h(j_M), deg(D_M)}, where j_M is a certain invariant of the isomorphism class of M, and D_M is the minimal discriminant of M. As an application, we make some observations about specializations of one-parameter families of Drinfeld modules.