Linear equations on Drinfeld modules

TL;DR

This paper characterizes linear relations among points on Drinfeld modules over function fields using explicit linear equations, providing bounds on generators of these relations, analogous to results for elliptic curves over number fields.

Contribution

It introduces an explicit system of linear equations to describe relations among points on Drinfeld modules and establishes bounds on the size of their generators.

Findings

Linear relations are characterized by solutions to explicit linear equations over _q[t].

An explicit upper bound for the size of generators of linear relations is provided.

The results are analogous to Masser's theorem for elliptic curves over number fields.

Abstract

Let $L$ be a finite extension of the rational function field over a finite field $\mathbb{F}_q$ and $E$ be a Drinfeld module defined over $L$. Given finitely many elements in $E(L)$, this paper aims to prove that linear relations among these points can be characterized by solutions of an explicitly constructed system of homogeneous linear equations over $\mathbb{F}_q[t]$. As a consequence, we show that there is an explicit upper bound for the size of the generators of linear relations among these points. This result can be regarded as an analogue of a theorem of Masser for finitely many $K$-rational points on an elliptic curve defined over a number field $K$.