Finite field elements of high order arising from modular curves

TL;DR

This paper presents elementary recursive methods to explicitly construct high-order elements in finite fields using modular tower formulas, applicable across all characteristics, and compares these results with recent work by Voloch.

Contribution

It introduces new elementary recursive constructions for high-order finite field elements based on Elkies' modular recursions, without requiring advanced modular curve knowledge.

Findings

High order elements are constructed explicitly in all characteristics.

The methods are elementary and do not rely on deep modular curve theory.

Results are compared and refined relative to Voloch's recent findings.

Abstract

In this paper, we recursively construct explicit elements of provably high order in finite fields. We do this using the recursive formulas developed by Elkies to describe explicit modular towers. In particular, we give two explicit constructions based on two examples of his formulas and demonstrate that the resulting elements have high order. Between the two constructions, we are able to generate high order elements in every characteristic. Despite the use of the modular recursions of Elkies, our methods are quite elementary and require no knowledge of modular curves. We compare our results to a recent result of Voloch. In order to do this, we state and prove a slightly more refined version of a special case of his result.