Finite field elements of high order arising from modular curves

Abstract

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.

Subgroups generated by rational functions in finite fields

“…We now assume that r > 8m 3 as otherwise there is nothing to prove. Therefore, there is λ ∈ G \ Λ such that (6) ψ(x) ≡ λψ(y) (mod p)…”

“…Thus the question of estimating N f (G, G) remains open. On the other hand, a number of results about points on curves and algebraic varieties with coordinates from small subgroups, in particular, in relation to the Poonen Conjecture, have been given in [6,8,9,10,17,18,20,21].…”

Elements of large order on varieties over prime finite fields

“…Another possible relaxation of the original problem is to construct elements x in a given field F q or in its extension of large order ord x, see [1,6,7,8,9,15,19,20,25,26] and references therein. We recall that for a non-zero element x ∈ F q in the algebraic closure F q of F q the order ord x is the smallest positive integer t with x t = 1.…”

Elements of High Order on Finite Fields From Elliptic Curves

“…Wiedemann's construction is an example of an iterative construction of finite fields. Other examples can be found in [2] and the references therein and also in [1], where the multiplicative order of elements thus obtained is estimated. When considering points of order s n on an elliptic curve for fixed s and varying n, other examples of iterative constructions can be found to which we can apply the theorems of this paper.…”