Iterative constructions of irreducible polynomials from isogenies

Abstract

Let $S$ be a rational fraction and let $f$ be a polynomial over a finite field. Consider the transform $T(f)=\mathrm{numerator}(f(S))$. In certain cases, the polynomials $f$, $T(f)$, $T(T(f))\dots$ are all irreducible. For instance, in odd characteristic, this is the case for the rational fraction $S=(x^2+1)/(2x)$, known as the $R$-transform, and for a positive density of all irreducible polynomials $f$. We interpret these transforms in terms of isogenies of elliptic curves. Using complex multiplication theory, we devise algorithms to generate a large number of other rational fractions $S$, each of which yields infinite families of irreducible polynomials for a positive density of starting irreducible polynomials $f$.