Iterative constructions of irreducible polynomials from isogenies

Let S be a rational fraction and let f be a polynomial over a finite field. Consider the transform T(f)=numerator(f(S)). In certain cases, the polynomials f, T(f), T(T(f))… are all irreducible. For instance, in odd characteristic, this is the case for the rational fraction S=(x2+1)/(2x), known as the R-transform, and for a positive density of 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 rational fractions S, each of which yields infinite families of irreducible polynomials for a positive density of starting irreducible polynomials f.