  Let $S$ be a rational fraction and let $f$ be a polynomial over a finite
field. Consider the transform $T(f)=\operatorname{numerator}(f(S))$. 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 $f$.
