  In 1923 Schur considered the following problem. Let f(X) be a polynomial with
integer coefficients that induces a bijection on the residue fields Z/pZ for
infinitely many primes p. His conjecture, that such polynomials are
compositions of linear and Dickson polynomials, was proved by M. Fried in 1970.
Here we investigate the analogous question for rational functions, also we
allow the base field to be any number field. As a result, there are many more
rational functions for which the analogous property holds. Some infinite series
come from rational isogenies of elliptic curves and deformations. There are
several sporadic examples which do not fit in any of the series we obtain.
First we translate the arithmetic property to a question about finite
permutation groups, and classify those groups which fulfill the necessary
conditions. The proofs depend on the classification of the finite simple
groups. Then we use arithmetic arguments to either rule out many cases, or to
prove that the remaining cases indeed give rise to examples. This part is based
on Mazur's classical results about rational points on the modular curves X_0(p)
and X_1(p), results about Galois images in GL_2(p) coming from action of the
absolute Galois group of Q on p-torsion points of elliptic curves, the theory
of complex multiplication, and the techniques used in the inverse regular
Galois problem.
