The rational function analogue of a question of Schur and exceptionality of permutation representations

In 1923 Schur considered the following problem. Let f ∈ Z[X] be a polynomial 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, and 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. The new infinite series come from rational isogenies or endomorphisms of elliptic curves. Besides them, there are finitely many sporadic examples which do not fit in any of the series we obtain. The Galois theoretic translation, based on Chebotarëv’s density theorem, leads to a certain property of permutation groups, called exceptionality. One can reduce to primitive exceptional groups. While it is impossible to explicitly describe all primitive exceptional permutation groups, we provide certain reduction results, and obtain a classification in the almost simple case. The fact that these permutation groups arise as monodromy groups of covers of Riemann spheres f : P → P, where f is the rational function we investigate, provides genus 0 systems. These are generating systems of permutation groups with a certain combinatorial property. This condition, combined with the classification and reduction results of exceptional permutation groups, eventually gives a precise geometric classification of possible candidates of rational functions which satisfy the arithmetic property from above. Up to this point, we make frequent use of the classification of the finite simple groups. We do not directly obtain information about the number fields over which these examples exist, or if they exist at all. However, except for finitely many cases, these remaining candidates are connected to isogenies Partially supported by the NSF and NATO. Supported by the DFG. Partially supported by the NATO.