CHAPTER 1 Introduction In 1923 Schur [52] posed the following question. Let f(X) G Z[X] be a polyno- mial which induces a bijection on Z/pZ for infinitely many primes p. He proved that if / has prime degree, then / is, up to linear changes over Q, a cyclic polynomial Xk, or a Chebychev polynomial Tk(X) (defined implicitly by Tk(Z + l/Z) = Zk + l/Zk). He conjectured that without the degree assumption, the polynomial is a composi- tion of such polynomials. This was proved almost 50 years later by Fried [11]. The obvious extension of Schur's original question to rational functions over number fields then leads to the notion of arithmetical exceptionality as follows. DEFINITION 1.1. Let K be a number field, and / G K(X) be a non-constant rational function. Write / as a quotient of two relatively prime polynomials in if [X]. For a place p of K denote by Kp the residue field. Except for finitely many places p, we can apply p to the coefficients of / to obtain fp G KP(X). Regard fp as a function from the projective line F1(KP) = Kp U {oo} to itself in the usual way. We say that / is arithmetically exceptional if there are infinitely many places p of K such that fp is bijective on ¥1(KP). A slight extension (see [61]) of Fried's above result is the following. Here Dm(a,X) denotes the Dickson polynomial of degree m belonging to a G K, which is most conveniently defined by Dm(a1 Z + a/Z) = Z m + (a/Z)m. THEOREM 1.2 (Fried). Let K be a number field, and f G K[X] be an arith- metically exceptional polynomial. Then f is the composition of linear polynomials in K[X] and Dickson polynomials Dm(a,X). In this paper we generalize Schur's question to rational functions over number fields. Under the assumption that the degree is a prime, this has already been investigated by Fried [13], showing that there are non-polynomial examples over suitable number fields. In contrast to the polynomial case, the classification result for rational functions is quite complicated in terms of the associated monodromy groups as well as from the arithmetic point of view. It is immediate from the definition that if an arithmetically exceptional rational function is a composition of rational functions over K, then each of the composition factors is arithmetically exceptional. Thus we are interested in indecomposable arithmetically exceptional functions. Note however that the composition of arithmetically exceptional functions need not be arithmetically exceptional, see Theorem 1.6. In order to state the classification of arithmetically exceptional functions, we need to introduce some notation. Let K be a field of characteristic 0, and / G K{X) be a non-constant rational function. Write / = R/S with coprime polynomials R, S G -K"[-X]. Then the degree deg(/) of / is defined to be the maximum of l
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