2 1. INTRODUCTION deg(R) and deg(S). Note that deg(/) is also the degree of the field extension K(X)\K(f(X)). Let t be a transcendental over if, and denote by L a splitting field of R{X) tS(X) over K{t). Let K be the algebraic closure of K in L. Set A = Gal(L|if (£)), G Gal(L\K(t)). Throughout the paper, A and G are called the arithmetic and geometric monodromy group of / , respectively. Note that G is a normal subgroup of A, with A/G = Gal(if |if). Furthermore, A and G act transitively on the roots of R(X)-tS(X). Let K be an algebraic closure of if, pi,... , p r be the places of K(t) which ramify in KL, and let e$ be the ramification index of ?Pi|pi, where ^ is a place of KL lying above p^. The unordered tuple (ei,..., er) is a natural invariant associated to / , we call it the signature of / . Denote by ifmin a minimal (with respect to inclusion) field, such that / exists over K = Km{n with given geometric monodromy group G and given signature. Of course the field ifmin need not be unique, but it turns out to be unique in most cases in the theorem below. If we fix G and the signature, then there are usually several possibilities for A, and one can also ask for a minimal field of definition if one fixes A too. To keep the theorem below reasonably short, we have not included that. Results can be found in the sections where the corresponding cases are dealt with. A rough distinction between the various classes of indecomposable arithmeti- cally exceptional functions is the genus g of L. Note that (see also Section 2, equation (2.1)) (1.1) 2(\G\-l+g) = \G\^2(l-l/el). The cases g 0 and g = 1 belong to certain well-understood series, to be investigated in Sections 5 and 6, respectively. If g 1, then / belongs to one of finitely many possibilities (finite in terms of G and signature). These sporadic cases are investigated in Section 7. DEFINITION 1.3. Let if be a field, and f(X) e K(X) be a rational function. Then we say that / is equivalent to g(X) G K(X), if there are linear fractional functions £±J2 G K{X) with f(X) = £i(g(£2(X))). (Of course, the arithmetic and geometric monodromy group is preserved under this equivalence, and so is the property of being arithmetically exceptional.) The rough classification of indecomposable arithmetically exceptional rational functions is given in the following theorem. See Section 3.1 for notation, and Sections 5, 6, and 7 for more details. THEOREM 1.4. Let K be a number field, f £ K(X) be an indecomposable arithmetically exceptional function of degree n 1, G the geometric monodromy group of f, T (ei,..., er) the signature, and Kmin be a minimal field of definition as defined above. Let g be as defined in equation (1.1). (a) If g = 0, then either (i) n 3 is a prime, G = Cn, T = (n,ri), Kmin = Q or (ii) n 5 is a prime, G = Cn x C2, T = (2, 2, n), Kmin = Q or (hi) n = 4, G = C2 x C2, T = (2,2,2), Kmin = Q. In case (i), f is equivalent to Xn or to a Redei function in case (ii), f is equivalent to a Dickson polynomial (see Section 5).
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