1. INTRODUCTION 5 (iii) a is even, M n L is the dihedral group of order pa — 1 and B/G is generated by a field-diagonal automorphism or (iv) p — 3, a is odd, M D L is dihedral of order pa + 1 and B = Aut(G). (c) L = Sz(2a) with a 1 odd, and M C\ L = 5f2 (2a/b) witfi 6 a prime not 5 and a ^ 6 or M D L zs t/ie normalizer of a Sylow 5 -subgroup (which is the normalizer of a torus) or (d) L = Re(3a) with a 1 odd and M D L = i?e(3a/6) wztfi 6 a divisor of a other than 3 or 7 or (e) L = U3(pa) andLHM = U3(pa/b) with b a prime not dividing p(p2 — 1) or (f) L = C/3(2a) it ^ft a 1 odd and MnL is the subgroup preserving a subspace decomposition into the direct sum of 3 orthogonal nonsingular 1-spaces or (g) L has Lie rank 2, L ^ Sp4(2y (= PSL2(9), a case covered in (b)). Then M fl L is a subfield group, the centralizer in L of a field automorphism of odd prime order r. Moreover, (i) r / p (with p the defining characteristic of L), (ii) if r = 3, then L is of type Sp4 with q even, and (iii) there are no Aut(L)-stable L-conjugacy classes of r-elements. REMARK. Lemma 3.16 describes the cases when (g) (iii) holds. In particular, if p and the type of L are fixed, then (g) (iii) holds for all but finitely many primes r. Together with Theorem 3.15 we obtain a precise existence result in the situation of (g)- If we now consider the primitive arithmetically exceptional groups which are not affine and add the genus 0 condition on G, then only two groups survive, accounting for the non-prime-power degrees 28 and 45 in Theorem 1.4. In the affine case, we first use the genus 0 condition (except for noting that exceptionality implies that G is not doubly transitive), and then check which ones give arithmetically exceptional configurations. We end up with six infinite series (and a few sporadics). Two of the series are well known and correspond to Redei functions and Dickson polynomials, respectively. The other four series have an interesting connection with elliptic curves, to be investigated in Section 6. Section 7 takes care of the sporadic cases. There we will also see that there is a big variety of arithmetically exceptional functions of degree 4 over number fields. This will have the following surprising consequence. THEOREM 1.6. Let K be an arbitrary number field. There are four arithmeti- cally exceptional rational functions of degree 4 over K, such that their composition is not arithmetically exceptional.

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