1. INTRODUCTION 3 (b) If g = 1, then n = p or p2 for an odd prime p. Set N = Cp in the former case, N = Cp x Cp in the latter case. Then one of the following holds. (i) n 5, G = T V x G2, T = (2, 2, 2, 2), Kmin = Q i f and only if n = p2 orne {5, 7,11,13,17,19,37,43,67,163} or (ii) n = 1 (mod 6), G = T V x G6, T = (2,3,6), K m m = Q ifn = p2, and Kmin = Q(v/II3) i/ n = p or (hi) n = 1 (mod 6), G = iV x G3, T = (3,3,3), K m m = Qifn = p2y and Kmin = Q(\ / Z 3) ifn = p or (iv) n = 1 (mod 4), G = T V x G4, T = (2,4,4), K m m = Q if n = p2, and Kmin = Q(v /r T) ifn = p. The functions f arise from isogenics or endomorphisms of elliptic curves as described in Section 6. (c) If g 1, then (i) n = ll 2 , G = G2! x GL2(3), T = (2,3,8) mt/i g = 122, K m i n = Q ( A / = 2 ) or (ii) n = 52, G = G2 x S3, T = (2,3,10) wtffc p = 6, Kmin = Q or (iii) n = 52, G = G2 x i i witft iif a Sylow 2-subgroup of the subgroup of index 2 in GL2(5), T = (2, 2, 2,4) wtfft # = 51, ifmin = Q(V=rT) or (iv) n = 52, G = G2 x (G6 x G2), T = (2, 2,2,3) mt/i ? = 26, Kmin = Q or (v) n = 32, G = G| x (C4 x G2), T = (2,2,2,4) wztfi g = 10, or T = (2, 2, 2, 2, 2) with g = 19, and Kmin = Q in both cases or (vi) n = 28, G = PSL2(8), T - (2,3, 7) w#ft # = 7, or T = (2,3,9) with g = 15, or T = (2, 2, 2,3) id£/i # = 43, and i^ m j n — Q in all three cases or (vii) n = 45, G = PSL2(9), T = (2,4, 5) with g = 10, and Kmin = Q. The main part of the proof is group theoretic. Arithmetic exceptionality trans- lates to a property of finite permutation groups, see Section 2. Let A and G be finite groups acting transitively on ft with G a normal subgroup of A. Then the triple (A, G, Q) is said to be exceptional if the only common orbit of A and G act- ing on ft x Q is the diagonal. We do not completely classify such triples - we may return to this in a future paper. Instead, we use further properties coming from the arithmetic context. Namely we get a group B with G B A with B/G cyclic such that (JB, G, Q) is still exceptional. We say that the pair (A, G) is arithmetically exceptional if such a group B exists. The notion of exceptionality has arisen in different context. It typically comes in the following situation: Let K be a field, and / : X — Y be a separable branched cover of projective curves which are absolutely irreducible over K. Let A and G denote the arithmetic and geometric monodromy group of this cover, respectively. Then (A, G) is exceptional if and only if the diagonal is the only absolutely ir- reducible component of the fiber product X x^ X. The arithmetic relevance lies in the well-known fact that an irreducible, but not absolutely irreducible curve, has only finitely many rational points. Thus if the fiber product has the excep- tionality property, then there are only finitely many P,Q 6 X(K) with P =^ Q and 4(P) = 0(Q), so (f) is essentially injective on K-rational points. If K has a procyclic Galois group, for instance if K is a finite field, then A/G is cyclic, and exceptionality automatically implies arithmetic exceptionality.

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