8 2. ARITHMETIC-GEOMETRIC PREPARATION PROOF. Without loss assume that p is a finite place, so K[t] is contained in the valuation ring. Let G\ be the integral closure of K[t] in L. Then D/I is the Galois group of (Oil*$)\(K[t]/p), see [53, Chapter I, §7, Prop. 20]. On the other hand, K embeds into Oi/*#, so D/I surjects naturally to A/G Qdl(K\K). Furthermore, if (j) G /, then u u^ G ^3 for all u G K, hence fi is trivial on K, so / D n G. D 2.5. Weak rigidity We use the weak rigidity criterion in order to prove existence of certain regular extensions E\K{t) with given ramification structure and geometric monodromy group. The main reference of the material in this section is [41] and [62]. DEFINITION 2.6. Let G b e a finite group, and H = (Ci,C2,... ,Cr) be an r- tuple of conjugacy classes of G. Consider the set of r-tuples (Ti, 02, 0) with o~i G Ci which generate G and o\&2 o~r = 1. We say that Ti is weakly rigid if this set is not empty, and if any two r-tuples in this set are conjugate under Aut(G). DEFINITION 2.7. Let K be a subfield of C, and B C KU {00} be an unordered tuple of distinct elements. Associate to each p G B a conjugacy class Cp of a finite group G of order n, and let Ti be the tuple of these classes. Set Y := Gal(K\K), and let £n be a primitive n-th root of unity. For each 7 G Y let 771(7) De a n integer with 7 -HCn)=C(7)- (a) The pair (23, W) is called K-rational, if 23 is T-invariant and C7(P) = Cp for each 7 G T and p G 23. (b) The pair (23, W) is called weakly K-rational, if 23 is T-invariant and C7(P) = a(Cp 7 ') for each 7 G T and p G 6, where a G Aut(G) depends on 7, but not on p. PROPOSITION 2.8. Let K be a field of characteristic 0, G be a finite group which is generated by 7i,... ,crr with o\G2 -o~r 1. Let B {pi,p2, ,Pr} C ¥1(K), and let CPi be the conjugacy class of Oi. Suppose that it '•— (Cp1? •.,Cpr) is weakly rigid and is weakly K -rational. Furthermore, let H G be a subgroup which is self-normalizing and whose G-conjugacy class is Aut(G) -invariant. Suppose that one of the pi is rational, and that Cp. intersects H non-trivially for this index i. Also, suppose that the Oi are a genus 0 system with respect to the action on G/H. Then there exists f G K(X) with geometric monodromy group G acting on G/H and ramification structure (B,H). PROOF. By Proposition 2.2, there exists a Galois extension L\K(i) with group G and ramification structure (B,H). Weak rigidity implies that L is Galois over K(t), see e.g. [62, Remark 3.9(a)]. Set A := Gal(L|if(£)), so G is the fixed group of K(t). Let J be the normalizer of H in A. As G is transitive on the ^-conjugates of H we obtain A = GJ. Also, GP\J = H because H is self-normalizing in G. Let E be the fixed field of J in L. Then E is linearly disjoint from K(t) over K(t), and KE is the fixed field of H. By assumption, E has genus 0, and the hypothesis about the nontrivial intersection of some class CPi with H with p^ being K-rational
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