CHAPTER 2 Arithmetic—Geometric Preparation 2.1. Arithmetic and geometric monodromy groups Let K be a field of characteristic 0 and let t be a transcendental. Fix a regular extension E of K(t) of degree n. (Regular means that K is algebraically closed in E.) Denote by L a Galois closure of E\K(t). Then A := Gal(L\K(t)) is the arithmetic monodromy group of E\K(t), where we regard A as a permutation group acting transitively on the n conjugates of a primitive element of E\K(t). Denote by K the algebraic closure of K in L. Then G := Gal(L\K(t)) is the geometric monodromy group of E\K(i). Note that A/G = Gal(K\K), and that G still permutes the n conjugates of E transitively, as E and K(i) are linearly disjoint over K(t). If f(X) G K(X) is a non-constant rational function, then we use the terms arithmetic and geometric monodromy group for the extension E K{x) over K(t), where x is a solution of f(X) = t in some algebraic closure of K(t). In this case, we call K the field of constants of / . 2.2. Distinguished conjugacy classes of inertia generators We continue to use the above notation. Let K be an algebraic closure of K. We identify the places p : t i— b (or 1/t i— 0 if 6 = oo) of J^(t) with K U {oc}. Let p be a ramified place of L\K(t). Set y := t p (or y := 1/t if p = oo). There is a minimal integer e such that L embeds into the power series field K^y1^)). For such an embedding let a be the restriction to L of the automorphism of K^y1^)) which is the identity on the coefficients and maps yxle to Cy1^^ where £ is a primitive eth root of unity. The embedding of L is unique only up to a ^(t)-automorphism of KL, that is a is unique only up to conjugacy in G Gal(L\K(t)) = Gal(KL\K(t)). We call the G-conjugacy class C of a the distinguished conjugacy class associated to p. Note that each a in C is the generator of an inertia group of a place of L lying above p, and that e is the ramification index of p. Let the tuple B := (pi,..., pr) consist of the places of K(t) which are ramified in KL, and let H = (Ci,..., Cr) be the tuple of the associated distinguished conjugacy classes of G. We call the pair (B,H) the ramification structure of L\K{t). 2.3. Branch cycle descriptions A fundamental fact, following from Riemann's existence theorem and a topo- logical interpretation of the geometric monodromy group as described below is the following. See [62] for a selfcontained proof. PROPOSITION 2.1. With the notation from above, let pi,.. . ,p r be the places of K(t) which ramify in L, and let Ci be the distinguished conjugacy class associated 6
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