2.4.THE BRANCH CYCLE ARGUMENT 7 to pi. Then there are o~i G Ci, such that the &i generate G, and the product relation o~\G2 o~r = 1 holds. We call such a tuple (7i,..., ay) a branch cycle description of L\K(t). The branch cycle description allows us to compute the genus of the field E. First note that this genus is the same as the genus of KE. For a G G, let ind(cr) be [E : K(t)\ minus the number of cycles of a in the given permutation representation. Then the Riemann-Hurwitz genus formula yields (2.1) Ylind(^) = 2 (lE : K (t)\ ~ l + genus(£)). i If the genus of E from above is 0, then we call (o"i,..., oy) a genus 0 system. A sort of converse to Proposition 2.1 is the following algebraic version of Rie- mann's existence theorem, see [62]. PROPOSITION 2.2. Let K be an algebraically closed field of characteristic 0, (Ji,..., ar be a generating system of a finite group G with o~\o2 &r 1, and pi,.. . ,p r distinct elements from K U {oo}. Then there exists a Galois extension L\K(t) with group G, ramified only in pi,.. . ,p r such that the distinguished con- jugacy class associated with pi is the conjugacy class erf. 2.4. The branch cycle argument We have seen that we can realize any given ramification structure over K(t). A situation we will encounter frequently is whether we can get a regular extension E\K(t) with given ramification structure, and given pair of geometric and arith- metic monodromy group. A powerful tool to either exclude possible pairs, or to determine A if G is known, is provided by Fried's branch cycle argument (see [62, Lemma 2.8], [41, Section 1.2.3], [55, Part 1]), of which the following proposition is an immediate consequence. PROPOSITION 2.3. LetK be afield of characteristic 0, L\K{t) be a finite Galois extension with group A, K the algebraic closure of K in L, and G := Gal(L\K(t)). Let (S,W) be the ramification structure of L\K(t). Let (n be a primitive n-th root of unity, where n \G\. Let 7 G Gdl{K\K) =: V be arbitrary, and m with with 7_1(Cn) = CT- Let a G A be the restriction to L of an extension of 7 to Gal(KL\K(t)). Then B is Y-invariant and C7(p) - (C™)a for each p G B. A special case is. COROLLARY 2.4. Let L\Q(t) be a finite Galois extension with group A, Q the algebraic closure of Q in L, and G := Gal(L|Q(t)). Let p be a rational place of Q(t), and a be in the associated distinguished conjugacy class of G. Then for each m prime to the order of a, the element a171 is conjugate in A to a. Often the branch cycle argument does not help in excluding certain pairs (A, G) and given ramification structure. Then the following observation is sometimes helpful. LEMMA 2.5. Let K be a field of characteristic 0, t a transcendental, and L\K(t) be a finite Galois extension with group A. Let K be the algebraic closure of K in L, and set G = Gal(L\K(t)). Let p be a rational place of K(t), and ^ be a place of L lying above p. Denote by D and I the decomposition and inertia group ofty, respectively. Then I D P\G and A = GD. In particular, A = GN^(J).
Previous Page Next Page