Nonrigid Constructions in Galois Theory

The context for this paper is the Inverse Galois Problem. First we give an if and only if condition that a finite group is the group of a Galois regular extension of R(X) with only real branch points. It is that the group is generated by elements of order 2 (Theorem 1.1 (a)). We use previous work on the action of the complex conjugation on covers of P [FrD]. We also use Fried and Völklein [FrV] and Pop [P] to show each finite group is the Galois group of a Galois regular extension of Q(X). Here Q is the field of all totally real algebraic numbers (Theorem 5.7). §1, §2 and §3 discuss consequences, generalizations and related questions. The second part of the paper, §4 and §5, concerns descent of fields of definition from R to Q. Use of Hurwitz families reduces the problem to finding Q-rational point on a special algebraic curve. Our first application considers realizing the symmetric group Sm as the group of a Galois extension of Q(X), regular over Q, satisfying two further conditions. These are that the extension has four branch points, and it also has some totally real residue class field specializations. Such extensions exist for m = 4, 5, 6, 7, 10 (Theorem 4.11). Suppose that m is a prime larger than 7. Theorem 5.1 shows that the dihedral group Dm of order 2m isn’t the group of a Galois regular extension of Q(X) with fewer than 6 branch points. The proof interprets realization of certain dihedral group covers as corresponding to rational points on modular curves. We then apply Mazur’s Theorem. New results of Kamienny and Mazur [KM] suggest that no bound on the number of branch points will allow realization of all Dms. §0.1. Description of Theorem 1.1: Throughout, C denotes the complex number field, X an indeterminate and C(X) a fixed algebraic closure of C(X). Let k be a subfield of C. We say a finite extension Y/k(X) with C(X) ⊃ Y is regular over k if k̄ ∩ Y = k. Equivalently [Y : k(X)] = [Y C : C(X)]. Denote this degree by n. Regard the degree n field extension Y C/C(X) as the function field extension of a degree n cover φ : YC → P. Here P is the complex projective line and YC is an irreducible non-singular curve. The map φ is ramified over a finite number of points x1, . . . , xr. We call these the branch points of the cover (or of the extension Y/k(X)). Our first result (Theorem 1.1 (a)) shows exactly when a finite group G is the group of a Galois regular extension of R(X) with only real branch points. This happens if and only if G is generated by involutions. Grants: ∗ Support from University of California (Irvine) and NSF grant DMS-8702150 ∗∗ Supported by BSF grant #87-00038, NSA grant MDA 904-91-H-0057, the Institute for Advanced Studies in Jerusalem and IFR Grant #90/91-15. AMS Subject classification: 11G35, 12F10, 14E20, 14G05, 20B25, 20C25