Galois Groups and Complex Multiplication

The Schur problem for rational functions is linked to the theory of complex multiplication and thereby solved. These considerations are viewed as a special case of a general problem, prosaically labeled the extension of constants problem. The relation between this paper and a letter of J. Herbrand to E. Noether (published posthumously) is speculatively summarized in a conjecture that may be regarded as an arithmetic version of Riemann's existence theorem. 0. Introduct.ion. Let F be a perfect field, and F a fixed algebraic closure of F. We consider the finite Galois extensions of F that come from certain types of geometric situations. Let W (vw)V be a cover (finite, flat morphism) of quasi-projective varieties such that W, V, and p(V, W) are defined over F, and W and V are absolutely irreducible. For X, a variety defined over F, we let F(X) be the field of rational functions on X defined over F. Therefore, we have a field extension F(W) over F(V) (by abuse, F(W)/F(V)). If ( V, W) is a separable morphism, then F(W)/F(V) is a finite separable extension, and we obtain (0.1) I G (.''_W) /F' ( V)) -->G (F( IF( V)) 's>tG (F/lF) 1> where: 9W is the smallest Galois extension of F(V) containing F(W) (the Galois closure of F(W)/F(V)); F = F( W, F) is the algebraic closure of F in F(W , and; rest denotes the restriction of elements of the Galois group G( I)/F(V)) to F. We call F the extension of constants obtained from W/V. The problem of the description of G (F/F) arises in several well-known problems. For example. let G be a finite group which we desire to realize as the Galois group of some Galois extension of the rational field Q. Suppose tha: = Q; V is a Zariski open subset of P' (projective n-space); and G(Q(W)/Q( V)), the (geometric) monodromy group is equal to G (a fact that may have come to us from analytic considerations, say in the manner of [Fr, 1]). In this case the limitation theorems of [Fr, 1, ?2] sometimes serve to show Received by the editors February 23, 1976. AMS (MOS) subject classifications (1970). Primary lOB15, lOD25, 12A55, 14D20, 14G05, 14H05, 14H10; Secondary 10M05, 14D05, 14H25, 14H30, 14K22. (1) The research for this paper was partially supported by an Alfred P. Sloan Foundation Grant and by a grant from the Institute for Advanced Study (Princeton) in Spring 1973. r American Mathematical Society 1978