Automorphism groups of algebraic function fields

Abstract

The conformal mappings of a compact Riemann surface R onto itself form a finite group, if the genus g of the surface is greater than 1. This is a classical result commonly attributed to Poincar6 and Klein. It was first suggested by H.A. Schwarz. There are several proofs, including those by Weierstrass [13], M. Noether [7] and Hurwitz [5]. In fact, Hurwitz proved that the order of this group cannot exceed 84(g-1) and that every finite group is a group of conformal mappings for a suitable compact Riemann surface. Considering the well-known correspondence between compact Riemann surfaces and fields of algebraic functions of one variable over the field of complex numbers, the above results can also be stated as follows: The group of automorphisms of a field of algebraic functions in one variable over the field of complex numbers is a finite group, provided its genus g is greater than one. Its order does not exceed 84(g -1). Every finite group is a group of automorphisms of a suitable function field.To extend these results to function fields F/K of characteristic p, it was natural to look for the analogue of Weierstrass points on a compact Riemann surface. This was done by F.K. Schmidt [11]. The theory of Weierstrass points in characteristic p is more complicated. Nevertheless, H.L. Schmid [10] was able to prove the finiteness of the group G of automorphisms. Concerning the order of G, Roquette [8] showed that the Hurwitz bound is valid if p > g + 1, with one exception. Henn [-4] has recently proved that [GI _-< 3(2g) ~/2, if F does not belong to one of four exceptional classes. The question concerning the realizability of a given group as a full group of automorphisms of a suitable function field is an open problem. Our principal result in this paper is an affirmative solution of this problem in the case of abelian groups.An abelian group is a direct product of cyclic groups of prime power order. Corresponding to the direct factors, we construct cyclic extensions of a rational function field K(x). The ramification is chosen to give arithmetically disjoint fields. This insures that the Galois group of the composite over K(x) is the direct product of the Galois groups of the factors. The ramification is further specified to produce Weierstrass points with suitable gap sequences. (See [1] or [11] for the definitions and basic properties.) This guarantees that any automorphism of the composite fixes K(x) elementwise.