Automorphisms of function fields

Abstract

1. Let K be an algebraic function field of one variable over the constant field k and let g > 0 be the genus of K. Let 9 be the group of all automorphisms of K that leave the elements of k fixed (and that leave a given place Po of K/k fixed if g = 1). A classical theorem due to Schwartz-Klein-NoetherWeierstrass-Poincar&-Hurwitz when g>1 (and older for g =1) says that 57 is finite if k is the field of complex numbers. From this one can easily deduce the same result if k is any field of characteristic zero. The theorem for k an algebraically closed field of characteristic p= 0 was proved by H. L. Schmid in 1938 [5], and a less computational proof for any algebraically closed k was given recently by Iwasawa and Tamagawa [3]. We intend to show how this result can be very easily proved by one of the classical arguments (given in essence, but somewhat imprecisely, in [1]) if we replace integration on the Riemann surface R of K by use of its jacobian variety J, and finally we shall show what the corresponding result is when k is an arbitrary field. The reasons for including here the easy case g =1 will become apparent in the last section. The analytic proof we have in mind runs as follows: 57 is naturally isomorphic to the group of complex analytic homeomorphisms of R (that leave Po fixed if g = 1). First consider the special case in which R is elliptic or hyperelliptic. R can then be considered (in one and only one way) as a two-sheeted covering surface of a Riemann sphere S (such that, if g =1, Po is a branch point of this covering). The elements of 57 give rise to analytic homeomorphisms of S that permute the ramification points of S. Since g > 0, the ramification points are in finite number > 2. The finiteness of 57 then follows from (1) any analytic homeomorphism of S leaving three distinct points fixed is the identity, and (2) any element of 5' that leaves all points of S fixed is either the identity or merely interchanges the sheets of R. On the other hand if K is not elliptic or hyperelliptic, then the ratios of the differentials of the first kind of K give rise to the canonical embedding of R in S,-1, the complex projective space of dimension (g 1), and the automorphisms of K/k correspond one-one to projective transformations of S,.1 that map R onto itself. It follows that G can be considered as a Lie group with a finite number of components that acts analytically on R (see the second lemma of ?2 for details), so it remains only to show that the component of the identity G of G has only one point. Hence we have to show that if ai, a2C G are homotopic (as maps of R), then al = 02. So let w be any differential of the first kind on R