Reduction of an Algebraic Function Field Modulo a Prime in the Constant Field

Abstract

The problem solved here is related to an old problem in algebraic geometry. Suppose f(x, y) is an irreducible polynomial in two variables and that the coefficients involve certain parameters 2, 22-, * I )rand the coefficient field k. Let the algebraic curve defined by f(x, y) = 0 over the field of coefficients k(21, 2, *-. = k(2) be of genus g. If the At are given special values 2i' in an algebraic extension of k, f(x, y) may factor in k(2'). Each irreducible factor defines an algebraic curve with genus g,. What is the relation between g and the gi? For an extended discussion of this problem, see Picard et Simart [10, Volume II, Ch. III]. The problem actually solved here is algebraic and is stated in Section 4. The solution, Theorem 4.19, of the algebraic problem gives the solution of the geometric problem if the powers of a solution of f(x, y) 0, considered as an equation with coefficients in k(2, x), form the integral basis mentioned in Lemma 4.1. Otherwise, components which are considered in the algebraic formulation may not appear at all in the geometric formulation. Sections 2 and 3 are devoted to a discussion of the tools needed to solve the main problem. The results stated there are well known in so far as they apply to fields. These results must be restated for rings in a form applicable to the main problem. In no case, even where we carry it out, is the extension any more than an exercise. It is also necessary to replace the trace by an arbitrary linear mapping, as a means for obtaining differentials in an algebraic extension of a field in which differentials are given. This is given in Artin [1, Ch. 13]. Therefore, we shall prove only a few statements which require amplification. It is with great pleasure that I express my appreciation to Professor Emil Artin, for without his guidance, advice, and inspiration this work would have been quite impossible; and to Professor Solomon Lefschetz