Let K be an algebraic function field in one variable over an algebraically closed field of positive characteristic p. We give an explicit upper bound for the number of rational points of genus-changing curves over K defined by y p =r(x) and show that every genus-changing curve of absolute genus 0 has finitely many K-rational points. We thus prove that every algebraic curve over K that admits genus change under base-field extensions has finitely many K-rational points.
1997 Academic Press
1. Introduction
Throughout the paper, unless otherwise specified, let K be an algebraic function field in one variable over an algebraically closed field k of characteristic p>0. In other words, K is a function field of a non-singular curve X of genus g defined over k. Let C be an algebraic curve defined over K.
The genus of a curve C relative to K can be defined as the integer g K such that the Riemann Roch formula holds; that is, for any K-divisor D of C of sufficiently large degree deg(D), the dimension l(D) of the K-vector space L(D) of functions of K(C) whose polar divisor is bounded by D is deg(D)+1& g K . We can also define the absolute genus of C, denoted by gΓ , to be the genus of C relative to the algebraic closure K of a base field K. We then note the inequality gΓ g K , as the relative genus g K of C does not change under separable extensions of a base field K, but may decrease under inseparable extensions of K (see [2] and [9]).
An algebraic curve CΓK is defined to be non-conservative (or genuschanging) if its relative genus g K is different from the absolute genus gΓ . Otherwise, C is said to be conservative. Typical examples of non-conservative curves are given by the equation x&ax p = y p (a Γ K p , p 3) with relative genus ( p&1)( p&2)Γ2 and absolute genus 0, and the equation y 2 =x p +a (a Γ K p , p 3) with relative genus ( p&1)Γ2 and absolute genus article no. NT972176