Rational Points on Picard Groups of Some Genus-Changing Curves of Genus at Least 2 Original Research Article

For an algebraic curve C/K defined by y2=xp+a (anegated set membershipKp) with relative genus (p−1)/2 and absolute genus 0, we prove that the Picard group of divisors of degree 0, denoted Pic0K(C), of a curve C/K fixed by the action of the Galois group G=gal(Ksep/K) has a finite number of K-rational points as a variety, where K is a function field in one variable with a finite constant field of characteristic pgreater-or-equal, slanted5 and Ksep is the separable closure of K.