A FUNCTIONAL RELATION FOR ACCESSORY PARAMETERS FOR GENUS 2 ALGEBRAIC CURVES WITH AN ORDER 4 AUTOMORPHISM

Let C be a genus 2 algebraic curve defined by an equation of the form y2 =x(x2 −1)(x− a)(x−1/a). As is well known, the five accessory parameters for such an equation can all be expressed in terms of a and the accessory parameter b corresponding to a. The main result of the paper is that if a = √ 1−a2, which in general yields a non-isomorphic curve C , then b a (a 2 −1) =−3 8 −ba (a2 −1). This is proven by it being shown how the uniformizing function from the unit disk to C can be explicitly described in terms of the uniformizing function for C.