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

Abstract

Let $C$ be a genus 2 algebraic curve defined by an equation of the form $y^2\,{=}\,x(x^2\,{-}\,1)(x\,{-} a)(x\,{-}\,1/a)$. As is well known, the five accessory parameters for such an equation can all be expressed in terms of $a$ and the accessory parameter $\frak b$ corresponding to $a$. The main result of the paper is that if $a'\,{=}\,\sqrt{1\,{-}\,{a}^2}$, which in general yields a non-isomorphic curve $C'$, then $\acb'a'({a'}^2\,{-}\,1) {=}\,{-}\frac38\,{-}\,\acb\,a\,({a}^2\,{-}\,1)$.

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 $C$.