A JACOBI-TYPE FORMULA FOR A FAMILY OF HYPERELLIPTIC CURVES OF GENUS 3 WITH AUTOMORPHISM OF ORDER 4

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In this paper we show a two-dimensional variant of the classical Jacobi formula between a theta constant and the Gauss hypergeometric function. We use the family of algebraic curves given in the form w4 = z2(z-1)2(z-λ1)(z-λ2) with two complex parameters λ1,λ2 and the modular functions for them. Our result is an exact extension of the classical formula that is contained as a degenerated case. As an application we give a new proof for the extended Gauss arithmetic geometric mean theorem in two variables obtained by Koike and Shiga (J. Number Theory 128 (2008), 2029-2126).