On cyclic trigonal Riemann surfaces. I

Abstract

Definition. Call the Riemann surfaces for the equation y 3 = P ( x ) {y^3} = P(x) cyclic trigonal. For one case of genus 4 4 ( 2 2 distinct g 3 1 g_3^1 ’s) and all genera greater than 4 4 , cyclic trigonal Riemann surfaces are characterized by the vanishing properties of the theta function at certain ( 1 / 6 ) (1/6) -periods of the Jacobian. Also for trigonal Riemann surfaces of genera 5 5 , 6 6 , and 7 7 , homogeneous theta relations are derived using the fact that Prym varieties for trigonal Riemann surfaces are Jacobians.