On the Thomae Formula for <b>Z</b>_<i>N</i> Curves

Abstract

We shall give an elementary and rigorous proof of the Thomae formula for b>Z_N_ curves which was discovered by Bershadsky and Radul [1, 2]. Instead of using the determinant of the Laplacian we use the traditional variational method which goes back to Riemann, Thomae, Fuchs. In the proof we made explicit the algebraic expression of the chiral Szego kernels and prove the vanishing of zero values of derivatives of theta functions with b>Z_N_ invariant 1/2_N_ characteristics.