Sigma functions for telescopic curves

Abstract

In this paper we consider a symplectic basis of the first cohomology group and the sigma functions for algebraic curves expressed by a canonical form using a finite sequence $(a_{1}, \ldots, a_{t})$ of positive integers whose greatest common divisor is equal to one (Miura [13]). The idea is to express a non-singular algebraic curve by affine equations of $t$ variables whose orders at infinity are $(a_{1}, \ldots, a_{t})$. We construct a symplectic basis of the first cohomology group and the sigma functions for telescopic curves, i.e., the curves such that the number of defining equations is exactly $t-1$ in the Miura canonical form. The largest class of curves for which such construction has been obtained thus far is $(n, s)$-curves ([4] [15]), which are telescopic because they are expressed in the Miura canonical form with $t=2$, $a_{1}=n$, and $a_{2}=s$, and the number of defining equations is one.