Sigma Functions for Telescopic Curves

TL;DR

This paper develops sigma functions for a broad class of algebraic curves called telescopic curves, using a canonical form based on positive integer sequences, extending previous work on simpler curve classes.

Contribution

It constructs sigma functions and a symplectic basis for telescopic curves, generalizing the known results from (n,s)-curves and expanding the class of curves with such structures.

Findings

Constructed sigma functions for telescopic curves

Established a symplectic basis of the first cohomology group

Extended the class of curves with known sigma function structures

Abstract

In this paper, we consider the sigma functions for algebraic curves expressed by a canonical form using a finite sequence $(a_1,...,a_t)$ 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 $(a_1,...,a_t)$. 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 ([3][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.