On algebraic expressions of sigma functions for (\(n, s\)) curves.

*(English)*Zbl 1214.14028One of the prominent features of Weierstrass’ elliptic sigma functions is their algebraic nature directly related to the defining equation of the elliptic curve. Klein extended the elliptic sigma functions to the case of hyperelliptic curves from this point of view. Since they are defined, it had been one of the central problems to determine the coefficients of the series expansion of the sigma functions.

Recently Klein’s sigma function is further generalized to the case of more general plane algebraic curves called \((n, s)\)-curves by Buchstaber, Enolski and Leykin. They made an important observation that the first term, with respect to certain degree introduced in the theory of soliton equations, of the series expansion of the sigma function, which corresponds to the most singular characteristic, is described by Schur function. In order to establish the connection to Schur functions Buchstaber, Enolski and Leykin have developed the rational theory of abelian integrals and characterized Schur functions by Riemann’s vanishing theorem.

The purpose of this paper is to generalize Klein’s algebraic formulas for the hyperelliptic sigma functions to the case of \((n,s)\)-curves and is to establish the relation with Schur functions directly. The sigma function, in this paper, signifies the sigma function with Riemann’s constant as its characteristic. An expression of the multivariate sigma function associated with an \((n,s)\)-curve is given in terms of algebraic integrals. As a corollary the first term of the series expansion around the origin of the sigma function is directly proved to be Schur function determined from the gap sequence at infinity.

Recently Klein’s sigma function is further generalized to the case of more general plane algebraic curves called \((n, s)\)-curves by Buchstaber, Enolski and Leykin. They made an important observation that the first term, with respect to certain degree introduced in the theory of soliton equations, of the series expansion of the sigma function, which corresponds to the most singular characteristic, is described by Schur function. In order to establish the connection to Schur functions Buchstaber, Enolski and Leykin have developed the rational theory of abelian integrals and characterized Schur functions by Riemann’s vanishing theorem.

The purpose of this paper is to generalize Klein’s algebraic formulas for the hyperelliptic sigma functions to the case of \((n,s)\)-curves and is to establish the relation with Schur functions directly. The sigma function, in this paper, signifies the sigma function with Riemann’s constant as its characteristic. An expression of the multivariate sigma function associated with an \((n,s)\)-curve is given in terms of algebraic integrals. As a corollary the first term of the series expansion around the origin of the sigma function is directly proved to be Schur function determined from the gap sequence at infinity.

Reviewer: V. V. Chueshev (Kemerovo)

##### MSC:

14H42 | Theta functions and curves; Schottky problem |

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

14H70 | Relationships between algebraic curves and integrable systems |