Sigma function associated with a hyperelliptic curve with two points at infinity

Baker constructed basic meromorphic functions on the Jacobian variety of a hyperelliptic curve with two points at infinity. We call them Baker functions. The construction is based on the Abel-Jacobi map, which allows us to identify the field of meromorphic functions on the Jacobian variety of the curve with the field of meromorphic functions on the symmetric product of the curve. In our previous paper, a solution to the KP equation was constructed in terms of the Baker function. This paper is devoted to the properties of the Baker functions. In this paper, we construct an entire function whose second logarithmic derivatives are the Baker functions. We prove that the power series expansion of the entire function around the origin is determined only by the coefficients of the defining equation of the curve and a branch point of the curve algebraically. We also describe the quasi-periodicity of the entire function and express the entire function in terms of the Riemann theta function.

💡 Research Summary

The paper addresses a long‑standing gap in the theory of hyperelliptic sigma functions: while the classical sigma function σ(u) associated with a hyperelliptic curve having a single point at infinity is well understood (its power‑series coefficients are explicit polynomials in the curve’s defining coefficients, and its second logarithmic derivatives ℘_{i,j}(u) provide the basic meromorphic Baker functions), the analogous construction for curves with two points at infinity has remained incomplete.

The authors begin by reviewing the standard sigma function for a one‑point hyperelliptic curve C: Y² = M(X) of genus g, introducing the holomorphic differentials ω_i, the normalized period matrices ω′, ω″, and the Riemann theta function with characteristics. They recall that σ(u) = ε exp(½ uᵀη′(ω′)⁻¹u) θ