The sigma function over a family of cyclic trigonal curves with a singular fiber

TL;DR

This paper studies the behavior of the sigma function over a family of cyclic trigonal curves with a singular fiber, comparing it to the normalized curve at the singular point and revealing its limiting properties.

Contribution

It provides an explicit analysis of the sigma function's limit over a family of cyclic trigonal curves with a singular fiber, highlighting differences from the theta function.

Findings

Sigma function has a well-defined limit at the singular fiber.

Explicit description of the theta divisor and theta characteristics at the limit.

Comparison between sigma and theta functions' behavior near the singularity.

Abstract

In this paper we investigate the behavior of the sigma function over the family of cyclic trigonal curves $X_s$ defined by the equation $y^3 =x(x-s)(x-b_1)(x-b_2)$ in the affine $(x,y)$ plane, for $s\in D_\varepsilon:=\{s \in \mathbb{C} | |s|<\varepsilon\}$. We compare the sigma function over the punctured disc $D_\varepsilon^*:=D_\varepsilon\setminus\{0\}$ with the extension over $s=0$ that specializes to the sigma function of the normalization $X_{\hat{0}}$ of the singular curve $X_{s=0}$ by investigating explicitly the behavior of a basis of the first algebraic de Rham cohomology group and its period integrals. We demonstrate, using modular properties, that sigma, unlike the theta function, has a limit. In particular, we obtain the limit of the theta characteristics and an explicit description of the theta divisor translated by the Riemann constant.