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