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

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 $s\in D_{\epsilon }:=\{s\in \mathbb{C}||s|<\epsilon \}$. We compare the sigma function over the punctured disc $D_{\epsilon }^{\ast }:=D_{\epsilon }\setminus \{0\}$ 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.

48 pages

References in corpus (4)

• Thomae type formulae for singular Z_N curves
• Theory of Heat Equations for Sigma Functions
• Some addition formulae for Abelian functions for elliptic and hyperelliptic curves of cyclotomic type
• The al function of a cyclic trigonal curve of genus three