2019/09/09 by Yu. V. Fedorov, Jiyro Komeda, Fedorov, Yuri +7
Computer Science · Mathematics · #Advanced Differential Equations and Dynamical Systems #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1909.03858
openalex publication_date 2019/09/09 · openalex created_date 2019/09/12 · openalex updated_date 2026/07/28
In this paper we investigate the behavior of the sigma function over the family of cyclic trigonal curves Xs defined by the equation y3 =x(x-s)(x-b1)(x-b2) in the affine (x,y) plane, for s∈ Dε:=\s ∈ ℂ | |s|<ε\. We compare the sigma function over the punctured disc Dε^*:=Dε∖\0\ with the extension over s=0 that specializes to the sigma function of the normalization X_0 of the singular curve Xs=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.