2022/08/22 by Takanori Ayano, Ayano, Takanori
Mathematics · #14H42(Primary) 14K25 #32A05(Secondary) #Algebraic Geometry (math.AG) #Algebraic Geometry and Number Theory #Complex Variables (math.CV) #FOS: Mathematics #Holomorphic and Operator Theory #Meromorphic and Entire Functions
paper · pdf · doi:10.48550/arxiv.2208.10079
openalex publication_date 2022/08/22 · openalex created_date 2022/08/24 · openalex updated_date 2026/07/28
A telescopic curve is a certain algebraic curve defined by m-1 equations in the affine space of dimension m, which can be a hyperelliptic curve and an (n,s) curve as a special case. The sigma function σ(u) associated with the telescopic curve of genus g is a holomorphic function on ℂg. For a subring R of ℂ and variables u=t(u1,…, ug), let R⟨⟨ u ⟩⟩=\∑i1,…,ig≥0κi1,…,ig\fracu1i1⋯ ugigi1!⋯ ig! \middle| κi1,…,ig∈ R\. If the power series expansion of a holomorphic function f(u) on ℂg around the origin belongs to R⟨⟨ u ⟩⟩, then f(u) is said to be Hurwitz integral over R. In this paper, we show that the sigma function σ(u) associated with the telescopic curve is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve and (1)/(2) over ℤ. Further, we show that σ(u)2 is Hurwitz integral over the ring generated by the coefficients of the defining equations of the curve over ℤ. Our results are a generalization of the results of Y. Ônishi for (n,s) curves to telescopic curves.