Sigma Functions for Telescopic Curves

2012/01/03 by Takanori Ayano, Ayano, Takanori · 1 citation
Computer Science · Mathematics · #14H42 #14H50 #14H55 #Algebraic Geometry (math.AG) #Algebraic and Geometric Analysis #Cryptography and Residue Arithmetic #FOS: Mathematics #Polynomial and algebraic computation

paper · doi:10.48550/arxiv.1201.0644

openalex publication_date 2012/01/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we consider the sigma functions for algebraic curves expressed by a canonical form using a finite sequence (a₁,...,a_t) of positive integers whose greatest common divisor is equal to one (Miura [13]). The idea is to express a non-singular algebraic curve by affine equations of t variables whose orders at infinity are (a₁,...,a_t). We construct a symplectic basis of the first cohomology group and the sigma functions for telescopic curves, i.e., the curves such that the number of defining equations is exactly t-1 in the Miura canonical form. The largest class of curves for which such construction has been obtained thus far is (n,s)-curves ([3][15]), which are telescopic because they are expressed in the Miura canonical form with t=2, a₁=n, and a₂=s, and the number of defining equations is one.

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• Sigma function associated with a hyperelliptic curve with two points at infinity

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