Kleinian hyperelliptic funtions of weight 2 associated with curves of genus 2

TL;DR

This paper introduces a new class of special functions related to genus 2 algebraic curves, extending Kleinian hyperelliptic functions, with broad applicability without restrictions on the curve's properties.

Contribution

It defines a novel collection of weight 2 functions associated with genus 2 curves, generalizing Kleinian hyperelliptic functions and relating to theta functions without restrictions.

Findings

Functions are well-defined for all genus 2 curves

No need for curves to have a Weierstrass point at infinity

Establishes a new link between these functions and theta functions

Abstract

We introduce a new collection of special functions associated to a complex curve of genus 2 similar to Kleinian hyperelliptic $\sigma$-function. These functions are related to weight 2 $\theta$-functions in the same fashion as $\sigma$-function is related to the classical $\theta$-function. A key feature of the introduced functions is the fact that they are well-defined for genus 2 algebraic curves without any restrictions (in particular it is not needed to assume that the curve has a Weierstrass point at infinity).