Klein's Fundamental 2-Form of Second Kind for the $C_{ab}$ Curves

TL;DR

This paper derives an explicit formula for Klein's fundamental 2-form of second kind for $C_{ab}$ curves, extending classical results and enabling applications in algebraic geometry and physics.

Contribution

It provides the first exact formula for Klein's 2-form for the general class of $C_{ab}$ curves, advancing beyond previous numerical solutions.

Findings

Derived an explicit formula for the 2-form of second kind for $C_{ab}$ curves

Extended Klein's classical results to a broader class of algebraic curves

Facilitates applications in sigma functions, algebraic geometry, and physics.

Abstract

In this paper, we derive the exact formula of Klein's fundamental 2-form of second kind for the so-called $C_{ab}$ curves. The problem was initially solved by Klein in the 19th century for the hyper-elliptic curves, but little progress had been seen for its extension for more than 100 years. Recently, it has been addressed by several authors, and was solved for subclasses of the $C_{ab}$ curves whereas they found a way to find its individual solution numerically. The formula gives a standard cohomological basis for the curves, and has many applications in algebraic geometry, physics, and applied mathematics, not just analyzing sigma functions in a general way.