Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 13 (2017), 017, 13 pages      arXiv:1701.00931      https://doi.org/10.3842/SIGMA.2017.017

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

Joe Suzuki
Department of Mathematics, Osaka University, Machikaneyama Toyonaka, Osaka 560-0043, Japan

Received January 05, 2017, in final form March 11, 2017; Published online March 16, 2017

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.

Key words: $C_{ab}$ curves; Klein's fundamental 2-form of second kind; cohomological basis; symmetry.

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