
SIGMA 13 (2017), 017, 13 pages arXiv:1701.00931
https://doi.org/10.3842/SIGMA.2017.017
Klein's Fundamental 2Form of Second Kind for the $C_{ab}$ Curves
Joe Suzuki
Department of Mathematics, Osaka University, Machikaneyama Toyonaka, Osaka 5600043, 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 2form of second kind for the socalled $C_{ab}$ curves. The problem was initially solved by Klein in the 19th century for the hyperelliptic 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 2form of second kind; cohomological basis; symmetry.
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