Sigma functions for a space curve of type \((3, 4, 5)\). (English) Zbl 1311.14033

Matsutani, Shigeki; Komeda, Jiryo

J. Geom. Symmetry Phys. 30, 75-91 (2013).

The paper under review deals with the construction of a generalized Kleinian sigma function for an affine \((3,4,5)\) space curve of genus two. Recall that the semigroup generated by 3, 4 and 5 is Weierstrass. In order to get that sigma function the authors use the so-called EEL construction discovered by Enolskii, Eilbeck and Leykin [J. C. Eilbeck et al., in: SIDE III – Symmetries and integrability of difference equations. Proceedings of the 3rd conference, Sabaudia, Italy, May 16–22, 1998. Providence, RI: American Mathematical Society (AMS). 121–138 (2000; Zbl 1003.14008)]. They define a curve \(X_2 = C(H_2)\) where \(H_2 = \langle 3,4,5 \rangle\), such that its one-point compactification \(X = X_2 \cup \{ \infty\}\) is non-singular. Then by using the EEL construction they obtain differentials of second and third kinds on \(X\).

Reviewer: José Javier Etayo (Madrid)

MSC:

14H55

Riemann surfaces; Weierstrass points; gap sequences

Keywords:

sigma function; affine space curve; Weierstrass semigroup