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

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences
##### Keywords:
sigma function; affine space curve; Weierstrass semigroup
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