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Sigma functions for telescopic curves. (English) Zbl 1328.14057
The author constructs sigma functions explicitly for a class of algebraic curves. Let $$C$$ be a compact Riemann surface of genus $$g$$. In order to express defining equations of $$C$$, he uses a canonical form for expressing non-singular algebraic curves introduced by S. Miura [“Linear codes on affine algebraic curves”, IEICE Trans. J81-A, 1398–1421 (1998)]. Given a finite sequence $$(a_{1},\dots, a_{t})$$ of positive integers whose greatest common divisor is equal to one, Miura introduced a non-singular algebraic determined by the sequence $$(a_{1},\dots, a_{t})$$. The idea is to express a non-singular algebraic curve by affine equations of $$t$$ variables whose orders at infinity are $$(a_{1},\dots, a_{t})$$.
F. Klein [Math. Ann. 32, 351–380 (1888; JFM 20.0491.01)] extended the elliptic sigma functions to the case of hyperelliptic curves of genus $$g$$, which are expressed in the Miura canonical form with $$t = 2, a_{1} = 2$$, and $$a_{2} = 2g + 1$$. V. M. Bukhshtaber, D. V. Leikin, V. Z. Enol’skii [Russ. Math. Surv. 54, No. 3, 628–629 (1999); translation from Usp. Mat. Nauk 54, No. 3, 155–156 (1999; Zbl 1081.14519)] and A. Nakayashiki [Asian J. Math. 14, No. 2, 175–212 (2010; Zbl 1214.14028)] extended Klein’s sigma functions to the case of more general plane algebraic curve called $$(n, s)-$$curves, which are expressed in the Miura canonical form with $$t = 2, a_{1} = n,$$ and $$a_{2} = s$$.
In this paper author consider a symplectic basis of the first cohomology group and give an explicit construction of sigma functions for telescopic curves, i.e., the curves such that the number of defining equations is exactly $$t - 1$$ in the Miura canonical form. The telescopic curves contain the $$(n, s)-$$curves as special cases.

##### MSC:
 14H55 Riemann surfaces; Weierstrass points; gap sequences 14H42 Theta functions and curves; Schottky problem 14H50 Plane and space curves
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##### References:
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