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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.

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.

Reviewer: V. V. Chueshev (Kemerovo)

### MSC:

14H55 | Riemann surfaces; Weierstrass points; gap sequences |

14H42 | Theta functions and curves; Schottky problem |

14H50 | Plane and space curves |

### Keywords:

sigma functions; compact Riemann surface of genus; canonical form for algebraic curves introduced by Miura; \((n; s)-\)curves; telescopic curves### References:

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[2] | V.M. Bukhshtaber, D.V. Leĭ kin and V.Z. Ènol’skiĭ: Rational analogs of Abelian functions , Funct. Anal. Appl. 33 (1999), 83-94. |

[3] | V.M. Bukhshtaber, D.V. Leĭ kin and V.Z. Ènol’skiĭ: Uniformization of Jacobi varieties of trigonal curves and nonlinear differential equations , Funct. Anal. Appl. 34 (2000), 159-171. · Zbl 0978.58012 |

[4] | V.M. Bukhshtaber, D.V. Leĭ kin and V.Z. Ènol’skiĭ: \(\sigma\)-functions of \((n, s)\)-curves , Russian Math. Surveys 54 (1999), 628-629. · Zbl 1081.14519 |

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[8] | F. Klein: Ueber hyperelliptische Sigmafunctionen , Math. Ann. 27 (1886), 431-464. · JFM 18.0418.02 |

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[11] | S. Matsutani: Sigma functions for a space curve \((3, 4, 5)\) type with an appendix by J. Komeda , arXiv: · Zbl 1311.14033 |

[12] | S. Miura: Linear codes on affine algebraic curves , IEICE Trans. J81-A , (1998), 1398-1421, in Japanese. |

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[14] | A. Nakayashiki: On algebraic expressions of sigma functions for \((n,s)\) curves , Asian J. Math. 14 (2010), 175-211. · Zbl 1214.14028 |

[15] | A. Nijenhuis and H.S. Wilf: Representations of integers by linear forms in nonnegative integers , J. Number Theory 4 (1972), 98-106. · Zbl 0226.10057 |

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[18] | H. Stichtenoth: Algebraic Function Fields and Codes, Universitext, Springer, Berlin, 1993. · Zbl 0816.14011 |

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