# zbMATH — the first resource for mathematics

The sigma function for trigonal cyclic curves. (English) Zbl 1441.14145
The Weierstrass semigroup of a complex pointed curve $$(X,p)$$ is the complement of the Weierstrass gap sequence at $$p$$. It determines the Weierstrass normal form of $$X$$ with respect to $$p$$.
In the present article, the authors use the Weierstrass normal form of a given pointed trigonal cyclic curve $$(X,p)$$ to obtain a number of detailed results about differentials and the sigma function. For instance, they obtain an explicit description of a basis for the vector space of differential one forms and thus coordinates for the canonical embedding. They also obtain detailed results which apply to differentials of the second and third kinds.
The authors then turn to the study of the Jacobian of $$X$$. As one result, they study the sigma function and present a solution to the Jacobi inversion problem. Many of the authors methods are sufficiently general so as to apply to the more general case of $$k$$-gonal cyclic covers of $$\mathbb{P}^1$$.

##### MSC:
 14K25 Theta functions and abelian varieties 14H40 Jacobians, Prym varieties 14H55 Riemann surfaces; Weierstrass points; gap sequences
Full Text:
##### References:
 [1] Accola, RDM, On cyclic trigonal Riemann surfaces. I, Trans. Am. Math. Soc., 283, 423-449, (1984) · Zbl 0584.14016 [2] Ayano, T., Sigma functions for telescopic curves, Osaka J. Math., 51, 459-481, (2014) · Zbl 1328.14057 [3] Ayano, T.: On Jacobi Inversion Formulae for Telescopic Curves SIGMA 12: Paper No. 086, p. 21 (2016) · Zbl 1348.14084 [4] Baker, H.F.: Abelian Functions: Abel’s Theorem and the Allied Theory of Theta Functions. Cambridge University Press, Cambridge (1995). (Reprint of the 1897 original) · Zbl 0848.14012 [5] Baker, H.F.: An Introduction to the Theory of Multiply-Periodic Functions, vol. XVI. University Press, Cambridge (1907) · JFM 38.0478.05 [6] Bras-Amorós, M.; Martinez-Moro, E. (ed.), Numerical semigroups and codes, (2013), Singapore [7] Buchstaber, VM; Enolskiĭ, VZ; Leĭkin, DV, Kleinian functions, hyperelliptic Jacobians and applications, Rev. Math. Math. Phys., 10, 1-103, (1997) · Zbl 0911.14019 [8] Buchstaber, V.M., Leĭkin, D.V., Enolskiĭ, V.Z.: $$\sigma$$-functions of $$(n, s)$$-curves. Uspekhi Mat. Nauk 54(3)(327), 155-156 (1999). (trans: Russ. Math. Surv. 54(3), 628-629) [9] Bukhshtaber, VM; Ènol’skiĭ, VZ; Leĭkin, DV, Rational analogues of Abelian functions, Funct. Anal. Appl., 33, 83-94, (1999) · Zbl 1056.14049 [10] Eilbeck, J.C., Enolskii, V.Z., Leykin, D.V.: On the Kleinian construction of Abelian functions of canonical algebraic curves. In: SIDE III—Symmetries and Integrability of Difference Equations (Sabaudia, 1998), CRM Proceedings, Lecture Notes, vol. 25, pp. 121-138. American Mathematical Society, Providence (2000) · Zbl 1003.14008 [11] Eisenbud, D.; Harris, J., Existence, decomposition, and limits of certain Weierstrass points, Invent. Math., 87, 495-515, (1987) · Zbl 0606.14014 [12] Farkas, H.M., Zemel, S.: Generalizations of Thomae’s Formula for $$Z_n$$ Curves. Springer, Berlin (2011) · Zbl 1222.14001 [13] Fay, J.D.: Theta Functions on Riemann Surfaces. Lectures Notes in Mathematics, vol. 352. Springer, Berlin (1973) · Zbl 0281.30013 [14] Fulton, W., Harris, J.: Representation Theory, Graduate Texts in Mathematics, vol. 129. Springer, Berlin (1991) [15] Herzog, J., Generators and relations of Abelian semigroup and semigroup ring, Manuscr. Math., 3, 175-193, (1970) · Zbl 0211.33801 [16] Kato, T., Weierstrass normal form of a Riemann surface and its applications, Sûgaku, 32, 73-75, (1980) [17] Klein, F., Ueber hyperelliptische sigmafunctionen, Math. Ann., 27, 431-464, (1886) · JFM 18.0418.02 [18] Klein, F., Ueber hyperelliptische sigmafunctionen (Zweite Abhandlung), Math. Ann., 32, 351-380, (1888) · JFM 20.0491.01 [19] Klein, F., Zur theorie der Abel’schen functionen, Math. Ann., 36, 1-83, (1890) · JFM 22.0498.01 [20] Komeda, J.; Matsutani, S.; Previato, E., The sigma function for Weierstrass semigroups $$\langle 3,7,8\rangle$$ and $$\langle 6,13,14,15,16\rangle$$, Int. J. Math., 24, 1350085, 58, (2013) · Zbl 1284.14045 [21] Komeda, J.; Matsutani, S.; Previato, E., The Riemann constant for a non-symmetric Weierstrass semigroup, Arch. Math. (Basel), 107, 499-509, (2016) · Zbl 1360.14090 [22] Korotkin, D.; Shramchenko, V., On higher genus Weierstrass sigma-function, Physica D, 241, 2086-2094, (2012) · Zbl 1262.14033 [23] Lang, S.: Introduction to Algebraic and Abelian Functions. Graduate Texts in Mathematics, vol. 89, 2nd edn. Springer, New York (1982) · Zbl 0513.14024 [24] Lewittes, J., Riemann surfaces and the theta functions, Acta Math., 111, 37-61, (1964) · Zbl 0125.31803 [25] Macdonald, I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs, 2nd edn. Oxford University Press, Oxford (1985) · Zbl 0672.20007 [26] Matsutani, S.; Komeda, J., Sigma functions for a space curve of type (3,4,5), J. Geom. Symmetry Phys., 30, 75-91, (2013) · Zbl 1311.14033 [27] Matsutani, S.; Previato, E., Jacobi inversion on strata of the Jacobian of the $$C_{rs}$$ curve $$y^r=f(x)$$ I, J. Math. Soc. Jpn., 60, 1009-1044, (2008) · Zbl 1160.14018 [28] Matsutani, S.; Previato, E., Jacobi inversion on strata of the Jacobian of the $$C_{rs}$$ curve $$y^r=f(x)$$ II, J. Math. Soc. Jpn., 66, 647-692, (2014) · Zbl 1297.14050 [29] Miura, S., Linear codes on affine algebraic curves, IEICE Trans., J81-A, 1398-1421, (1998) [30] Nakayashiki, A., On algebraic expansions of sigma functions for $$(n, s)$$ curves, Asian J. Math., 14, 175-212, (2010) · Zbl 1214.14028 [31] Nakayashiki, A., Tau function approach to theta functions, Int. Math. Res. Not. IMRN, 2016, 5202-5248, (2016) · Zbl 1404.32028 [32] Pinkham, HC, Deformation of algebraic varieties with $$G_m$$ action, Astérisque, 20, 1-131, (1974) · Zbl 0304.14006 [33] Ramírez Alfonsín, J.: The Diophantine Frobenius Problem. Oxford University Press, Oxford (2005) · Zbl 1134.11012 [34] Suzuki, J.: Klein’s fundamental 2-form of second kind for the $$C_{ab}$$ curves. In: SIGMA Symmetry, Integrability and Geometry: Methods and Applications 13, Paper No. 017, p. 13 (2017) · Zbl 1386.14116 [35] Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis. Cambridge University Press, Cambridge (1927) · JFM 45.0433.02
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.