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This article is cited in 19 scientific papers (total in 20 papers)
Solution of the Problem of Differentiation of Abelian Functions over Parameters for Families of $(n,s)$-Curves
V. M. Buchstabera, D. V. Leikinb
a Steklov Mathematical Institute, Russian Academy of Sciences
b Institute of Magnetism, National Academy of Sciences of Ukraine
Abstract:
We consider a wide class of models of plane algebraic curves, so-called $(n,s)$-curves. The case $(2,3)$ is the classical Weierstrass model of an elliptic curve. On the basis of the theory of multivariate sigma functions, for every pair of coprime $n$ and $s$ we obtain an effective description of the Lie algebra of derivations of the field of fiberwise Abelian functions defined on the total space of the bundle whose base is the parameter space of the family of nondegenerate $(n,s)$-curves and whose fibers are the Jacobi varieties of these curves. The essence of the method is demonstrated by the example of Weierstrass elliptic functions. Details are given for the case of a family of genus 2 curves.
Keywords:
sigma function, differentiation with respect to parameters, universal bundle of Jacobi varieties, $(n,s)$-curve, vector field tangent to the discriminant of a singularity
DOI:
https://doi.org/10.4213/faa2926
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English version:
Functional Analysis and Its Applications, 2008, 42:4, 268–278
Bibliographic databases:
    
UDC:
517.958+515.178.2
Received: 03.09.2008
Citation:
V. M. Buchstaber, D. V. Leikin, “Solution of the Problem of Differentiation of Abelian Functions over Parameters for Families of $(n,s)$-Curves”, Funktsional. Anal. i Prilozhen., 42:4 (2008), 24–36; Funct. Anal. Appl., 42:4 (2008), 268–278
Citation in format AMSBIB
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E. Yu. Bunkova, V. M. Buchstaber, “Heat Equations and Families of Two-Dimensional Sigma Functions”, Proc. Steklov Inst. Math., 266 (2009), 1–28
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Buchstaber V.M., “Heat equations and sigma functions”, Geometric methods in physics, AIP Conf. Proc., 1191, 2009, 46–58
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Nakayashiki A., “Sigma function as a tau function”, Int. Math. Res. Not. IMRN, 2010, no. 3, 373–394
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Eilbeck J.C., Enolski V.Z., Gibbons J., “Sigma, tau and Abelian functions of algebraic curves”, J. Phys. A, 43:45 (2010), 455216, 20 pp.
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J. Harnad, V. Z. Enolski, “Schur function expansions of KP $\tau$-functions associated to algebraic curves”, Russian Math. Surveys, 66:4 (2011), 767–807
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E. Yu. Bun'kova, “The differential-geometric structure of the universal bundle of elliptic curves”, Russian Math. Surveys, 66:4 (2011), 816–818
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Sebbar A., Sebbar A., “functions and integrals of elliptic functions”, Geod. Dedicata, 160:1 (2012), 373–414
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A. M. Vershik, A. P. Veselov, A. A. Gaifullin, B. A. Dubrovin, A. B. Zhizhchenko, I. M. Krichever, A. A. Mal'tsev, D. V. Millionshchikov, S. P. Novikov, T. E. Panov, A. G. Sergeev, I. A. Taimanov, “Viktor Matveevich Buchstaber (on his 70th birthday)”, Russian Math. Surveys, 68:3 (2013), 581–590
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Trans. Moscow Math. Soc., 74 (2013), 245–260
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E. Yu. Netay, “Geometric differential equations on bundles of Jacobians of curves of genus 1 and 2”, Trans. Moscow Math. Soc., 74 (2013), 281–292
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V. M. Buchstaber, “Polynomial dynamical systems and the Korteweg–de Vries equation”, Proc. Steklov Inst. Math., 294 (2016), 176–200
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T. Ayano, V. M. Buchstaber, “The field of meromorphic functions on a sigma divisor of a hyperelliptic curve of genus 3 and applications”, Funct. Anal. Appl., 51:3 (2017), 162–176
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Dynnikov I.A., Glutsyuk A.A., Mironov A.E., Taimanov I.A., Vesnin A.Yu., “The Conference “Dynamics in Siberia”, Novosibirsk, February 26 - March 4, 2017”, Sib. Electron. Math. Rep., 14 (2017), A7–A30
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Nakayashiki A., “Degeneration of Trigonal Curves and Solutions of the KP-Hierarchy”, Nonlinearity, 31:8 (2018), 3567–3590
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Bunkova E.Yu., “Differentiation of Genus 3 Hyperelliptic Functions”, Eur. J. Math., 4:1, 1, SI (2018), 93–112
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Julia Bernatska, Dmitry Leykin, “On Regularization of Second Kind Integrals”, SIGMA, 14 (2018), 074, 28 pp.
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Bernatska J., Leykin D., “On Degenerate SIGMA-Functions in Genus 2”, Glasg. Math. J., 61:1 (2019), 169–193
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A. V. Domrin, “Uniqueness theorem for the two-dimensional sigma function”, Funct. Anal. Appl., 54:1 (2020), 21–30
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V. M. Buchstaber, E. Yu. Bunkova, “Lie Algebras of Heat Operators in a Nonholonomic Frame”, Math. Notes, 108:1 (2020), 15–28
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V. M. Bukhshtaber, E. Yu. Bunkova, “Sigma-funktsii i algebry Li operatorov Shredingera”, Funkts. analiz i ego pril., 54:4 (2020), 3–16
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