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Funktsional. Anal. i Prilozhen., 2008, Volume 42, Issue 4, Pages 24–36 (Mi faa2926)  

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

Full text: PDF file (250 kB)
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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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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. E. Yu. Bunkova, V. M. Buchstaber, “Heat Equations and Families of Two-Dimensional Sigma Functions”, Proc. Steklov Inst. Math., 266 (2009), 1–28  mathnet  crossref  mathscinet  zmath  isi  elib
    2. Buchstaber V.M., “Heat equations and sigma functions”, Geometric methods in physics, AIP Conf. Proc., 1191, 2009, 46–58  crossref  adsnasa  isi
    3. Nakayashiki A., “Sigma function as a tau function”, Int. Math. Res. Not. IMRN, 2010, no. 3, 373–394  crossref  mathscinet  zmath  isi
    4. 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.  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. J. Harnad, V. Z. Enolski, “Schur function expansions of KP $\tau$-functions associated to algebraic curves”, Russian Math. Surveys, 66:4 (2011), 767–807  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    6. E. Yu. Bun'kova, “The differential-geometric structure of the universal bundle of elliptic curves”, Russian Math. Surveys, 66:4 (2011), 816–818  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. Sebbar A., Sebbar A., “functions and integrals of elliptic functions”, Geod. Dedicata, 160:1 (2012), 373–414  crossref  mathscinet  zmath  isi
    8. 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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    9. Trans. Moscow Math. Soc., 74 (2013), 245–260  mathnet  crossref  mathscinet  zmath  elib
    10. 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  mathnet  crossref  mathscinet  zmath  elib
    11. V. M. Buchstaber, “Polynomial dynamical systems and the Korteweg–de Vries equation”, Proc. Steklov Inst. Math., 294 (2016), 176–200  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    12. 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  mathnet  crossref  crossref  mathscinet  isi  elib
    13. 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  mathnet  crossref  mathscinet  isi
    14. Nakayashiki A., “Degeneration of Trigonal Curves and Solutions of the KP-Hierarchy”, Nonlinearity, 31:8 (2018), 3567–3590  crossref  mathscinet  zmath  isi
    15. Bunkova E.Yu., “Differentiation of Genus 3 Hyperelliptic Functions”, Eur. J. Math., 4:1, 1, SI (2018), 93–112  crossref  mathscinet  zmath  isi
    16. Julia Bernatska, Dmitry Leykin, “On Regularization of Second Kind Integrals”, SIGMA, 14 (2018), 074, 28 pp.  mathnet  crossref
    17. Bernatska J., Leykin D., “On Degenerate SIGMA-Functions in Genus 2”, Glasg. Math. J., 61:1 (2019), 169–193  crossref  mathscinet  zmath  isi
    18. A. V. Domrin, “Uniqueness theorem for the two-dimensional sigma function”, Funct. Anal. Appl., 54:1 (2020), 21–30  mathnet  crossref  crossref  isi
    19. V. M. Buchstaber, E. Yu. Bunkova, “Lie Algebras of Heat Operators in a Nonholonomic Frame”, Math. Notes, 108:1 (2020), 15–28  mathnet  crossref  crossref  mathscinet  isi
    20. V. M. Bukhshtaber, E. Yu. Bunkova, “Sigma-funktsii i algebry Li operatorov Shredingera”, Funkts. analiz i ego pril., 54:4 (2020), 3–16  mathnet  crossref  mathscinet
  • Функциональный анализ и его приложения Functional Analysis and Its Applications
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