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Tr. Mat. Inst. Steklova, 2005, Volume 251, Pages 54–126 (Mi tm45)  

This article is cited in 38 scientific papers (total in 39 papers)

Addition Laws on Jacobian Varieties of Plane Algebraic 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: The paper is devoted to the theory of sigma functions defined on Jacobi varieties of plane algebraic curves. We develop this theory aiming at applications in the theory of nonlinear differential equations and mathematical physics. We propose a method for studying addition laws of Abelian functions which is based on polylinear functional equations that hold for sigma functions. The solutions to polylinear functional equations are constructed with the help of the following key tools: (1) a degenerate Baker–Akhiezer function with a unique singularity in the neighborhood of which this function behaves like $\xi ^{-g}\exp \{p(\xi ^{-1})\}(1+O(\xi ))$, wher $g$ is the genus of the curve and $p$ is a polynomial of degree at most $2g-1$; (2) entire rational functions $R_{kg}$ that have $kg$ zeros on the curve and define the operations of inversion, when $k=2$, and addition, when $k=3$, on the $g$th symmetric power of the curve. We give explicit addition formulas for hyperelliptic Abelian functions and present a construction of multidimensional heat equations in a nonholonomic frame that hold for sigma functions. We also establish a relation between the recursions that define the power series expansion of sigma functions and the Cauchy problems for systems of linear difference equations. The exposition includes several open problems and a large number of examples.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2005, 251, 49–120

Bibliographic databases:
UDC: 515.178.2+517.958
Received in July 2005

Citation: V. M. Buchstaber, D. V. Leikin, “Addition Laws on Jacobian Varieties of Plane Algebraic Curves”, Nonlinear dynamics, Collected papers, Tr. Mat. Inst. Steklova, 251, Nauka, MAIK Nauka/Inteperiodika, M., 2005, 54–126; Proc. Steklov Inst. Math., 251 (2005), 49–120

Citation in format AMSBIB
\Bibitem{BucLei05}
\by V.~M.~Buchstaber, D.~V.~Leikin
\paper Addition Laws on Jacobian Varieties of Plane Algebraic Curves
\inbook Nonlinear dynamics
\bookinfo Collected papers
\serial Tr. Mat. Inst. Steklova
\yr 2005
\vol 251
\pages 54--126
\publ Nauka, MAIK Nauka/Inteperiodika
\publaddr M.
\mathnet{http://mi.mathnet.ru/tm45}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=2234377}
\zmath{https://zbmath.org/?q=an:1132.14024}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2005
\vol 251
\pages 49--120


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    This publication is cited in the following articles:
    1. V. M. Buchstaber, I. M. Krichever, “Integrable equations, addition theorems, and the Riemann–Schottky problem”, Russian Math. Surveys, 61:1 (2006), 19–78  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    2. V. M. Buchstaber, D. V. Leikin, “Functional equations defining multiplication in a continuous Krichever–Novikov basis”, Russian Math. Surveys, 61:1 (2006), 165–167  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    3. L. O. Sheinman, “Sigma-functions and Krichever–Novikov bases”, Russian Math. Surveys, 61:6 (2006), 1183–1185  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    4. Eilbeck J.C., Enolski V.Z., Matsutani S., Ônishi Y., Previato E., “Abelian functions for trigonal curves of genus three”, Int. Math. Res. Not. IMRN, 2007, Art. ID rnm 140, 38 pp.  crossref  mathscinet  zmath  isi  scopus
    5. Baldwin S., Eilbeck J.C., Gibbons J., Ônishi Y., “Abelian functions for cyclic trigonal curves of genus 4”, J. Geom. Phys., 58:4 (2008), 450–467  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. P. G. Grinevich, S. P. Novikov, “Singular finite-gap operators and indefinite metrics”, Russian Math. Surveys, 64:4 (2009), 625–650  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. 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
    8. Buchstaber V.M., “Heat Equations and Sigma Functions”, Geometric Methods in Physics, AIP Conference Proceedings, 1191, 2009, 46–58  crossref  adsnasa  isi  scopus
    9. Nakayashiki A., “Sigma function as a tau function”, Int. Math. Res. Not. IMRN, 2010, no. 3, 373–394  crossref  mathscinet  zmath  isi  scopus
    10. Nakayashiki A., “On algebraic expressions of sigma functions for $(n,s)$ curves”, Asian J. Math., 14:2 (2010), 175–211  crossref  mathscinet  isi  scopus
    11. 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  scopus
    12. Buchstaber V.M., Bunkova E.Yu., “Addition Theorems, Formal Group Laws and Integrable Systems”, XXIX Workshop on Geometric Methods in Physics, AIP Conference Proceedings, 1307, 2010, 33–43  crossref  mathscinet  zmath  adsnasa  isi  scopus
    13. Nakayashiki A., “On hyperelliptic abelian functions of genus 3”, J. Geom. Phys., 61:6 (2011), 961–985  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    14. Enolski V.Z., Hackmann E., Kagramanova V., Kunz J., Laemmerzahl C., “Inversion of hyperelliptic integrals of arbitrary genus with application to particle motion in general relativity”, J. Geom. Phys., 61:5 (2011), 899–921  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    15. V. M. Buchstaber, E. Yu. Bun'kova, “Krichever Formal Groups”, Funct. Anal. Appl., 45:2 (2011), 99–116  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    16. 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
    17. Enolski V., Hartmann B., Kagramanova V., Kunz J., Laemmerzahl C., Sirimachan P., “Inversion of a general hyperelliptic integral and particle motion in Horava-Lifshitz black hole space-times”, J Math Phys, 53:1 (2012), 012504  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
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    19. Korotkin D., Shramchenko V., “On Higher Genus Weierstrass SIGMA-Function”, Physica D, 241:23-24 (2012), 2086–2094  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    20. 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
    21. Trans. Moscow Math. Soc., 74 (2013), 245–260  mathnet  crossref  mathscinet  zmath  elib
    22. 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
    23. Braden H.W., Enolski V.Z., Fedorov Yu.N., “Dynamics on Strata of Trigonal Jacobians and Some Integrable Problems of Rigid Body Motion”, Nonlinearity, 26:7 (2013), 1865–1889  crossref  mathscinet  zmath  isi  elib  scopus
    24. Ayano T., “SIGMA Functions For Telescopic Curves”, Osaka J. Math., 51:2 (2014), 459–480  mathscinet  zmath  isi  elib
    25. 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
    26. V. A. Bykovskii, “Hyperquasipolynomials and their applications”, Funct. Anal. Appl., 50:3 (2016), 193–203  mathnet  crossref  crossref  mathscinet  isi  elib
    27. A. A. Illarionov, “Functional Equations and Weierstrass Sigma-Functions”, Funct. Anal. Appl., 50:4 (2016), 281–290  mathnet  crossref  crossref  mathscinet  isi  elib
    28. M. D. Monina, “O range konechnogo nabora teta-funktsii”, Dalnevost. matem. zhurn., 16:2 (2016), 181–185  mathnet  elib
    29. Joe Suzuki, “Klein's Fundamental $2$-Form of Second Kind for the $C_{ab}$ Curves”, SIGMA, 13 (2017), 017, 13 pp.  mathnet  crossref
    30. P. G. Grinevich, S. P. Novikov, “Singular solitons and spectral meromorphy”, Russian Math. Surveys, 72:6 (2017), 1083–1107  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    31. A. A. Illarionov, “Solution of functional equations related to elliptic functions”, Proc. Steklov Inst. Math., 299 (2017), 96–108  mathnet  crossref  crossref  isi  elib
    32. A. A. Illarionov, M. A. Romanov, “Hyperquasipolynomials for the Theta-Function”, Funct. Anal. Appl., 52:3 (2018), 228–231  mathnet  crossref  crossref  isi  elib
    33. Julia Bernatska, Dmitry Leykin, “On Regularization of Second Kind Integrals”, SIGMA, 14 (2018), 074, 28 pp.  mathnet  crossref
    34. A. A. Illarionov, “Reshenie funktsionalnykh uravnenii, svyazannykh s ellipticheskimi funktsiyami. II”, Sib. elektron. matem. izv., 16 (2019), 481–492  mathnet  crossref
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    37. A. A. Illarionov, “Hyperelliptic systems of sequences of rank 4”, Sb. Math., 210:9 (2019), 1259–1287  mathnet  crossref  crossref  adsnasa  isi
    38. A. V. Domrin, “Teorema edinstvennosti dlya dvumernoi sigma-funktsii”, Funkts. analiz i ego pril., 54:1 (2020), 29–40  mathnet  crossref
    39. A. A. Illarionov, “On a Multilinear Functional Equation”, Math. Notes, 107:1 (2020), 80–92  mathnet  crossref  crossref
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