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Uspekhi Mat. Nauk, 2006, Volume 61, Issue 1(367), Pages 25–84 (Mi umn1715)  

This article is cited in 16 scientific papers (total in 17 papers)

Integrable equations, addition theorems, and the Riemann–Schottky problem

V. M. Buchstaberab, I. M. Krichevercd

a Steklov Mathematical Institute, Russian Academy of Sciences
b University of Manchester
c L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences
d Columbia University

Abstract: The classical Weierstrass theorem claims that, among the analytic functions, the only functions admitting an algebraic addition theorem are the elliptic functions and their degenerations. This survey is devoted to far-reaching generalizations of this result that are motivated by the theory of integrable systems. The authors discovered a strong form of the addition theorem for theta functions of Jacobian varieties, and this form led to new approaches to known problems in the geometry of Abelian varieties. It is shown that strong forms of addition theorems arise naturally in the theory of the so-called trilinear functional equations. Diverse aspects of the approaches suggested here are discussed, and some important open problems are formulated.


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English version:
Russian Mathematical Surveys, 2006, 61:1, 19–78

Bibliographic databases:

UDC: 517.9
MSC: Primary 14H42, 14H40; Secondary 14K20, 14K25, 37K10
Received: 20.12.2005

Citation: V. M. Buchstaber, I. M. Krichever, “Integrable equations, addition theorems, and the Riemann–Schottky problem”, Uspekhi Mat. Nauk, 61:1(367) (2006), 25–84; Russian Math. Surveys, 61:1 (2006), 19–78

Citation in format AMSBIB
\by V.~M.~Buchstaber, I.~M.~Krichever
\paper Integrable equations, addition theorems, and the Riemann--Schottky problem
\jour Uspekhi Mat. Nauk
\yr 2006
\vol 61
\issue 1(367)
\pages 25--84
\jour Russian Math. Surveys
\yr 2006
\vol 61
\issue 1
\pages 19--78

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    This publication is cited in the following articles:
    1. V. M. Buchstaber, D. V. Leikin, “Addition Laws on Jacobian Varieties of Plane Algebraic Curves”, Proc. Steklov Inst. Math., 251 (2005), 49–120  mathnet  mathscinet  zmath
    2. V. M. Buchstaber, “$n$-valued groups: theory and applications”, Mosc. Math. J., 6:1 (2006), 57–84  mathnet  crossref  mathscinet  zmath
    3. V. M. Buchstaber, E. V. Koritskaya, “Quasilinear Burgers–Hopf Equation and Stasheff Polytopes”, Funct. Anal. Appl., 41:3 (2007), 196–207  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    4. Dimakis A., Mueller-Hoissen F., “Weakly Nonassociative Algebras, Riceati and KP Hierarchies”, Generalized Lie Theory in Mathematics, Physics and Beyond, 2009, 9–27  crossref  mathscinet  zmath  isi  scopus
    5. Burban I., Henrich T., “Semi-stable vector bundles on elliptic curves and the associative Yang–Baxter equation”, J Geom Phys, 62:2 (2012), 312–329  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus
    6. Gaëtan Borot, Bertrand Eynard, “Geometry of Spectral Curves and All Order Dispersive Integrable System”, SIGMA, 8 (2012), 100, 53 pp.  mathnet  crossref  mathscinet
    7. 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
    8. V. A. Bykovskii, “Hyperquasipolynomials and their applications”, Funct. Anal. Appl., 50:3 (2016), 193–203  mathnet  crossref  crossref  mathscinet  isi  elib
    9. A. A. Illarionov, “Functional Equations and Weierstrass Sigma-Functions”, Funct. Anal. Appl., 50:4 (2016), 281–290  mathnet  crossref  crossref  mathscinet  isi  elib
    10. M. D. Monina, “O range konechnogo nabora teta-funktsii”, Dalnevost. matem. zhurn., 16:2 (2016), 181–185  mathnet  elib
    11. 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
    12. A. A. Illarionov, “Solution of functional equations related to elliptic functions”, Proc. Steklov Inst. Math., 299 (2017), 96–108  mathnet  crossref  crossref  isi  elib
    13. A. A. Illarionov, M. A. Romanov, “Hyperquasipolynomials for the Theta-Function”, Funct. Anal. Appl., 52:3 (2018), 228–231  mathnet  crossref  crossref  isi  elib
    14. A. A. Illarionov, “Reshenie funktsionalnykh uravnenii, svyazannykh s ellipticheskimi funktsiyami. II”, Sib. elektron. matem. izv., 16 (2019), 481–492  mathnet  crossref
    15. A. A. Illarionov, N. V. Markova, “Reshenie funktsionalnykh uravnenii, svyazannykh s ellipticheskimi funktsiyami. III”, Dalnevost. matem. zhurn., 19:2 (2019), 197–205  mathnet
    16. A. A. Illarionov, “Hyperelliptic systems of sequences of rank 4”, Sb. Math., 210:9 (2019), 1259–1287  mathnet  crossref  crossref  adsnasa  isi
    17. A. A. Illarionov, “On a Multilinear Functional Equation”, Math. Notes, 107:1 (2020), 80–92  mathnet  crossref  crossref  isi  elib
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