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

This article is cited in 39 scientific papers (total in 40 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
\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.
\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
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