Addition Laws on Jacobian Varieties of Plane Algebraic Curves

V. M. Buchstaber ; D. V. Leikin

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Nonlinear dynamics, Tome 251 (2005), pp. 54-126

Résumé

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.