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Addition laws on Jacobian varieties of plane algebraic curves. (English. Russian original) Zbl 1132.14024
Proc. Steklov Inst. Math. 251, 49-120 (2005); translation from Tr. Mat. Inst. Steklova 251, 54-126 (2005).
The authors study the sigma functions for Jacobians of plane algebraic curves and apply these results to the study of addition laws of abelian functions. They find explicit addition formulas for hyperelliptic abelian functions and give a construction of second-order linear differential equations that hold for sigma functions of high genera. Then equations can be interpreted as multidimensional heat equations in a nonholonomic frame.
For the entire collection see [Zbl 1116.34001].

MSC:
14H40 Jacobians, Prym varieties
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
14K20 Analytic theory of abelian varieties; abelian integrals and differentials
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