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Abelian functions for trigonal curves of genus three. (English) Zbl 1210.14032

Summary: We develop the theory of generalized Weierstrass \(\sigma\)- and \(\wp\)-functions defined on a general trigonal curve of genus three. In particular, we give a list of the associated partial differential equations satisfied by the \(\wp\)-functions, a proof that the coefficients of the power series expansion of the \(\sigma\)-function are polynomials of coefficients of the defining equation of the curve, and the derivation of two addition formulae.

MSC:

14H42 Theta functions and curves; Schottky problem
14H45 Special algebraic curves and curves of low genus
33E05 Elliptic functions and integrals
37K20 Relations of infinite-dimensional Hamiltonian and Lagrangian dynamical systems with algebraic geometry, complex analysis, and special functions
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