Abelian Functions for Trigonal Curves of Genus Three

Abstract

We develop the theory of generalized Weierstrass σ-and ℘-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 ℘-functions, a proof that the coefficients of the power series expansion of the σ-function are polynomials of coefficients of the defining equation of the curve, and the derivation of two addition formulae.

Periods of second kind differentials of $(n,s)$-curves

“…This method can be generalized to higher genera curves in the cases where σ-expansions are known. But only isolated cases of such expansions are elaboratedthe Buchstaber-Leykin recursion for genus two σ-functions [BL05], calculation of first few terms of σ-expansions of (3, 4)-curve presented in [EEMOP08], (3, 5) [BEGO08], (3, 7) and (3, 8)-curves are given in [Eng09] and (4, 5)-curve in [EE09]. We see an advantage of the method proposed, since the derivation is reduced to examining of series over one complex variable, the local coordinate, whilst the generalization of the Weierstrass method leads to a series in many variables.…”

“…Therefore the next cases that could be analyzed are the cases (3, s) -trigonal curves, (3.30) y 3 − a 2 (x)y 2 − a 1 (x)y − a 0 (x) = 0 with appropriate polynomials a 1 (x) and a 0 (x). Analytic expressions for the basic meromorphic differentials can be found in [BEL12], [EEMOP08], whilst expressions for the projective connection S KW (P ) are given in the course of the proof of Prop. 2.1.…”

“…In particular, the approach of Klein (for any Riemann surface of genus 3) and [KSh12] (for an arbitrary Riemann surface of any genus) to the theory of higher genus sigma-functions is based on resolving the generalized Legendre relations in terms of theta-constants. Using (n, s)-curves, it is possible to develop the construction of Abelian functions and the associated integrable PDEs in terms of the σ-function of the trigonal curve, [EEMOP08], [BEGO08], to develop the study of space curves [Mat13], [AN12], to consider τ -functions of integrable hierarchies as σ-functions, [Nak10a], [HE11], to develop the description of classical surfaces like Kummer, Coble surfaces [BEL12], [EGOP13], to describe Jacobi inversion on the strata of non-hyperelliptic Jacobians [MP08], [BEF12], to develop number-theoretical problems [BEH05], [KMP12] and others.…”

Solving the polynomial equations by algorithms of power geometry

“…Здесь также существует унифор-мизация через тэта-функции. Но как её находить, пока неясно [16,17]. По-видимому, существуют кривые, которые не являются суперэллиптиче-скими.…”

Addition formulae for Abelian functions associated with specialized curves

“…One is by simplifying the expression in terms of ℘ and its derivative given by taking the product of modified formulae from equation (1.5) or (1.6), or their higher genus generalizations. The other is similar to the method used in Eilbeck et al [7], that is, by balancing an expression such as equation (1.11) with a linear combination of suitable ℘-functions, in which the derivations of the correct coefficients are aided by algebraic computing software.…”