Sigma, tau and Abelian functions of algebraic curves

@article{Eilbeck2010SigmaTA,
  title={Sigma, tau and Abelian functions of algebraic curves},
  author={John Christopher Eilbeck and V. Z. Enolski and Jos Gibbons},
  journal={Journal of Physics A: Mathematical and Theoretical},
  year={2010},
  volume={43},
  pages={455216}
}
We compare and contrast three different methods for the construction of the differential relations satisfied by the fundamental Abelian functions associated with an algebraic curve. We realize these Abelian functions as logarithmic derivatives of the associated sigma function. In two of the methods, the use of the tau function, expressed in terms of the sigma function, is central to the construction of differential relations between the Abelian functions. 

Abelian functions associated with genus three algebraic curves

We develop the theory of Abelian functions associated with algebraic curves. The growth in computer power and an advancement of efficient symbolic computation techniques has allowed for recent

Sigma Function as A Tau Function

The tau function corresponding to the affine ring of a certain plane algebraic curve, called (n, s)-curve, embedded in the universal Grassmann manifold is studied. It is neatly expressed by the

Deriving Bases for Abelian Functions Matthew England

We present a new method to explicitly define Abelian functions associated with algebraic curves, for the purpose of finding bases for the relevant vector spaces of such functions. We demonstrate the

Derivatives of Schur, Tau and Sigma Functions on Abel-Jacobi Images

We study derivatives of Schur and tau functions from the view point of the Abel-Jacobi map. We apply the results to establish several properties of derivatives of the sigma function of an (n,s)

ON DEGENERATE SIGMA-FUNCTIONS IN GENUS 2

Abstract We obtain explicit expressions for genus 2 degenerate sigma-function in terms of genus 1 sigma-function and elementary functions as solutions of a system of linear partial differential

Sato Grassmannian and Degenerate Sigma Function

The degeneration of the hyperelliptic sigma function is studied. We use the Sato Grassmannian for this purpose. A simple decomposition of a rational function gives a decomposition of Plücker

The sigma function for Weierstrass semigroups and

Compact Riemann surfaces and their abelian functions are instrumental to solve integrable equations; more recently the representation theory of the Monster and related modular form have pointed to

On the Expansion Coefficients of KP Tau Function

We study the expansion coefficients of the tau function of the KP hierarchy. If the tau function does not vanish at the origin, it is known that the coefficients are given by Giambelli formula and

Tau Function Approach to Theta Functions

We study theta functions of a Riemann surface of genus g from the view point of tau function of a hierarchy of soliton equations. We study two kinds of series expansions. One is the Taylor expansion

Schur function expansions of KP τ-functions associated to algebraic curves

The Schur function expansion of Sato-Segal-Wilson KP -functions is reviewed. The case of -functions related to algebraic curves of arbitrary genus is studied in detail. Explicit expressions for the

References

SHOWING 1-10 OF 72 REFERENCES

Abelian Functions for Trigonal Curves of Genus Three

We develop the theory of generalized Weierstrass sigma- and \wp-functions defined on a trigonal curve of genus three. In particular we give a list of the associated partial differential equations

On the Kleinian construction of Abelian functions of canonical algebraic curves

The discovery of classical and quantum completely integrable systems led to an increase in interest in the theory of Abelian functions in theoretical physics and applied mathematics. This area was

On Algebraic Expressions of Sigma Functions for (n,s) Curves

An expression of the multivariate sigma function associated with a (n,s)-curve is given in terms of algebraic integrals. As a corollary the first term of the series expansion around the origin of the

Sigma Function as A Tau Function

The tau function corresponding to the affine ring of a certain plane algebraic curve, called (n, s)-curve, embedded in the universal Grassmann manifold is studied. It is neatly expressed by the

Solution of the problem of differentiation of Abelian functions over parameters for families of (n, s)-curves

We consider a wide class of models of plane algebraic curves, so-called (n, s)-curves. The case (2, 3) is the classical Weierstrass model of an elliptic curve. On the basis of the theory of

METHODS OF ALGEBRAIC GEOMETRY IN THE THEORY OF NON-LINEAR EQUATIONS

CONTENTSIntroduction § 1. The Akhiezer function and the Zakharov-Shabat equations § 2. Commutative rings of differential operators § 3. The two-dimensional Schrodinger operator and the algebras

Schur function expansions of KP τ-functions associated to algebraic curves

The Schur function expansion of Sato-Segal-Wilson KP -functions is reviewed. The case of -functions related to algebraic curves of arbitrary genus is studied in detail. Explicit expressions for the

Abelian Functions: Abel's Theorem and the Allied Theory of Theta Functions

1. The subject of investigation 2. The fundamental functions on a Riemann surface 3. The infinities of rational functions 4. Specification of a general form of Riemann's integrals 5. Certain forms of

Theta Functions on Riemann Surfaces

Riemann's theta function.- The prime-form.- Degenerate Riemann surfaces.- Cyclic unramified coverings.- Ramified double coverings.- Bordered Riemann surfaces.

Hyperelliptic solutions of KdV and KP equations: re-evaluation of Baker's study on hyperelliptic sigma functions

Explicit function forms of hyperelliptic solutions of Korteweg-de Vries (KdV) and Kadomtsev-Petviashvili (KP) equations are constructed for a given curve y2 = f(x) whose genus is three. This paper is
...