34 research outputs found
Multi-Dimensional Sigma-Functions
In 1997 the present authors published a review (Ref. BEL97 in the present
manuscript) that recapitulated and developed classical theory of Abelian
functions realized in terms of multi-dimensional sigma-functions. This approach
originated by K.Weierstrass and F.Klein was aimed to extend to higher genera
Weierstrass theory of elliptic functions based on the Weierstrass
-functions. Our development was motivated by the recent achievements of
mathematical physics and theory of integrable systems that were based of the
results of classical theory of multi-dimensional theta functions. Both theta
and sigma-functions are integer and quasi-periodic functions, but worth to
remark the fundamental difference between them. While theta-function are
defined in the terms of the Riemann period matrix, the sigma-function can be
constructed by coefficients of polynomial defining the curve. Note that the
relation between periods and coefficients of polynomials defining the curve is
transcendental.
Since the publication of our 1997-review a lot of new results in this area
appeared (see below the list of Recent References), that promoted us to submit
this draft to ArXiv without waiting publication a well-prepared book. We
complemented the review by the list of articles that were published after 1997
year to develop the theory of -functions presented here. Although the
main body of this review is devoted to hyperelliptic functions the method can
be extended to an arbitrary algebraic curve and new material that we added in
the cases when the opposite is not stated does not suppose hyperellipticity of
the curve considered.Comment: 267 pages, 4 figure
Periods of second kind differentials of (n,s)-curves
For elliptic curves, expressions for the periods of elliptic integrals of the
second kind in terms of theta-constants, have been known since the middle of
the 19th century. In this paper we consider the problem of generalizing these
results to curves of higher genera, in particular to a special class of
algebraic curves, the so-called -curves. It is shown that the
representations required can be obtained by the comparison of two equivalent
expressions for the projective connection, one due to Fay-Wirtinger and the
other from Klein-Weierstrass. As a principle example, we consider the case of
the genus two hyperelliptic curve, and a number of new Thomae and
Rosenhain-type formulae are obtained. We anticipate that our analysis for the
genus two curve can be extended to higher genera hyperelliptic curves, as well
as to other classes of non-hyperelliptic curves.Comment: 21 page
Sigma, tau and Abelian functions of algebraic curves
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.Comment: 25 page
Algebraic description of Jacobians isogeneous to certain Prym varieties with polarization (1,2)
For a class of non-hyperelliptic genus 3 curves C which are 2-fold coverings
of elliptic curves E, we give an explicit algebraic description of all
birationally non-equivalent genus 2 curves whose Jacobians are degree 2
isogeneous to the Prym varieties associated to such coverings. Our description
is based on previous studies of Prym varieties with polarization (1,2) in
connection with separation of variables in a series of classical and new
algebraic integrable systems linearized on such varieties.
We also consider some special cases of the covering C -> E, in particular,
when the corresponding Prym varieties contain pairs of elliptic curves and the
Jacobian of C is isogeneous (but not isomorphic) to the product of 3 different
elliptic curves.
Our description is accompanied with explicit numerical examples.Comment: 51 pages, 3 figures, 3 diagram
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 satisfied by the \wp-functions, a
proof that the coefficients of the power series expansion of the sigma-function
are polynomials of moduli parameters, and the derivation of two addition
formulae.Comment: 32 pages, no figures. Revised version has the a fuller description of
the general (3,4) trigonal curve results, the first version described only
the "Purely Trigonal" cas