The theory of Abelian functions was a central topic of the 19th century mathematics. In mid-seventies of the last century a new wave arose of investigation in this field in response to the discovery that Abelian functions provide solutions of a number of challenging problems of modern Theoretical and Mathematical Physics.
In a cycle of our joint papers with V. Enolskii and D. Leykin we have developed a theory of multivariate sigma-function, an analogue of the classic Weierstrass sigma-function.
A sigma-function is defined on a cover of U , where U is the space of a bundle p : U ! B defined by a family of plane algebraic curves of fixed genus. The base B of the bundle is the space of the family parameters and a fiber Jb over b 2 B is the Jacobi variety of the curve with the parameters b. A second logarithmic derivative of the sigma-function along the fiber is an Abelian function on Jb.
Thus, one can generate a ring F of fiber-wise Abelian functions on U. The problem to find derivations of the ring F along the base B is a reformulation of the classic problem of differentiation of Abelian functions over parameters. Its solution is relevant to a number of topical applications.
The talk presents a solution of this problem recently found by the speaker and D. Leykin.
Contact A.P. Fordy (email: allan@maths.leeds.ac.uk ) for further details.
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Last updated October 19th, 2007.