Functional Analysis and Its Applications

, Volume 42, Issue 4, pp 268-278

First online:

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

  • V. M. BuchstaberAffiliated withSteklov Mathematical Institute Email author 
  • , D. V. LeykinAffiliated withInstitute of Magnetism

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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 multivariate sigma functions, for every pair of coprime n and s we obtain an effective description of the Lie algebra of derivations of the field of fiberwise Abelian functions defined on the total space of the bundle whose base is the parameter space of the family of nondegenerate (n, s)-curves and whose fibers are the Jacobi varieties of these curves. The essence of the method is demonstrated by the example of Weierstrass elliptic functions. Details are given for the case of a family of genus 2 curves.

Key words

sigma function differentiation with respect to parameters universal bundle of Jacobi varieties (n, s)-curve vector field tangent to the discriminant of a singularity