Abelian function fields on Jacobian varieties

Abstract

In this paper the fields of multiply periodic, or Kleinian $\wp$-functions are exposed. Such a field arises on the Jacobian variety of an algebraic curve, and provides natural algebraic models of the Jacobian and Kummer varieties, possesses the addition law, and accommodates dynamical equations with solutions. All this will be explained in detail for plane algebraic curves in their canonical forms. Example of hyperelliptic and non-hyperelliptic curves are presented.

35 pages

References in corpus (5)

• Abelian Functions for Cyclic Trigonal Curves of Genus Four
• Identities for hyperelliptic P-functions of genus one, two and three in covariant form
• Sigma, tau and Abelian functions of algebraic curves
• Reality conditions for the KdV equation and exact quasi-periodic solutions in finite phase spaces
• Solution of the Jacobi inversion problem on non-hyperelliptic curves