On Regularization of Second Kind Integrals

Abstract

We obtain expressions for second kind integrals on non-hyperelliptic (n,s)-curves. Such a curve possesses a Weierstrass point at infinity, which is a branch point where all sheets of the curve come together. The infinity serves as the base point for Abel's map, and the base point in the definition of the second kind of integrals. We define second kind differentials as having a pole at infinity; therefore, the second kind integrals need to be regularized. We propose the regularization consistent with the structure of the field of Abelian functions on the Jacobian of the curve. In this connection, we introduce the notion of a regularization constant, a uniquely defined free term in the expansion of the second kind integral over a local parameter in the vicinity of infinity. This is a vector with components depending on parameters of the curve; the number of components is equal to the genus of the curve. The presence of the term guarantees the consistency of all relations between Abelian functions constructed with the help of the second kind integrals. We propose two methods of calculating the regularization constant, and obtain these constants for (3,4), (3,5), (3,7), and (4,5)-curves. By the example of (3,4)-curve, we extend the proposed regularization to the case of second kind integrals with the pole at an arbitrary fixed point. Finally, we propose a scheme for obtaining addition formulas, where the second kind integrals, including the proper regularization constants, are used.