On Regularization of Second Kind Integrals

TL;DR

This paper develops a method for regularizing second kind integrals on non-hyperelliptic algebraic curves, introducing regularization constants to ensure consistency of Abelian functions, and provides explicit calculations for specific curves.

Contribution

It introduces a novel regularization scheme for second kind integrals on non-hyperelliptic curves, including methods to compute the regularization constants and extends the approach to arbitrary points.

Findings

Derived explicit expressions for second kind integrals on specific curves.

Defined a unique regularization constant ensuring consistency of Abelian functions.

Proposed a scheme for addition formulas involving regularized 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 basepoint for Abel's map, and the basepoint in the definition of the second kind integrals. We define second kind differentials as having a pole at the 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 Jacobian of the curve. In this connection we introduce the notion of regularization constant, a uniquely defined free term in the expansion of the second kind integral over a local parameter in the vicinity of the infinity. This is a vector with components depending on parameters of the curve, the number of components is equal to genus of the curve. Presence of the term guarantees 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 of obtaining addition formulas, where the second kind integrals, including the proper regularization constants, are used.