On regularization of second kind integrals
Julia Bernatska (National University of Kyiv-Mohyla Academy)
Thursday 21st September, 2017 16:00-17:00 seminar room
We obtain expressions for second kind integrals on nonhyperelliptic (n, s)-curves. The curves possess a Weierstrass point at infinity which is a branch point where all sheets of the curve come together. The infinity of an (n, s)-curve serves as the base point for Abel's map, and the base point in the definition of the second kind integrals. Since the second kind integrals introduced in this way are singular, we propose the regularization consistent with the structure of the field of Abelian functions on the Jacobian of the curve. We introduce the notion of regularization constant, a uniquely defined free term in the expansion of the second kind integrals over a local parameter in the vicinity of the infinity. This is a vector of dimension equal to genus of the curve, depending on parameters of the curve only. The presence of the term ensures correctness of all relations between Abelian functions on the Jacobian of an (n, s)-curve.
We propose two methods of calculating the regularization constant, and obtain these constants for (3,4), (3,5), (3,7) and (4,5)-curves. Also we show how to extend the proposed regularization to the case when the pole of second kind integrals is moved from infinity to an arbitrary point. We show how to derive addition formulas, computation of which requires the second kind integrals, including correct regularization constants.
joint work with Dmitry Leykin