On Calculation of Abelian Differentials

This paper considers the construction of the fundamental function and Abelian differentials of the third kind on a plane algebraic curve over the field of complex numbers that has no singular points. The algorithm for constructing differentials of the third kind was described in Weierstrass’s lectures. The paper discusses its implementation in the Sage computer algebra system. The specifics of this algorithm, as well as the very concept of the differential of the third kind, implies the use of both rational numbers and algebraic numbers, even when the equation of a curve has integer coefficients. Sage has a built-in tool for computations in algebraic number fields, which allows the Weierstrass algorithm to be implemented almost literally. The simplest example of an elliptic curve shows that it requires too many resources, far beyond the capabilities of an office computer. A symmetrization of the method is proposed and implemented, which makes it possible to solve the problem while saving a significant amount of computational resources.