Miura: Divisor Class Group Arithmetic

TL;DR

Miura is a software package that efficiently computes divisor class group arithmetic for nonsingular curves by leveraging ideal class group isomorphisms, simplifying complex algebraic computations.

Contribution

The package streamlines divisor class group calculations using ideal class group techniques and shortens the implementation with MaCaulay2, enhancing both understanding and efficiency.

Findings

Efficient computation of divisor class group arithmetic.

Reduction of complex calculations to ideal class group operations.

Shorter, more effective implementation using MaCaulay2.

Abstract

The Package Miura contains functions that compute divisor class group arithmetic for nonsingular curves. The package reduces computation in a divisor class group to that in the ideal class group via the isomorphism. The underlying quotient ring should be over the ideal given by a nonsingular curve in the form of Miura. Although computing the multiplication of two integral ideals is not hard, we need to obtain an ideal such that the shortest Groebner basis component is the minimum in order to obtain the representative of the ideal class. Although the basic procedure is due to Arita, the source code has become much shorter using MaCaulay2. The package is useful not just for computation itself but also for understanding the divisor class group arithmetic from the ideal point of view.