An extension of Kedlaya's algorithm for hyperelliptic curves

TL;DR

This paper extends Kedlaya's algorithm for hyperelliptic curves to even degree models, improving computational efficiency and addressing theoretical subtleties in eigenvalue analysis.

Contribution

It generalizes Kedlaya's algorithm to even degree hyperelliptic models and analyzes the benefits of using an alternative differential basis for better computational properties.

Findings

The $x^idx/y^3$ basis yields an integral transformation matrix.

The algorithm extension handles even degree models.

Eigenvalue analysis requires discarding redundant eigenvalues.

Abstract

In this paper we describe a generalisation and adaptation of Kedlaya's algorithm for computing the zeta function of a hyperelliptic curve over a finite field of odd characteristic that the author used for the implementation of the algorithm in the Magma library. We generalise the algorithm to the case of an even degree model. We also analyse the adaptation of working with the $x^idx/y^3$ rather than the $x^idx/y$ differential basis. This basis has the computational advantage of always leading to an integral transformation matrix whereas the latter fails to in small genus cases. There are some theoretical subtleties that arise in the even degree case where the two differential bases actually lead to different redundant eigenvalues that must be discarded.